Unit 3: Writing Equations
Lesson 1: Writing Equations in Slope Intercept Form
Slope Intercept Form:
Example 1
Write an equation for line A and Line B.
Line A:
slope_____
y-intercept______
Line B:
slope_____
y-intercept______
Key Words When Working with Real World Problems
Rate: _______________________________________________________________________
Fee: ____________________________________________________________________
Rate - Slope (m)
Fee - Y-intercept (b)
Y = mx + b
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Unit 3: Writing Equations
Example 2
A local amusement park charges $45.00 per person for admission plus an additional $15 fee for parking.
Write an equation that you could use to find the cost of admission into the park for a family or group of friends.
Let x represent the number of people in the group and y represent the total cost.
•
How much would admission cost for a group of 5 people visiting the park?
Rate - Slope (m)
Fee- Y-intercept (b)
Example 3
The appliance repairman charges a fee of $75 to make a house call. There is also a charge of $35 an hour for labor.
Write an equation that you could use to find the amount the repairman charges for a house call based on the number of
hours of labor. Let x represent the number of hours for labor and y represent the total cost.
•
How much would it cost for a house call that requires 1.75 hours of labor?
Rate - Slope (m)
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Fee- Y-intercept (b)
Unit 3: Writing Equations
Lesson 1: Writing Equations in Slope Intercept Form
1. Which equation represents a line with a slope of 4 and a y-intercept of -2?
y = mx + b
y = ___x + ____
slope y-intercept
A. y= -2x + 4
B. 4x +-2y = 0
C. y = 4x -2
D. -2x +4y = 0
2. Which equation represents a line with a slope of ½ and passes through the origin?
A. y= 1/2x
B. y = x +1/2
C. y = 1/2
D. 1/2x +y = 0
3. Write the equation of a line that has a y-intercept of -4 and a slope of 3.
4. Write the equation of a line that has a rise of 3 and a run of 4 and passes through the
origin.
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Unit 3: Writing Equations
Directions: For problems 5-8, write an equation in slope intercept form that represents the
line in the graph.
5.
m = ____
b = ____
7.
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6. m = ___
8.
b = _____
Unit 3: Writing Equations
9. The U-Haul Company charges $29.99 a day to rent a truck. In addition, they charge
$0.30 per mile driven. Which equation could you use to find the cost of using their truck
for one day?
Let x represent the number of miles driven. Let y represent the total cost.
Slope (m)
Y-intercept (b)
A. y = 29.99x +0.30
B. y = 0.30x + 29.99
C. 0.30x +y = 29.99
D. 29.99x +0.30 = y
10. A new ice cream shop charges $2.50 for a cone of ice cream plus $0.55 per topping.
Write an equation that you could use to determine the cost of buying an ice cream cone
with toppings. Let x represent the number of toppings and y represent the total cost.
Slope (m)
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Y-intercept (b)
Unit 3: Writing Equations
11. You are throwing a party for a friend’s birthday. The hall rental is a flat fee of $150.
The price for catering is $8.50 per person. Write an equation that you could use to
determine the total cost based on the number of people coming to the party. Let x
represent the number of people attending the party and y represent the total cost.
• Suppose 20 people respond that they will attend the party. How much will the party
cost?
Slope (m)
Y-intercept (b)
TIP:
Did you notice that in every one of these problems the following occurred?
• The variable y always represented the total.
• The word per was used and it was always associated with the slope.
• The “flat fee” was always the y – intercept.
1. Write an equation that has a slope of -4 and a y-intercept of 6.
(1 point)
2. Write the equation of a line that has a rise of 3 and a run of -2 and passes through the origin.
(1 point)
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Unit 3: Writing Equations
3. Write an equation that represents the line on each graph. (1 point each)
a. ____________________
b. _________________________
4. The Christmas ornament store charges $12.50 per personalized ornament plus $3.50 for
shipping. Write an equation that could be used to determine the cost of buying x amount of
ornaments. Let x represent the number of ornaments and y represent the total amount. (2 points )
5. The kids Pottery Palace charges $12.00 per piece of pottery plus a $6.20 glazing charge. Write
an equation that could be used to determine the cost of buying p pieces of pottery and having them
glazed. Let p represent the number of pieces of pottery and t represent the total cost. (2 points)
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Unit 3: Writing Equations
Lesson 1 - Writing Equations in Slope Intercept Form – Answer Key
1. Which equation represents a line with a slope of 4 and a y-intercept of -2?
y = mx +b
slope y intercept
y = 4x - 2
A. y= -2x + 4
B. 4x +-2y = 0
C. y = 4x -2
D. -2x +4y = 0
2. Which equation represents a line with a slope of ½ and passes through the origin?
y = mx +b
The origin is (0,0), so the y-intercept is 0.
slope y intercept
y = 1/2x +0
A. y= 1/2x
B. y = x +1/2
C. y = ½
D. 1/2x +y = 0
3. Write the equation of a line that has a y-intercept of -4 and a slope of 3.
y = mx + b
slope
y= 3x
y intercept
y = 3x - 4
-4
4. Write the equation of a line that has a rise of 3 and a run of 4 and passes through the
origin.
Slope = rise = 3
Run
4
Slope (m) = ¾
y = mx +b
y=¾x
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y-intercept (b) = 0
Unit 3: Writing Equations
Directions: For problems 5-8, write an equation in slope intercept form that represents the
line in the graph.
5.
m = 3/2
b= -5
y = 3/2x -5
7.
m = -1/1
y = -x +5
6. m = -4/1
b=3
y = -4x +3
b= 5
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8. m = 2/5
y = 2/5x -7
b= -7
Unit 3: Writing Equations
9. The U-Haul Company charges $29.99 a day to rent a truck. In addition, they charge
$0.30 per mile driven. Which equation could you use to find the cost of using their truck
for one day. Let x represent the number of miles driven. Let y represent the total cost.
Slope (m)
Y-intercept (b)
.30
29.99
Y=
Total cost = y
Flat fee = 29.99 (Y-int)
Charge per mile = 0.30 (slope)
number of miles driven = x
mx + b
Y = 0.30(x) + 29.99
A. y = 29.99x +0.30
B. y = 0.30x + 29.99
C. 0.30x +y = 29.99
D. 29.99x +0.30 = y
Tip: You can eliminate A and D immediately because they are the same answer – just written in a
different order! Plus you know that 0.30 is not the flat rate or the constant in the problem!
10. A new ice cream shop charges $2.50 for a cone of ice cream plus $0.55 per topping.
Write an equation that you could use to determine the cost of buying an ice cream cone
with toppings. Let x represent the number of toppings and y represent the total cost.
Slope (m)
Y-intercept (b)
.55
2.50
Y = mx +b
y= 0.55x +2.50
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Total cost = y
Flat fee = 2.50 per cone
Charge per topping = 0.55
Number of toppings = x
Unit 3: Writing Equations
11. You are throwing a party for a friend’s birthday. The Hall rental is a flat fee of $150.
The price for catering is $8.50 per person. Write an equation that you could use to
determine the total cost based on the number of people coming to the party. Let x
represent the number of people attending the party and y represent the total cost.
Slope (m)
Y-intercept (b)
8.50
150
Total cost = y
Flat fee = 150 hall rental
Charge per person= 8.50
Number of people = x
Y=mx+b
y = 8.50x +150
• Suppose 20 people respond that they will attend the party. How much will the party
cost?
Substitute 20 for the number of people (x).
y = 8.50(20) +150
y = $320
The party will cost $320.
1. Write an equation that has a slope of -4
and a y-intercept of 6. (1 point)
Slope (m) = -4
y-intercept (b) = 6
Y = mx + b
Y = -4x +6
2. Write the equation of a line that has a rise of 3 and a run of -2 and passes through the origin.
(1 point)
Rise = 3
Slope (m) = -3/2
Origin (0,0) - This is the y-intercept (b)
Run -2
Y = mx + b
Y = -3/2
(Since b = 0, we don’t need to write anything for b)
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Unit 3: Writing Equations
3. Write an equation that represents the line on each graph. (1 point each)
a. y = 3x+4
b. y = 1/2x - 4
The y-intercept is -4 because the line
crosses the y axis at y = -4 or (0,-4).
Therefore, b = 4
The y-intercept is 4 because the line crosses
the y axis at: y = 4 or (0,4). Therefore, b = 4
The slope from point to point:, the rise is 3 and
the run is 1, so the slope is 3. Therefore, m=3
Y = mx+ b
The slope from point to point: the rise is 1
and the run is 2, so the slope is ½. M = ½
and y = 3x + 4
Y = mx+ b
and y = 1/2x - 4
4. The Christmas ornament store charges $12.50 per personalized ornament plus $3.50 for
shipping. Write an equation that could be used to determine the cost of buying x amount of
ornaments. Let x represent the number of ornaments and y represent the total amount. (2 points )
Slope (m) = 12.5 (12.50 per ornament)
y-intercept (b) = 3.50 - this is the fixed amount
Y = mx + b
Y = 12.50x + 3.50
5. The kids Pottery Palace charges $12.00 per piece of pottery plus a $6.20 glazing charge. Write
an equation that could be used to determine the cost of buying p pieces of pottery and having them
glazed. Let p represent the number of pieces of pottery and t represent the total cost. (2 points)
Slope (m) = 12 ($12 per piece of pottery)
y = mx + b
or
y- intercept (b) = 6.20 (this is the fixed amount)
t = mp + b
y = 12x + 6.20
t = 12p+ 6.20
(given the assigned variables)
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Unit 3: Writing Equations
Lesson 2: Writing Equations in Standard Form
Standard Form
Special Rule #1 for Standard Form:
The lead coefficient (A) must be _______________________________.
Example 1
Rewrite the equation y = 6x – 8 in standard form.
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Unit 3: Writing Equations
Special Rule #2 for Standard Form:
Equations that are written in standard form CANNOT contain _____________________________.
Ax + By = C
A, B, and C must be ___________________________.
Example 2
Rewrite the equation y = 1/3x +
½ in standard form.
Step 1: Multiply all terms by ____ in order to get rid of the fractions.
Step 2: Write the equation in standard form.
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Unit 3: Writing Equations
Lesson 2: Writing Equations in Standard Form
1. Rewrite the equation y = 3x -5 in standard form.
A.
B.
C.
D.
3x+y = -5
3x -y = 5
-3x –y = -5
3x+y = 5
2. Rewrite the equation y = -2x +6 in standard form.
3. Rewrite the equation y = 8x in standard form.
4. In the equation 2x +1/2y = 7, which term does not have an integer coefficient?
A.
B.
C.
D.
2x
1/2y
7
None of the above.
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Unit 3: Writing Equations
5. Write the equation y = -2/3x +2 in standard form with integer coefficients.
Step 1: Multiply all terms by _____ in order to get rid of the fraction.
Step 2: Write the equation in standard form.
A.
B.
C.
D.
2x +3y = 6
-2x +3y = 6
2/3x +y = 2
2x +y = 2
6. Write the equation y = 3/4x -5 in standard form with integer coefficients.
Step 1: Multiply all terms by _____ in order to get rid of the fraction.
Step 2: Write the equation in standard form.
7. Write the equation y = -3/5x – 2 in standard form with integer coefficients.
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Unit 3: Writing Equations
8. Which equation represents y = 1/2x +3/4 in standard form with integer coefficients?
Step 1: Multiply all terms by _____ in order to get rid of the fractions.
Step 2: Write the equation in standard form.
A.
B.
C.
D.
9.
10.
2x+4y = 3
x + 4y = 3
2x -4y = -3
-2x +y = 3
Write the equation y = -2/3x +5/6 in standard form.
Write the equation y = 3/4x +3/8 in standard form.
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Unit 3: Writing Equations
11. Write the equation (in standard form) that represents the line on the graph.
12.
Write the equation (in standard form) that represents the line on the graph.
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Unit 3: Writing Equations
Part 1: Rewrite each equation in standard
form with only integer coefficients. (2 points each)
1. y = 4x – 3
2. y = - ¾ x + 5
3. y = 1/2x – 2/3
Part 2: Write an equation in standard form that represents the line on the graph. (3 points)
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Unit 3: Writing Equations
Lesson 2: Writing Equations in Standard Form – Answer Key
1. Rewrite the equation y = 3x -5 in standard form.
y = 3x -5
-3x + y = 3x – 3x -5
Subtract 3x from both sides
-3x + y = -5
-1[-3x +y = -5]
Multiply all terms by -1 to make the lead
3x -y = 5
coefficient positive
A.
3x+y = -5
B. 3x -y = 5
C. -3x –y = -5
D. 3x+y = 5
2. Rewrite the equation y = -2x +6 in standard form.
y = -2x +6
2x +y = -2x +2x +6
Add 2x to both sides.
2x +y = 6
Equation written in standard form.
3. Rewrite the equation y = 8x in standard form.
y = 8x
y-y = 8x –y
Subtract y from both sides.
0 = 8x –y or 8x – y = 0
Equation written in standard form.
4. In the equation: 2x +1/2y = 7, which term does not have an integer coefficient?
½ is the coefficient of y, but it is not an integer. It is a fraction. You cannot have
fractions in your standard form equations.
A. 2x
B. 1/2y
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C. 7
D. None of the above.
Unit 3: Writing Equations
5. Write the equation y = -2/3x +2 in standard form with integer coefficients.
Step 1: Multiply all terms by 3 in order to get rid of the fraction.
3[y = -2/3x +2]
Tip: You can eliminate B and C
3y = -2x +6
from your answers.
B. Has a lead coefficient that is
negative. It must be positive.
Step 2: Write the equation in standard form.
3y = -2x +6
C. Has a coefficient that is a
fraction, and it must be an
integer.
2x +3y = -2x +2x + 6
Add 2x to both sides.
2x +3y = 6
Equation in standard form.
A. 2x +3y = 6
B. -2x +3y = 6
C. 2/3x +y = 2
D. 2x +y = 2
6. Write the equation y = 3/4x -5 in standard form with integer coefficients.
Step 1: Multiply all terms by 4 in order to get rid of the fraction.
4[y = 3/4x – 5]
4y = 3x - 20
Step 2: Write the equation in standard form.
4y = 3x - 20
-3x + 4y =3x – 3x -20
Subtract 3x from both sides.
-3x +4y = -20
-1[-3x +4y = -20]
3x -4y = 20
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Multiply all terms by -1 to make the lead
coefficient positive.
Equation written in standard form.
Unit 3: Writing Equations
7. Write the equation y = -3/5x – 2 in standard form with integer coefficients.
y = -3/5x - 2
5[y= -3/5x – 2]
Multiply all terms by 5 to get rid of the fraction.
5y = -3x -10
3x +5y = -3x +3x – 10
Add 3x to both sides.
3x +5y = -10
Equation written in standard form.
8. Which equation represents y = 1/2x +3/4 in standard form with integer coefficients?
Step 1: Multiply all terms by
4[y = 1/2x +3/4]
4y = 2x +3
4
in order to get rid of the fractions.
Step 2: Write the equation in standard form.
4y = 2x +3
-2x + 4y = 2x – 2x +3
Subtract 2x from both sides.
-2x +4y = 3
-1[-2x +4y = 3]
Multiply all terms by -1 to make the lead
coefficient positive.
2x – 4y = -3
A.
B.
C.
D.
2x+4y = 3
x + 4y = 3
2x -4y = -3
-2x +y = 3
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Equation written in standard form.
Unit 3: Writing Equations
9. Write the equation y = -2/3x +5/6 in standard form.
6[y = -2/3x + 5/6]
Multiply all terms by 6 to get rid of the fractions.
6y = -4x +5
4x +6y =-4x +4x + 5
Add 4x to both sides.
4x +6y = 5
Equation written in standard form.
10. Write the equation y = 3/4x +3/8 in standard form.
8[y = 3/4x +3/8]
Multiply all terms by 8 to get rid of the fractions.
8y = 6x +3
-6x +8y = 6x -6x +3
Subtract 6x from both sides.
-6x +8y = 3
-1[-6x +8y = 3]
Multiply all terms by -1 to make the lead
coefficient positive.
6x – 8y = -3
Equation written in standard form.
11. Write the equation (in standard form) that represents the line on the graph.
1. Write the equation in slope intercept form:
Slope (m) = 3/4
Y-intercept (b) = 3
y= mx +b
y = 3/4x + 3
2. Get rid of the fraction by multiplying every term by 4.
4[y = 3/4x + 3]
4y=3x +12
3. Move the x term to the left side by subtracting 3x from
both sides.
-3x +4y = 3x -3x +12
-3x +4y = 12
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4. Multiply all terms by -1 to make the lead coefficient
positive.
-1[-3x +4y = 12]
3x -4y = -12
Unit 3: Writing Equations
12. Write the equation (in standard form) that represents the line on the graph.
1. Write the equation in slope intercept form:
Slope (m) = -2/3
Y-intercept (b) = -3
y= mx +b
y = -2/3x - 3
2. Get rid of the fraction by multiplying every term by 3.
3[y = -2/3x - 3]
3y= -2x -9
3. Move the x term to the left side by adding 2x to both
sides.
2x+3y = -2x+ 2x -9
2x +3y = -9
Part 1: Rewrite each equation in standard form with
only integer coefficients. (2 points each)
1. y = 4x – 3
In order to rewrite the equation in standard form, we must have the variables, x and y on the same
side and the constant on the opposite side: Ax + By = C
-4x + y = 4x- 4x – 3
Subtract 4x from both sides
-4x + y = - 3
Simplify
-1(-4x + y) = -3(-1)
Multiply all terms by -1
4x – y = 3
Simplify
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
2. y = - ¾ x + 5
Start by getting rid of the fraction by multiplying all terms by 4.
4(y) = 4(-3/4x+5)
4y = -3x + 20
Simplify
Now we want x and y to be on the same side and the constant, 20 to be on the opposite side.
4y + 3x = -3x +3x +20
Add 3x to both sides
4y +3x = 20 or 3x +4y = 20
Simplify and rewrite with variables in order.
3x +4y = 20
Final answer
3. y = 1/2x – 2/3
In order to get rid of the fraction, we must multiply by 6 on both sides
6(y) = 6[1/2x – 2/3]
6y = 3x – 4
-3x + 6y = 3x-3x – 4
Subtract 3x from both sides
-3x + 6y = -4
Simplify
-1[-3x + 6y] = -4(-1)
Multiply by -1 to make the lead coefficient positive.
3x -6y = 4
Simplify: Final answer
Part 2: Write an equation in standard form that represents the line on the graph. (3 points)
It’s easiest to first write the equation in slope intercept form and
then rewrite it in standard form.
We can see that the y-intercept is -2, and the slope is -1/2, so…
m = -1/2
b = -2
y = mx + b
y = -1/2x -2 is the equation in slope intercept form
Now, rewrite in standard form. Let’s start by getting rid of the
fraction. We’ll multiply all terms by 2.
Copyright© 2009 Algebra-class.com
2(y) = 2(-1/2x -2)
Multiply by 2.
2y = -x – 4
Simplify
2y +x = -x+x – 4
Add x to both sides.
2y +x = -4
Simplify
x + 2y = -4
Standard form equation.
Unit 3: Writing Equations
Lesson 3: Standard Form Real World Problems
When you write an equation in standard form, you should read the problem and look for:
•
•
Information about 2 different things that when added together will give you a total.
A total
Example 1
Jamie is planning a dinner party. Chicken entrees cost $15 per head and fish entrees cost $18
per head. Jamie has a budget of $225 for the dinner party.
•
•
Write an equation that represents Jamie’s situation.
If five people request fish entrees, how many chicken entrees can she buy?
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Unit 3: Writing Equations
Word Problems Organizer
What I Know
Define Your Variable(s).
Write a Verbal Model & Substitute
Solve
Solution
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Unit 3: Writing Equations
Lesson 3: Real World Problems & Standard Form
1. At your local market, ground beef costs $2 per pound and chicken breasts cost $3 per
pound. You have a total of $30 to spend on ground beef and chicken for a party. Which
equation represents the amounts of ground beef and chicken you can buy?
Let x represent the number of pounds of ground beef and y represent the number of pounds
of chicken breasts.
A. 3x +2y = 30
B. 2x +3y = 30
C. y = 2x +3
D. 30x +3y = 2
2. Use the equation from problem 1 above to solve:
If you buy 3 pounds of ground beef, how many pounds of chicken can you buy for $30?
A. 8 pounds of chicken
B. 10.5 pounds of chicken
C. 18.5 pounds of chicken
D. None of the above.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
3. You are in the running to be elected a member of your town council You have $275 to
spend on advertising. It costs $2 to make a button and $1.50 to make a sign.
• Write an equation that represents the different number of buttons, x, and signs, y you
could make.
• If you make 50 buttons, how many signs can you make?
4. Jessica made 240 oz of jam. She has two types of jars. The first holds 10 oz and the
second holds 12 oz.
• Write an equation that represents the different numbers of 10 oz jars, x, and 12 oz jars,
y, that will hold all of the jam.
• If Jessica used 16, 10 oz jars, how many 12 oz jars would she be able to make?
Your next assignment will be a quiz. You will need to know:
•
How to write an equation in slope intercept form and how to write an equation in standard form.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
1. You are purchasing “photo books” as holiday gifts this year. A 5 page
photo book costs $10.50 and a 10 page photo book costs $17.50. You
have a total of $217 to spend on holiday gifts.(4 points)
•
•
•
Write an equation that you could use to determine how many 5 page photo books (x) you
could purchase and how many 10 page photo books (y) you can purchase.
If you decide that you need four 5 page photo books, how many ten page photo books can
you purchase?
Justify your answer mathematically.
2. A youth group is purchasing tickets to a local amusement park. Adult tickets cost $15.50 and
child tickets cost $10.25. They have a total of $545 to spend on tickets. (4 points)
•
•
•
Write an equation that the youth group could use to determine the total number of adult
tickets (a) and the total number of child tickets (c) that they can purchase.
They have 32 children going to the amusement park. How many adult tickets can they buy
as chaperones?
Justify your answer mathematically.
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Unit 3: Writing Equations
Lesson 3 - Real World Problems & Standard Form – Answer Key
1. At your local market, ground beef costs $2 per pound and chicken breasts cost $3 per
pound. You have a total of $30 to spend on ground beef and chicken for a party. Which
equation represents the amounts of ground beef and chicken you can buy?
Let x represent the number of pounds of ground beef and y represent the number of pounds
of chicken breasts.
Ground Beef
+ Chicken
= Total Amount
A. 3x +2y = 30
Price • # of pounds + Price • # of pounds = Total Amount
B. 2x +3y = 30
2• x
+
3 • y
= 30
C. y = 2x +3
2x +3y = 30
D. 30x +3y = 2
2. Use the equation from problem 1 above to solve:
If you buy 3 pounds of ground beef, how many pounds of chicken can you buy for $30?
A. 8 pounds of chicken
2x +3y = 30
x = 3 (# of pounds of ground
beef)
C. 18.5 pounds of chicken
2(3) + 3y = 30
Substitute 3 for x
D. None of the above.
6 + 3y = 30
Simplify: 2(3) = 6
B. 10.5 pounds of chicken
Solve for Y:
6-6 +3y = 30 -6
Subtract 6 from both sides
3y = 24
Simplify: 30 – 6 = 24
3y = 24 = 8
3
3
You could buy 8 lbs of
chicken.
y=8
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Unit 3: Writing Equations
3. You are in the running to be elected a member of your town council. You have $275 to
spend on advertising. It costs $2 to make a button and $1.50 to make a sign.
• Write an equation that represents the different number of buttons, x, and signs, y you
could make.
• If you make 50 buttons, how many signs can you make?
Advertising Costs:
Buttons
Signs
= Total
Price • # of buttons
+ Price • # of signs
= Total
2
+
• x
+
1.50 • y
= 275
2x +1.50y = 275
• 2x +1.50y = 275 is the equation that represents the different number of buttons and
signs you could make with $275.
Part 2: If you make 50 buttons, how many signs can you make?
If you know you have to make 50 buttons, then you know your value for x, the number
of buttons. Let’s substitute 50 for x and solve for y.
2x +1.50y = 275
2(50) + 1.50y = 275
Substitute 50 for x, the number of buttons.
100 +1.50y = 275
Simplify: 2(50) = 100
Solve for Y:
100 -100 +1.50y = 275-100
Subtract 100 from both sides.
1.50y = 175
Simplify: 275 - 100
1.50y = 175
1.50
1.50
Divide both sides by 1.50
• You could make 116 signs if you make 50 buttons.
Y = 116.6
*You can’t make .6 of a sign, and you can’t round up
because you would go over your limit of $275. Therefore, you must round down.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
4. Jessica made 240 oz. of jam. She has two types of jars. The first holds 10 oz. and the
second holds 12 oz.
• Write an equation that represents the different numbers of 10 oz jars, x, and 12 oz jars,
y, that will hold all of the jam.
• If Jessica used 16, 10 oz jars, how many 12 oz jars would she be able to make?
10 oz. jars
+ 12 oz. jars
= total number of oz.
10 oz• # of jars + 12oz. • # of jars = total number of oz.
10 •
x
+ 12
• y
= 240 oz
10x +12y = 240
• The equation that represents the different number of 10 oz jars and 12 oz jars that
will hold all of the jam is: 10x +12x = 240.
Part 2: If Jessica used 16, 10 oz. jars, how many 12 oz. jars would she be able to make?
The variable that represents 10 oz. jars is x, so let’s substitute 16 for x and solve for y.
10x +12y = 240
10(16) + 12y = 240
160 +12y = 240
Solve for y:
160 – 160 +12y = 240 – 160
12y = 80
Substitute 16 for x.
Simplify: 10(16) = 160
Subtract 160 from both sides
Simplify: 240 – 160 = 80
12y = 80
12
12
Divide by 12 on both sides
y = 6.67
Simplify: 80/12 = 6.67
• If Jessica used 16, 10 oz. jars, she could make 6, 12 oz. jars.
**y = 6.67 but you can’t really make .67 of a jar. You can’t round up because then you
would exceed 240 oz., so you must round down in this case.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
1. You are purchasing “photo books” as holiday gifts this year. A 5 page
photo book costs $10.50 and a 10 page photo book costs $17.50. You have
a total of $217 to spend on holiday gifts.(4 points)
•
Write an equation that you could use to determine how many 5 page photo books (x) you
could purchase and how many 10 page photo books (y) you can purchase.
Cost of 5 page·(# of 5 page) + Cost of 10 page·(# of 10 page) = Total
10.50x
+
17.50y
= 217
The equation that can be used to determine the total number of photo books that can be purchased is:
10.50x + 17.50y = 217
•
If you decide that you need four 5 page photo books, how many ten page photo books can
you purchase?
I know that I need four 5 page photo books, therefore, I can substitute 4 for x into the equation and solve for
y.
10.50(4) + 17.50y = 217
Substitute 4 for x.
42 + 17.50y =217
Simplify: 10.5(4) = 42
42 – 42 + 17.50y = 217-42
Subtract 42 from both sides
17.5y = 175
Simplify: 217-42 = 175
17.5y/17.5 = 175/17.5
Divide by 17.5 on both sides
Y = 10
Simplify: 175/17.5 = 10
If you buy 4, 5 page photo books, then you will be able to buy 10, 10 page photo books for $217.
•
Justify your answer mathematically.
In order to justify your answer mathematically, you must substitute for x and y to determine if the
equation is mathematically correct.
X=4
y = 10
10.5(4) + 17.5(10) = 217
42 + 175 = 217
217 = 217
Yes, my equation is correct!
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
2. A youth group is purchasing tickets to a local amusement park. Adult tickets cost $15.50 and
child tickets cost $10.25. They have a total of $545 to spend on tickets. (4 points)
•
Write an equation that the youth group could use to determine the total number of adult
tickets (a) and the total number of child tickets (c) that they can purchase.
Price of adult(# of adult) + Price of child(# of child) = Total
15.50a
+
10.25c =
545
The equation that can be used to determine the total number of adult and child tickets is:
15.50a + 10.25c = 545
•
They have 32 children going to the amusement park. How many adult tickets can they buy
as chaperones?
Since I know that I have 32 children going, I can substitute 32 for c and solve for a.
15.50a + 10.25(32) = 545
Substitute 32 for c.
15.50a + 328 = 545
Simplify: 10.25(32) = 328
15.50a + 328 – 328 = 545 – 328
Subtract 328 from both sides
15.50a = 217
Simplify: 545 – 328 = 217
15.50a/15.50 = 217/15.50
Divide by 15.50 on both sides
a = 14
If they take 32 children to the amusement park, then they can take 14 adults as chaperones
for $545.
•
Justify your answer mathematically.
In order to justify mathematically, we must substitute for a and c to determine if the equation is
mathematically correct.
15.50a + 10.25c = 545
15.50(14) + 10.25(32) = 545
217 + 328 = 545
545 = 545
Yes! My equation is mathematically correct!
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
Writing Equations Quiz
1. Write the following equations in standard form:
•
•
A. y = -4x + 10
B. y = 2/3x – 7
2. Write an equation in slope intercept form that represents each line on the graph.
Line A:
Line B:
3. Peppe’s Pizza charges $10.50 per cheese pizza plus $0.85 per topping. Which equation could
be used to determine the cost of buying a pizza with x number of toppings?
A. y = 10.50x + 0.85
B. 10.50x + y = 0.85
C. 0.85x + y = 10.50
D. y = 0.85x + 10.50
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Unit 3: Writing Equations
4. JOJO the clown sells balloons. “Character” balloons cost $3 apiece and plain balloons cost
$1.50 a piece. JOJO sold $169.50 worth of balloons at the school fair.
•
•
Write an equation that represents how many “Character” balloons and how many plain
balloons JOJO sold.
JOJO sold 55 plain balloons. How many “character” balloons did she sell?
5. A moving company charges $150 an hour plus a $75 local gas fee.
•
•
Write an equation that could be used to find the total cost of using the moving company.
Suppose you utilize the moving company for 8 hours. What would be your total cost?
Justify your answer.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
Writing Equations Quiz- Answer Key
1. Write the following equations in standard form: (2 points each - 4 points total)
•
•
A. y = -4x + 10
B. y = 2/3x – 7
B. (3)y = 3[2/3x – 7]
3y = 2x – 21
-2x + 3y = 2x -2x – 21
-2x + 3y = -21
-1[-2x + 3y] = -1(-21)
2x – 3y = 21
A. 4x + y = -4x + 4x + 10
4x + y = 10
Add 4x to both sides
Equation written in standard form
Multiply by 3 on both sides to remove the fraction.
Multiply/Distribute the 3
Subtract 2x from both sides
Multiply by -1 in order to make the lead coefficient positive.
Equation written in standard form.
2. Write an equation in slope intercept form that represents each line on the graph. (2 points each
–
4 points total)
Line A: y-intercept = 5
Slope = 4/1
Equation: y = 4x + 5
Line B: y-intercept = -8
Slope = -3/4
Equation: y = -3/4x - 8
3. Peppe’s Pizza charges $10.50 per cheese pizza plus $0.85 per topping. Which equation could
be used to determine the cost of buying a pizza with x number of toppings? (1 point)
A. y = 10.50x + 0.85
B. 10.50x + y = 0.85
C. 0.85x + y = 10.50
D. y = 0.85x + 10.50
Copyright© 2009 Algebra-class.com
The flat fee is $10.50 per pizza. Therefore, this is the yintercept.
The rate is $0.85 per topping (key word: per). This is the
slope in the equation.
Since we know the y-intercept and slope, we can write the
equation in slope intercept form.
Unit 3: Writing Equations
4. JOJO the clown sells balloons. “Character” balloons cost $3 apiece and plain balloons cost
$1.50 a piece. JOJO sold $169.50 worth of balloons at the school fair. Write an equation that
represents how many “Character” balloons and how many plain balloons JOJO sold. (3 points)
JOJO sold 55 plain balloons. How many “character” balloons did she sell?
•
Character balloons + plain balloons = Total cost
Let x = the number of character balloons.
•
•
3x + 1.50y = 169.50
3x +1.50(55) = 169.50
Let y = the number of plain balloons.
Substitute 55 for y into the equation since we know
he sold 55 plain balloons.
Simplify: 1.50(55) = 82.50
Subtract 82.50 from both sides.
Simplify: 169.50 – 82.50 = 87
Divide by 3 on both sides.
3x + 82.50 = 169.50
3x + 82.50 -82.50 = 169.50 – 82.50
3x = 87
3x/3 = 87/3
x= 29
If JOJO sold 55 plain balloons, then he sold 29 character balloons.
3(29) +1.50(55) = 169.50
169.50 = 169.50
5. A moving company charges $150 an hour plus a $75 local gas fee.
•
•
(3 points)
Write an equation that could be used to find the total cost of using the moving company.
Suppose you utilize the moving company for 8 hours. What would be your total cost?
Justify your answer.
$150 is the rate; therefore, this is the slope.
Y = mx + b
Let x = the number of hours.
•
•
•
$75 is the fee; therefore, this is the y-intercept.
m = 150
Let y = total cost
b = 75
Y = 150x + 75 is the equation that represents this problem.
Y = 150(8) + 75
Substitute 8 for x since the company was utilized for 8
hours.
Y = $1275
If the moving company is utilized for 8 hours, the total cost is $1275.
Justify:
1275 = 150(8) + 75
1275 = 1275
This quiz is worth 15 points.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
Lesson 4: Writing Equations Given Slope and a Point
Slope Intercept Form:
In order to write an equation in slope intercept form, we need to know:
If we are given slope and a point, we need to find ___________ before we can write our
equation.
Example 1
Write the equation of a line, in slope intercept form, that passes through the point
(4, -6) and has a slope of 2.
Rate (Slope) (m)
X Value of Ordered
Pair
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Y Value of Ordered
Pair
Y-intercept (b)
Unit 3: Writing Equations
Example 2
While visiting Disney World, Jonathan took a cab ride from the Orlando Airport to his hotel in
Kissimmee. The total cab ride cost $46.50 and the hotel was located 30 miles from the airport.
The cabbie charged $1.35 per mile for the entire trip.
•
•
Write a linear equation that could be used to determine the cost of a cab ride to anywhere
in the Orlando area.
Suppose you wanted to take a cab ride that totaled 22 miles. What would the total cab
fare cost?
Rate (Slope) (m)
X Value of Ordered
Pair
Copyright© 2009 Algebra-class.com
Y Value of Ordered
Pair
Y-intercept (b)
Unit 3: Writing Equations
Lesson 4: Writing Equations Given Slope and a Point
1. Write the equation of a line that has a slope of 3 and passes through the point (2,1).
Step 1: Identify
Slope (m)
X Coordinate
Y Coordinate
Y-Intercept(b)
Step 2: Substitute into y = mx +b and solve for b.
Step 3: Write the equation in slope intercept form. (Substitute for m and b)
y = mx +b
2. Which equation represents a line that has a slope of -5 and passes through the point
(5,-2).
A.
B.
C.
D.
y = -5x +23
y = -5x -27
y = -2x +5
y = -5x -23
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
3. Write the equation of a line that has a slope of ½ and passes through the point (-8,-3)
4. Given the equation: y = 3x -1. Identify the point that this line passes through.
A.
B.
C.
D.
(-1,2)
(3,9)
(2,5)
(-2,5)
5. A school group ordered t-shirts on the internet at a rate of $6.50 per shirt. The total
cost for an order of 100 shirts was $670.
• Write an equation that the class can use to determine its’ total for any number
of t-shirts ordered.
• What do you think the y-intercept most likely represents in this problem?
Rate (Slope) (m)
Copyright© 2009 Algebra-class.com
X Value of related pair
Y Value of related pair
Y-intercept (b)
Unit 3: Writing Equations
6. Your new cell phone company sent you a bill stating that you owed $220 for 1500
minutes of use. You know that your current charge is $0.13 a minute.
• Write an equation that can be used to find the total of your cell phone bill for
any given month.
• The next month you used 900 minutes. How much will you owe the cell phone
company?
Rate (Slope) (m)
X Value of related pair
Y Value of related pair
Y-intercept (b)
7. John took a new job for a company that offered him an average raise in pay of $3000
per year. After 5 years on the job, John’s annual salary was $58000.
• Write an equation that John could use to calculate his annual salary,( y) over
any given number of years (x).
•
What does the y-intercept represent in the context of this problem?
Rate (Slope) (m)
Copyright© 2009 Algebra-class.com
X Value of related pair
Y Value of related pair
Y-intercept (b)
Unit 3: Writing Equations
1. Write the equation of a line that has a slope of -4 and passes through
the point (-6, 27) (2 points)
2. A catering service charges $32 per dish. Sam’s order for 5 dishes totaled $190. (3 points)
•
•
Write an equation that can be used to determine the total cost for any number of dishes
ordered.
What do you think the y-intercept most likely represents in this problem?
3. After ordering 22 tickets from a wholesale ticket agency for $1983, you realized that there must
be a service fee since each ticket cost $89 apiece. What is the service fee? (3 points)
•
•
Write an equation that could be used to determine the total cost for any number of tickets
ordered.
What is the service fee for ordering the tickets through the wholesaler?
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
Lesson 4 - Writing Equations Given Slope and a Point – Answer Key
1. Write the equation of a line that has a slope of 3 and passes through the point (2,1).
Step 1: Identify
Slope (m)
X Coordinate
Y Coordinate
Y-Intercept(b)
3
2
1
-5
Step 2: Substitute into y = mx +b and solve for b.
y = mx + b
1 = 3(2) + b
Substitute for y, m,& x.
1 = 6 +b
Simplify: 3(2) = 6
1 – 6 = 6-6 +b
Subtract 6 from both sides
-5 = b
Simplify: 1 – 6 = -5
(**Record -5 in the chart for b!)
Step 3: Write the equation in slope intercept form. (Substitute for m and b)
y = mx +b
y = 3x - 5
2. Write the equation of a line that has a slope of -5 and passes through the point (5,-2).
Slope (m)
X Coordinate
-5
Y Coordinate
5
y = mx + b
-2
Y-Intercept(b)
23
Tip:
A. y = -5x +23
-2 = -5(5) +b
Substitute
-2 = -25 +b
Simplify: 5(-5) = -25
C. y = -2x +5
-2 +25 = -25 +25 +b
Add 25 to both sides.
D. y = -5x -23
23 = b
Simplify; -2 +25 = 23
B. y = -5x -27
The Y – Intercept (b) is 23. Complete your table.
Write the equation: y = -5x +23
Copyright© 2009 Algebra-class.com
You can eliminate
answer C because
you know the
slope is -5 (not -2)
Unit 3: Writing Equations
3. Write the equation of a line that has a slope of ½ and passes through the point (-8,-3)
Slope (m)
X Coordinate
1/2
Y Coordinate
-8
y = mx + b
y = mx + b
-3 = ½(-8) +b
y = 1/2x + 1
-3
Y-Intercept(b)
1
-3 = -4 +b
-3+4 = -4 +4 +b
1=b
4. Given the equation: y = 3x -1. Identify the point that this line passes through.
For this problem, it is easiest to substitute the answer into the equation to find out
which one is a solution!
y = 3x – 1
y = 3x – 1
y = 3x – 1
2 = 3(-1) -1
9 = 3(3) -1
5 = 3(2) -1
2 = -3 – 1
9=9–1
5=6–1
2 = -4 NO!
9 = 8 NO!
5 = 5 YES!
A. (-1,2)
B. (3,9)
C. (2,5)
Copyright© 2009 Algebra-class.com
D. (-2,5)
Unit 3: Writing Equations
5. A school group ordered t-shirts on the internet at a rate of $6.50 per shirt. The total
cost for an order of 100 shirts was $670.
• Write an equation that the class can use to determine its’ total for any number
of t-shirts ordered.
• What do you think the y-intercept most likely represents in this problem?
Rate (Slope) (m)
X Value of related pair
6.50
100
Y Value of related pair
670
Y-intercept (b)
20
The problem says “a rate of 6.50 per shirt”. Rate and per are both key words for slope.
Record 6.50 as the slope in your chart.
Then the problem says that 100 shirts was $670. These two numbers are related;
therefore they make an ordered pair.
Let x = the number of t-shirts
Let y – the total cost
y= mx + b
y = mx + b
670 = 6.50(100) +b
y = 6.50x +20
670 = 650 +b
670 -650 = 650 – 650 + b
20 = b
• The equation that can be used to determine the total for any number of tshirts ordered is: y = 6.50x +20
**The y-intercept is the value when x = 0. So, what does it mean when the
number of t-shirts ordered is 0?
• In this problem, the y-intercept most likely represents the cost of shipping.
Since the y intercept is the cost when 0 shirts are ordered, this is the amount it
costs to ship an order.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
6. Your new cell phone company sent you a bill stating that you owed $220 for 1500
minutes of use. You know that your current charge is $0.13 a minute.
• Write an equation that can be used to find the total of your cell phone bill for
any given month.
• The next month you used 900 minutes. How much will you owe the cell phone
company?
Rate (Slope) (m)
.13
X Value of related pair
1500
Y Value of related pair
Y-intercept (b)
220
25
Since the charge is $0.13 a minute, this considered a “rate” or the slope in the
problem.
You are also told that 1500 minutes costs $220. These two numbers are related.
They are your ordered pair. Even though $220 comes first in the problem, this is
your y value. Y is usually always your “total”.
Let x = the number of minutes used
Let y = the total cost
y = mx + b
220 = 1500(.13) + b
220 = 195 +b
220 – 195 = 195 – 195 +b
25 = b
y = mx + b
y = 0.13x + 25
• The equation that can be used to find the total of the cell phone bill for any month is:
y = 0.13x+25
• If I use 900 minutes, then I will owe the cell phone company $142.00
y = 0.13x +25
142 = 0.13(900) +25
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
7. John took a new job for a company that offered him an average raise in pay of $3000
per year. After 5 years on the job, John’s annual salary was $58000.
• Write an equation that John could use to calculate his annual salary,( y) over
any given number of years (x).
•
What does the y-intercept represent in the context of this problem?
Rate (Slope) (m)
3000
X Value of related pair
Y Value of related pair
5
58000
Y-intercept (b)
The phrase “3000 per year” indicates the slope in the problem.
5 years and $58000 are related, so this is your ordered pair.
Let x = the number of years on the job.
y = mx +b
58000 = 3000(5) +b
58000 = 15000 +b
58000 – 15000 = 15000 – 15000 + b
43000 = b
Let y = Annual Salary
y = mx + b
y = 3000x + 43000
• The equation that could be used to calculate John’s annual salary is:
y = 3000x + 43000.
**The y-intercept is the value when x = 0. So, what does it mean when the
number of years on the job is 0?
• Since the y-intercept is John’s annual salary at 0 years, this would be his
starting salary. The salary that they offered him when he took the job.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
1. Write the equation of a line that has a slope of -4 and passes through the point (-6, 27) (2 points)
Slope (m) = -4
X = -6
Y = 27
Y-intercept (b) = ???
Y = mx + b
Formula for writing an equation in slope intercept form
27 = -4(-6) + b
Substitute for m, x, and y. Solve for b.
27 = 24 + b
Simplify: -4(-6) = 24
27-24 = 24-24 + b
Subtract 24 from both sides
3=b
The y-intercept (b) = 3
The equation of the line with a slope of -4 that passes through (-6,27) is: y = -4x + 3.
(The slope is -4 and the y-intercept is 3. These are the two things we must know to write
an equation in slope intercept form.
2. A catering service charges $32 per dish. Sam’s order for 5 dishes totaled $190. (3 points)
•
Write an equation that can be used to determine the total cost for any number of dishes
ordered.
I know that the charge was $32 per dish. Per is the key word for slope, so the slope (m)= 32.
The total for 5 dishes was $190. This is an ordered pair, since 5 can only be paired with 190:
(5, 190). Therefore I know: slope (m) = 32
x=5
y = 190
y-intercept (b) = ???
Y = mx + b
Formula for writing an equation in slope intercept form
190 = 32(5) + b
Substitute for m, x, and y. Solve for b.
190 = 160 + b
Simplify: 32(5) = 160
190 – 160 = 160 – 160 + b
Subtract 160 from both sides
30 = b
Simplify: 190-160 = 30
The equation that represents this situation is: y = 32x + 30
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
•
What do you think the y-intercept most likely represents in this problem?
The y-intercept is the constant and since this is a catering service, the y-intercept most likely
represents a set-up/delivery fee for the service.
3. After ordering 22 tickets from a wholesale ticket agency for $1983, you realized that there must
be a service fee since each ticket cost $89 apiece. What is the service fee? (3 points)
•
Write an equation that could be used to determine the total cost for any number of tickets
ordered.
We know that 22 tickets cost $1983. Since 22 can only be paired with 1983, this is an ordered pair:
(22, 1983). We are also told that the tickets cost $89 apiece. This is the rate or slope(m): 89.
So, we know:
slope (m) = 89
x = 22
y =1983
y-intercept (b)= ???
Y = mx +b
Formula for writing an equation in slope intercept form.
1983 = 89(22) + b
Substitute for m, x, and y. Solve for b.
1983 = 1958 + b
Simplify: 89(22) = 1958
1983 – 1958 = 1958 – 1958 +b
Subtract 1958 from both sides
25 = b
Simplify: 1983-1958 = 25
m = 89
•
b = 25
so y = 89x + 25 is the equation that represents this situation.
What is the service fee for ordering the tickets through the wholesaler?
The service fee for ordering tickets through the wholesaler is $25. 25 is the y-intercept which is the
constant. Most times, a service fee is a set amount.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
Lesson 5: Writing Equations Given Two Points
To write an equation in slope intercept form given two points, we must find:
Example 1
Write an equation for the line that passes through the points (1, 6) & (3, -4)
Ordered Pair #1
(
)
Ordered Pair #2
(
Copyright© 2009 Algebra-class.com
)
Slope (m)
Y-Intercept (b)
Unit 3: Writing Equations
Example 2
In 2001 the cost of college tuition was $15000 a year. In 2010, the cost of college tuition was $23000 a year.
Let x = 0 represent 2000.
•
•
Write an equation that could be used to predict the college tuition for any given year.
Predict the cost of college tuition for the year 2015.
Ordered Pair #1
(
)
Ordered Pair #2
(
Copyright© 2009 Algebra-class.com
)
Slope (m)
Y-Intercept (b)
Unit 3: Writing Equations
Lesson 5: Writing the Equation of a Line Given Two Points
1.
A.
B.
C.
D.
2.
Find the slope of the line that passes
through the points (2,4) & (-4, -6).
1
-1
-5/3
5/3
Slope Formula for Two Points:
y2- y1
x2 – x1
Write the equation of the line that passes through the origin & (4,7).
Slope (m)
Y-Intercept (b)
Step 1: Find the slope of the two points. Record your answer in the chart.
Step 2: Use 1 point (doesn’t matter which one) and the slope to solve for the yintercept (b).
Step 3: Write your answer in slope intercept form: y = mx +b
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
3. Write the equation of the line that passes through (-10, 9) & (4,-9)
Slope (m)
Y-Intercept (b)
Step 1: Find the slope of the two points. Record your answer in the chart.
Step 2: Use 1 point (doesn’t matter which one) and the slope to solve for the y-intercept (b).
Step 3: Write your answer in slope intercept form: y = mx +b
4. Write an equation in slope intercept form that passes through (-2,3) & (5, -4)
Slope (m)
Y-Intercept (b)
Step 1: Find the slope of the two points. Record your answer in the chart.
Step 2: Use 1 point (doesn’t matter which one) and the slope to solve for the y-intercept (b).
Step 3: Write your answer in slope intercept form: y = mx +b
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
5. Write an equation for the line that passes through the points (-8,-4) & (-5, 11)
Slope (m)
A.
B.
C.
D.
Y-Intercept (b)
y = 5x +36
y = 5x – 14
y = -7/13x +4/13
y = 5x +44
6. In 2000 the cost of attending an Oriole’s game was $8.00 per person. In 2009 the cost
of attending an Oriole’s game is $18 per person. Let x = 0 represent the year 2000.
• Write an equation that can be used to predict the cost of attending an
Oriole’s game for any given year.
• Predict how much an Oriole game will cost in the year 2013.
Ordered Pair #1
(
)
Ordered Pair #2
(
Copyright© 2009 Algebra-class.com
)
Slope (m)
Y-Intercept (b)
Unit 3: Writing Equations
7.
A shoe store made a profit of $14510 in 1988 and a profit of $21260 in 1993. Write an
equation that can be used to predict the profit, y, in terms of the year, x. Let x=0
represent the year 1980.
• Predict the profit for the year 2009.
• What does the y-intercept represent in the context of this problem?
Ordered Pair #1
(
8.
)
Ordered Pair #2
(
Y-Intercept (b)
)
The local cable company has seen an increase in the amount of HDTV subscribers over
the past few months. In January there were 1 million HDTV subscribers and by June
there were 3.2 million subscribers.
• Write an equation that could be used to estimate the amount of HDTV subscribers
for any given month during that year.
• Predict how many HDTV subscribers there will be by December of that year.
Ordered Pair #1
(
Slope (m)
)
Ordered Pair #2
(
Copyright© 2009 Algebra-class.com
)
Slope (m)
Y-Intercept (b)
Unit 3: Writing Equations
9.
The results of a study on first time mothers found that in year 1 of the study the
median age of a first time mother was 24. In year 25 of the study, the median age of
first time mothers was 28.
• Write an equation in slope intercept form that could be used to estimate the
median age, y, of a first time mother, for any year, x, during the study.
• Predict the median age of a first time mother during year 15 of the study.
Ordered Pair #1
(
)
Ordered Pair #2
(
Slope (m)
Y-Intercept (b)
)
1. Find the slope of the line that passes through the points (2,7) and (-4,-23).
(1 point)
2. Write the equation for the line that passes through (-4, -6) and (2,-9). (2 points)
3. The cost of having a wedding at the Grand Ballroom depends on the number of guests attending.
For a wedding guest list of 150 people, the total is $10000. For a wedding guest list of 200 people,
the cost is $13250. (4 points)
•
•
•
Write an equation that can be used to determine the total cost for n number of guests invited
to the wedding.
What is the cost of a wedding with 275 guests?
What do you think the slope represents in this problem?
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
Lesson 5 - Writing the Equation of a Line Given Two Points – Answer Key
1. Find the slope of the line that passes
through the points (2,4) & (-4, -6).
Slope Formula for Two Points:
y2- y1
x2 – x1
Use the formula for slope:
A. 1
B. -1
y2- y1 = -6 – 4 = -10 = 5
x2 – x1 -4 – 2 -6 3
C. -5/3
5/3 is the slope for the line.
D. 5/3
2. Write the equation of the line that passes through the origin & (4,7).
(0,0)
Slope (m)
Y-Intercept (b)
7/4
0
Step 1: Find the slope of the two points. Record your answer in the cart.
y2- y1 = 7-0 = 7
x2 – x1 4-0 4
7/4 is the slope for the line.
Step 2: Use 1 point (doesn’t matter which one) and the slope to solve for the y-intercept.
Y= mx + b
(I’m going to use the point (0,0)
0 = 7/4(0) +b
0=b
Step 3: Write your answer in slope intercept form: y = mx + b
Copyright© 2009 Algebra-class.com
y = 7/4x
Unit 3: Writing Equations
3. Write the equation of the line that passes through (-10, 9) & (4,-9)
Slope (m)
Y-Intercept (b)
-9/7
-27/7
Step 1: Find the slope of the line that passes through the two points
y2- y1 = -9 – 9 = -18 = -9
x2 – x1 4 - -10 14
7
-9/7 is the slope of the line.
Step 2: Use 1 point (doesn’t matter which one) and the slope to solve for the y-intercept.
Y= mx + b
(I’m choosing (-10,9) as my point)
9 = -9/7(-10) + b
9 = 90/7 + b
9 -90/7 = 90/7 – 90/7 + b
-27/7 = b
Step 3: Write your answer in slope intercept form: y = mx + b
Y = -9/7x – 27/2
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
4. Write an equation in slope intercept form that passes through (-2,3) & (5, -4)
Slope (m)
Y-Intercept (b)
-1
1
Step 1: Find the slope of the two points. Record your answer in the chart.
y2- y1 = -4 – 3 = -7 = -1
x2 – x1 5 – (-2)
7
Step 2: Use 1 point (doesn’t matter which one) and the slope to solve for the y-intercept (b).
y = mx + b
(I’m going to use the point (-2,3)
3 = -1(-2) +b
3 = 2 +b
3 -2 = 2-2 +b
1=b
Step 3: Write your answer in slope intercept form: y = mx +b
y = mx +b
y = -x +1
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
5. Write an equation for the line that passes through the points (-8,-4) & (-5, 11)
Slope (m)
Y-Intercept (b)
5
36
Step 1: Find the slope. y2- y1 =
x 2 – x1
11 – (-4) = 15 = 5
-5 – (-8)
3
A. y = 5x +36
B. y = 5x – 14
C. y = -7/13x +4/13
D. y = 5x +44
Step 2: Use 1 point and the slope to solve for b.
y = mx + b
(I’m going to use the point (-5, 11)
11 = 5(-5) +b
11 = -25 +b
11 +25 = -25 +25 +b
36 = b
Step 3: Write your answer in slope intercept form:
y = mx +b
y = 5x +36
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
6. In 2000 the cost of attending an Oriole’s game was $8.00 per person. In 2009 the
cost of attending an Oriole’s game is $18 per person. Let x = 0 represent the year
2000.
• Write an equation that can be used to predict the cost of attending an
Oriole’s game for any given year.
• Predict how much an Oriole game will cost in the year 2013.
Ordered Pair #1
( 0, 8
)
Same as (2000,8)
Ordered Pair #2
( 9, 18
)
Slope (m)
Y-Intercept (b)
10/9
8
same as (2009,18)
Step 1: Find the slope:
y2- y1 = 18 – 8 = 10
Tip:
x 2 – x1
9-0
9
Step 2: Use 1 point and the slope to solve for b.
y = mx +b
(I’m going to use 9,18)
18 = 9(10/9) +b
18 = 10 +b
Since the first ordered pair was (0,8)
you may have automatically known
that the y-intercept was 8. Since the
x coordinate was 0, the y coordinate
is the y-intercept!
If you were able to figure this out –
Great Job – you didn’t have to do all
of this work!
18 -10 = 10-10 +b
8=b
Step 3: Write your equation: y = mx +b
Y = 10/9x +8
• The equation that can be used to predict the cost of attending an Oriole’s game for
any given year is: y = 10/9x +8
• In the year 2013, an Orioles’ game will cost about $22.44
Y = 10/9x +8
22.44 = 10/9(13) +8
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
7. A shoe store made a profit of $14510 in 1988 and a profit of $21260 in 1993. Write
an equation that can be used to predict the profit, y, in terms of the year, x. Let x=0
represent the year 1980.
• Predict the profit for the year 2009.
• What does the y-intercept represent in the context of this problem?
Ordered Pair #1
Ordered Pair #2
( 8, 14510 )
(13, 21260)
Slope (m)
Y-Intercept (b)
1350
3710
Same as (1988,14510) Same as (1993,21260)
Step 1: Find the slope:
y2- y1 = 21260 – 14510 = 6750 = 1350
x 2 – x1
13 – 8
5
Step 2: Find the y – intercept. Use the slope and 1 point.
y = mx +b
(I’m going to use the point (8, 14510)
14510 = 1350(8) +b
14510 = 10800 + b
14510-10800 = 10800-10800 + b
3710 = b
Step 3: y = mx+b
y = 1350x + 3710
• The profit for the year 2009 will be about $42860.
42860 = 1350(29) + 3710
• Since the y-intercept represents year 0, this would be the year 1980. That means
that the shoe store made a profit of $3710 in 1980. This is most likely the year when
the store opened.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
8. The local cable company has seen an increase in the amount of HDTV subscribers
over the past few months. In January there were 1 million HDTV subscribers and by
June there were 3.2 million subscribers.
• Write an equation that could be used to estimate the amount of HDTV
subscribers for any given month during that year.
• Predict how many HDTV subscribers there will be by December of that year.
Ordered Pair #1
( 1,1
Ordered Pair #2
Slope (m)
Y-Intercept (b)
(6, 3.2)
.44
.56
)
Let x = the number of the month: i.e. January = 1, February = 2 …
Let y = the number of HDTV subscribers (in millions)
Step 1: Find the slope:
y2- y1 = 3.2 – 1 = 2.2 = .44
x 2 – x1 6 – 1
5
Step 2: Find the y-intercept. Use the slope and 1 point.
y = mx +b (I am going to use the point (1,1)
1 = .44(1) +b
1 = .44 +b
1 - .44 = .44 -.44 +b
.56 = b
Step 3: Write the equation: y = mx +b
y = .44x +.56
• The equation that could be used to estimate the amount of HDTV subscribers for any
given month during that year is: y = .44x +.56
• In December of that year there will be about 5.84 million subscribers.
y = .44x + .56
5.84 = .44(12) +.56
Copyright© 2009 Algebra-class.com
I substituted 12 for x (the month) because December is the 12th
month.
Unit 3: Writing Equations
9. The results of a study on first time mothers found that in year 1 of the study the
median age of a first time mother was 24. In year 25 of the study, the median age of
first time mothers was 28.
• Write an equation in slope intercept form that could be used to estimate the
median age, y, of a first time mother, for any year, x, during the study.
• Predict the median age of a first time mother during year 15 of the study.
Ordered Pair #1
( 1, 24 )
Ordered Pair #2
Slope (m)
Y-Intercept (b)
.17
23.83
(25,28 )
Step 1: Find the slope:
y2- y1 = 28 – 24 = 4 = 1 = .17
x2 – x1 25 – 1 24
6
Step 2: Find the y-intercept. Use the slope and a point.
y= mx +b (I am going to use the point (1,24)
24 = .17(1) +b
24 = .17 +b
24 - .17 = .17 - .17 +b
23.83 = b
Step 3: Write the equation in slope intercept form:
y = mx +b
y = .17x +23.83
• The equation that could be used to estimate the median age of a first time mother
for any given year during the study is y = .17x +23.83
• The median age of a first time mother during year 15 is about 26 years old.
26.38 = .17(15) +23.83
**We usually don’t use decimals when referring to age,
so I rounded the age to 26 years.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
1. Find the slope of the line that passes through the points (2,7) and (-4,-23). (1 point)
We will use the slope formula in order to determine the slope.
y2- y1 = -23 –7 = -30 = 5
x2 – x1
-4 – 2
-6
The slope of the line that passes through (2,7) and (-4,-23) is 5
2. Write the equation for the line that passes through (-4, -6) and (2,-9). (2 points)
Step 1: Find the slope of the line.
y2- y1 = -9 – (-6) = -3 = -1/2 or -.5
x2 – x1 2 – (-4)
6
Step 2: Use the slope and 1 point to find the y-intercept.
Slope (m) = -1/2 (2,-9) where x = 2 and y = -9
Y = mx + b
-9 = -1/2(2) + b
-9 = -1 + b
-9+1 = -1+1 +b
-8 = b
Slope Intercept Form Equation
Substitute for m, x, and b.
Simplify: -1/2(2) = -1
Add 1 to both sides
Simplify: -9+1 = -8
Step 3: Write the equation in slope intercept form.
Now we know the slope (m) and y-intercept so we can write an equation in slope intercept form.
Slope (m) = -1/2
Y-intercept (b) = -8
Y = mx+ b
Y = - 1/2x - 8 is the equation for the line that passes through the points (-4, -6) and (2,-9).
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
3. The cost of having a wedding at the Grand Ballroom depends on the number of guests attending.
For a wedding guest list of 150 people, the total is $10000. For a wedding guest list of 200 people,
the cost is $13250. (4 points)
•
Write an equation that can be used to determine the total cost for n number of guests invited
to the wedding.
Since this is a real world problem, we must determine the two points. Which numbers are related?
(150, 10000) and (200, 13250)
Step 1: Find the slope using the slope formula.
y2- y1 = 13250 – 10000 = 3250 = 65
x2 – x1
200-150
50
The slope is 65.
Step 2: Use the slope and 1 point to find the y-intercept.
Slope (m) = 65
(150, 10000) where x = 150
y = 10000
Y = mx+ b
Slope Intercept form equation.
10000 = 65(150) + b
Substitute for m, x, and y.
10000 = 9750 + b
Simplify: 65(150) = 9750
10000 – 9750 = 9750 -9750 + b
Subtract 9750 from both sides
250 = b
Simplify: 10000-9750 = 250
250 is the y-intercept (b)
Y = mx+b
Y = 65x + 250 is the equation that can be used for this situation.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
•
What is the cost of a wedding with 275 guests?
Use the equation from above and substitute 275 for x.
Y = 65x + 250
Y = 65(275) + 250
Substitute 275 for x.
Y = 18125
A wedding with 275 guests would cost $18125.
•
What do you think the slope represents in this problem?
In this problem, the slope represents the cost per guest to attend the wedding.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
**Bonus Lesson – Solving Real World Problems**
Many algebra students can perform the “skills” to solve problems, but have difficulty
with solving real world problems (or word problems). The sad part is that the skills are
pretty much useless unless you can apply them to real situations.
Most algebra students need more practice with real world problems and need to be
taught “strategies” to help understand the problems! Hopefully I’ve given you ample
practice problems in this book. Now, I want to share a few strategies that might help
with these problems.
Within each lesson, you solved several word problems using that particular skill. This
may have seemed easy considering that you knew what skill you were to apply in order
to solve. What happens when you are presented with a problem, but you don’t know
which skill to apply? That’s what we are going to focus on in this lesson!
Let’s look at this example as a model.
Each year the attendance at the Haunted Dungeons increases by about 95 people.
During the second year, 545 people attended the event.
• Write an equation that represents this situation.
Step 1: Analyze and organize the information that is given to you in the problem
Highlighting is always a great technique to use. In order to analyze the information,
highlight what you feel is important and needed to solve the problem.
Each year the attendance at the Haunted Dungeons increases by about 95 people. During
the second year, 545 people attended the event.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
• Write an equation that represents this situation.
Attendance that increases by 95 people each year seems important. The second
year, there were 545 also seems important. I will analyze this information a little
more and figure out what it really means as I organize the information.
Then I need to organize the information. I like to use a table (as you’ve seen in all
of the previous lessons). This table gives you a place to write the information that is
needed to write a linear equation.
Remember, you might be given the following information in order to write an
equation:
• Slope and the y-intercept
• Slope and 1 point
• Two points
You will use the information given to you in the problem to fill in parts of the table.
Notice that the slope and y-intercept are colored green. Once you have these two
numbers, you can stop and write an equation in slope intercept form.
Slope
95
Y-intercept
Point 1
Point 2
(2, 545)
Let’s think about attendance that increases by 95 people each year. Does this
number fit into our table?
• Is it slope? Is it a rate of change? YES! Why? The phrase, “Increases by 95 people
each year” indicates a rate of change. The total attendance changes every year by
95 people. There are no key words (such as per) to indicate slope, because this
problem is worded a little different. You have to think about what the sentence
means. You can fill in 95 under slope.
• The other piece of information that you are given is in the second year, there were
545 people.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
For some of you who don’t read carefully, this might trick you because since
“second year” is spelled out, many don’t view it as a number. If this is the case,
you might assume that 545 is the y- intercept and go on to write the equation.
This is NOT the case!
Second (2nd) year and 545 are related numbers (they go together). Therefore, this is an
ordered pair. You can add this is on your chart. (2, 545)
You might be wondering how you know which is the x coordinate and which is the y
coordinate.
• If you are dealing with time, this is always your x coordinate.
• The total or end result is always the y coordinate.
In this situation 2nd year is a “time” so that is your x coordinate and 545 is the total after
the 2nd year, so that is your y coordinate.
Now you have slope and 1 point. From here you can find the y-intercept and write an
equation. Review the lesson on Writing Equations Given Slope and a Point if needed.
(This lesson is only to show you how to better understand the word problems in order to
solve them. Therefore, I am not going to find a solution, we are only going to analyze and
put the information into a table!)
Let’s look at another problem.
A jewelry company sells silver bracelets for $10. If you would like your name engraved,
each letter costs $0.50. Let x represent the number of letters in a name and y represent
the total cost. Write an equation could you use to calculate the cost of a bracelet with
engraving?
Slope
Y-intercept
0.50
10
Copyright© 2009 Algebra-class.com
Point 1
Point 2
Unit 3: Writing Equations
Highlight the important information.
$10 is the cost of the bracelet and each letter engraved costs $0.50.
Let’s first look at the $10. Ask yourself: “Is this number a rate of change?” Will it allow
for the total amount to change?
No, the bracelet costs $10 whether you get it engraved or not. $10 is not the slope.
Is it the y- intercept? Is this number a constant? When x = 0, does this number apply?
Let’s see: x is the number of letters in a name, so if there are 0 letters engraved, will the
bracelet cost $10? YES! Therefore, this number is the y-intercept. Record this number in
the table.
Now let’s look at $0.50. Let’s see if this number is the slope. Is it a rate of change? Will
the total amount change based on this term? YES. Each letter costs $0.50, so each time
you add a letter the total cost will change! This is the slope. Record this number in your
chart.
Now you have the two things you need to write an equation, the slope and y-intercept.
You can stop here and write your equation!
Let’s look at another example:
By the end of your fifth Spanish Lesson you have learned 20 vocabulary words. After 15
lessons, you have learned 50 vocabulary words. Write an equation that gives the
number of vocabulary words you know, y, in terms of the number of lessons that you’ve
had, x.
Slope
Y-intercept
Let’s highlight our important information:
Copyright© 2009 Algebra-class.com
Point 1
Point 2
(5, 20)
(15, 50)
Unit 3: Writing Equations
We have 5th Spanish lesson, 20 vocabulary words
15 lessons, 50 vocabulary words
X = the number of lessons
y = the number of vocabulary words.
You always want to look and see if you have a slope, or rate of change. In this case, I don’t
see a number that changes the total on a consistent basis.
Let’s look a little more closely at our definition of x and y. X is the number of lessons and
Y is the number of vocabulary words. Do I have information that pertains to lessons and
vocabulary words? Yes! Let’s see
(# of lesson, # of Vocabulary Words)
(5th lesson, 20 vocab words)
(15 lessons, 50 vocab words)
I have enough information to write 2 ordered pairs! Let’s write them in our table!
Now you will need to find the slope and y-intercept in order to solve the problem.
One last example to analyze…
You have $20 to spend to buy pretzels and chips for a party. The pretzels cost $2.25 a
bag and chips cost $3.50 a bag. Write an equation that represents the total number of
pretzels, p, and chips, c, you can purchase.
Slope
Y-intercept
Let’s look at our highlighted information
Copyright© 2009 Algebra-class.com
Point 1
Point 2
Unit 3: Writing Equations
$20 to spend – This looks like a total amount.
Pretzels cost $2.25 a bag
Chips cost $3.50 a bag
P = total number of pretzels
C = total number of chips
Do you see a number that could represent slope? Do you see a rate of change? It looks
like $2.25 a bag could represent slope because the total will change based on how many
bags of pretzels you buy… BUT….
We might have 2 numbers that look like slope because we also have chips that cost $3.50
a bag. So, our total will change based on how many bags of chips we buy too!
The difference here is that our total is set! Our total cannot change, but we can change
the number of bags of chips and pretzels that we buy to meet our total.
When you have a set total, and two things that you add together to get that total, then
you have a standard form equation. We really don’t have any of the information that
applies to the table, so we know that this will be written in standard form instead of slope
intercept form:
2.25p +3.50c = 20
So, how do you feel about analyzing the information to write an equation?
Two things that might help you the most:
1. Write a list of the important information that you highlight in your problem. This helps
you to better analyze the information and make sense of it!
2. Organize your information in a table. This helps you to see what information you are
given and what you need to find before writing your equation.
I hope that this Bonus Lesson helps you as you see a mixture of problems on your
assessments.
Copyright© 2009 Algebra-class.com
Algebra 1
Unit 3: Writing Equations
Lesson 6: Writing Linear Equations
New Focus: Point Slope Form
Writing an Equation Given Slope and the Y-Intercept
A discount CD store sells CD’s and DVD’s for $10 per disc. They also charge $4.50 for shipping (no matter
how many discs you order).
•
•
Write an equation that you could use to find the total price (y) for any number of discs (x) purchased.
How much would it cost to order 15 DVD’s from this company?
Writing an Equation Given Slope and a Point
A park ranger estimates that after his 3rd year working in the park, there are about 5500 deer that live on
park grounds. He estimates that the population increases by about 35 deer per year.
•
•
Write an equation that estimates how many deer will be in the park in x years.
About how many deer will be in the park after the 10th year?
Point-slope Form
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
Writing an Equation Given Two Points
John sells snack products for snack machines in college universities. He makes a daily salary plus
commission. When John’s sales reach $500, he makes $175. When John’s sales are $1000, he makes
$250.
•
•
•
Write an equation to calculate John’s total paycheck (y) based on his daily sales (x).
How much will John make if he sells $750 worth of products?
What is John’s daily salary and commission rate?
Rate of Change
An internet provider company began service in the year 2000. They had 1.1 million users in the year 2005
and 1.9 million users in the year 2010.
• What is the average rate of change in users per year for the internet provider?
• Estimate how many users there will be in the year 2020. Explain how you determined your answer.
Checking with the Graphing Calculator
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
Lesson 6: Writing Linear Equations Practice Problems
New Focus: Point Slope Form
1. Josie has a collection of stamps. She has 74 stamps and collects 8 per month.
• Write an equation that could be used to identify the total number (t) of stamps Josie has in her
collection based on the number of months (m) collecting.
• What is the slope and y-intercept and what do they represent?
• How many stamps will Josie have collected after 10 months?
2. Write an equation that has a slope of ½ and passes through the point (5,2)
3. Write an equation that passes through the points (2,7) and (-4, -8). Explain the steps that you used to
solve this problem.
4. Write an equation that has a slope of 4 and an x-intercept of 2. Explain how you solved this problem.
5. Write an equation that has a y-intercept of -3/2 and passes through the point (1/2,9)
6. Write an equation that has an x-intercept of -1 and a y-intercept of 10.
7. Write an equation for the line that passes through the points indicated in the table.
X
Y
-4
-1
0
2
4
5
8. Two florists are selling Valentine bouquets for Valentine’s Day. Their advertisements are below:
Judy’s Florist
Florist by the Bay
Valentines Special
Valentines Special
$35 Rose Arrangements
$50 Rose Arrangements
$2 per mile delivery fee
Free delivery
•
•
•
•
Write an equation for each florist that will allow them to determine the price (p) for any number
of miles (m) driven for delivery.
Create a table of values to display your data for the two florists.
If you live 10 miles away Florist by the Bay and 9 miles from Judy’s Florist, who would cheaper
to order from?
For what distance do the two florists charge the same amount?
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
9. Jake is trying to figure out the equation that a new cell phone company is using to determine the total cost
of his bill. This company charges based on the number of minutes used plus a monthly maintenance
fee. The first month, Jake used 210 minutes and his bill was $38.80. The next month, he used 330
minutes and his bill was $48.40.
• Write an equation that can be used to determine the total cost (t) of the bill based on the number of
minutes (m) used monthly. Explain how you determined your answer.
• Use the table function on your graphing calculator to determine how much a phone bill would cost for
a total of 550 minutes.
• If Jakes monthly budget for cell phone use is $50, what is the maximum number of minutes that he
can use to stay in budget?
• What is the monthly maintenance fee for this phone? How do you know?
• How much is Jake charged per minute? How do you know?
10. In the second month, a stock had an average price of $24.50. By month 10, the stock’s average price
was $47.25. What is the average rate of change in the stock’s price?
• What would you need to know in order to make a prediction about month five’s average price?
Estimate the average price for month number 5.
11. Carrie took a cab ride in NYC. The cost for driving 3 miles was $7.75. The taxi driver charges $1.25 per
mile. Write an equation that can be used to determine the total cost (t) for any number of miles (m) in NYC.
Explain how you came up with the equation.
• How much would a 10 mile taxi ride cost?
• If you only had $15 on you, what is the farthest distance you could travel in the taxi? Justify your
answer mathematically.
•
•
•
•
•
Anna has an online business selling hair bows. She sells each hair
bow for $4.50 plus a fixed shipping and handling price. One customer
paid $27.50 for 5 hair bows. (13 points total)
How much is Anna charging for shipping and handling? Explain how you determined your answer. (3)
How much should Anna charge for 3 hair bows? (2)
Lisa is Anna’s competitor. She sold 5 hair bows for $26.75 and to another customer she sold 7 hair
bows for $34.25. Does Anna charge more or less per hair bow? Explain how you determined your
answer. (3)
Use the table on the graphing calculator to determine how many hair bows need to be purchased in
order for the two companies to charge the same amount. (2)
If you are buying 15 hair bows, who would be the most cost effective choice? Explain your answer. (3)
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
Lesson 6: Writing Linear Equations Practice Problems
New Focus: Point Slope Form (Answer Key)
1. Josie has a collection of stamps. She has 74 stamps and collects 8 per month.
• Write an equation that could be used to identify the total number (t) of stamps Josie has in her
collection based on the number of months (m) collecting.
This problem tells us that she has 74 stamps. This is a fixed (constant) number. This is the number that she
is starting with. Therefore, this number is the y-intercept. (The y-intercept is always a constant number that
represents the value when x =0 or your “beginning/starting” number.
The problem then tells us that Josie collects 8 per month. The key word here is “per”. Since she collects 8
every month, this number represents the slope.
We are given the slope and y-intercept:
an equation in slope intercept form:
m=8
b = 74. With these two pieces of information, we can write
y = mx+b
y = 8x + 74
•
or
t = 8m +74
What is the slope and y-intercept and what do they represent?
The slope is 8 and this represents how much Josie’s stamp collection increases per month. She collects 8
more stamps per month.
The y-intercept is 74. This number represents the number of stamps that she started with when she began
the collection.
•
How many stamps will Josie have collected after 10 months?
After 10 months, Josie will have collected 154 stamps.
If you substitute 10 for x, you will find the total number of stamps collected.
T = 8m +74
T = 8(10)+74
T = 154
2. Write an equation that has a slope of ½ and passes through the point (5,2)
Since we know one point and the slope, we can use the Point-slope form to write the equation.
m = ½,
x1 = 5
y1 =2
y – y1 = m(x-x1)
y – 2 = ½(x-5)
y – 2 = 1/2x – 5/2
y -2 +2 = 1/2x – 5/2 +2
y = 1/2x -1/2
Substitute for m, x1 and y1
Distribute the ½ throughout the parenthesis
Add 2 to both sides
Simplify: -5/2 +2 = -1/2
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
3. Write an equation that passes through the points (2,7) and (-4, -8). Explain the steps that you used to
solve this problem.
Since we are given two points, we must first start by finding the slope of the line that passes through the two
points. We will use the slope formula to find the slope.
− −8 − 7
−15 15 5
=
=
=
= − −4 − 2
−6
6
2
Now we can use the point-slope form to write the equation. (*You can use either point for x1 and y1)
m = 5/2 ,
x1 = 2
y1 =7
Since the problem
y – y1 = m(x-x1)
asks for an
y – 7 = 5/2(x-2)
Substitute for m, x1 and y1
explanation, you
must write out the
y – 7 = 5/2x – 5
Distribute the 5/2 throughout the parenthesis
steps.
y -7 +7 = 5/2x -5 +7
Add 7 to both sides
y = 5/2x +2
Simplify: -5+7 = 2
4. Write an equation that has a slope of 4 and an x-intercept of 2. Explain how you solved this problem.
We know the slope is 4 and we know the x-intercept is 2. Can you write the x-intercept as an ordered pair?
Yes, remember, the x-intercept is when y = 0, so the ordered pair is (2,0). Now we know the slope and a
point, so we can use point-slope form.
m=4,
x1 = 2
y – y1 = m(x-x1)
y – 0 = 4(x-2)
y – 0 = 4x – 8
y = 4x - 8
y1 =0
Substitute for m, x1 and y1
Distribute the 4 throughout the parenthesis
This is the equation.
**Explanation: I first wrote the x-intercept of 2 as an ordered pair (2,0). Then I substituted the values for the
slope, and the x and y coordinates of the ordered pair into the formula for point-slope. I solved the formula for
y and found the equation: y = 4x – 8.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
5. Write an equation that has a y-intercept of -3/2 and passes through the point (1/2,9)
We know the y-intercept is -3/2. We can write this as an ordered pair: (0, -3/2). The two ordered pairs are:
(0, -3/2) and (1/2, 9). We need to find the slope and since we have fractions, using the slope formula could
get a little messy. (You can change the fractions to decimals to substitute into the slope formula.)
Since you know the y-intercept and another point, we can substitute these values into the equation y = mx+b
and solve for m.
Y = mx+b
x=1/2, y = 9, b = -3/2
9 = m(1/2) -3/2
9 = 1/2m – 3/2
9+3/2 = 1/2m – 3/2 +3/2
Add 3/2 to both sides
10.5 = 1/2m
Simplify
2(10.5) = 2(1/2m)
Multiply by 2 on both sides
21 = m
Now we know m= 21 and b = -3/2
Y = 21x – 1.5 is the equation.
You can use the slope formula and point-slope form. This is another way to do it.
-3/2 = -1.5 and ½ = .5 So our 2 points are: (0, -1.5) (.5, 9)
− 9 − (−1.5)
10.5
=
=
= 21
− . 5 − 0
.5
Point-slope form:
m = 21 ,
x1 = .5
y1 =9
y – y1 = m(x-x1)
y – 9 = 21(x-.5)
y – 9 = 21x – 10.5
y-9+9 = 21x – 10.5 +9
y = 21x -1.5 is the equation
Substitute for m, x1 and y1
Distribute the 4 throughout the parenthesis
Add 9 to both sides
6. Write an equation that has an x-intercept of -1 and a y-intercept of 10.
We know the y-intercept, so all we have to find is the slope. We could write the intercepts as 2 points and use
the slope formula or you can substitute into y=mx+b and solve for m. Either method is great. I will use the
slope formula since I have easy numbers to work with. (-1, 0) (0, 10)
− 10 − 0
10
=
=
= 10
− 0 − −1
1
Slope (m) = 10 and the Y-intercept (b) = 10
Y = mx + b
Y = 10x +10 is the equation
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
7. Write an equation for the line that passes through the points indicated in the table.
X
Y
-4
-1
0
2
4
5
If all three points are on the same line, then it won’t matter which two points I choose to find the slope. We
must find the slope and the y-intercept. I will choose (0,2) and (4,5) to find the slope.
= = the slope is ¾. Now we can use point-slope form.
Let’s choose the point: (0,2). So, m = ¾, x1 = 0 y1 = 2
m = 3/4 ,
x1 = 0
y1 =2
y – y1 = m(x-x1)
y – 2 =3/4(x-0)
Substitute for m, x1 and y1
y – 2 = 3/4x
Distribute the 3/4 throughout the parenthesis
y-2+2 = 3/4x+2
Add 2 to both sides
y = 3/4x +2
We know that (0,2) is on the line because we used that point in our point-slope formula.
Let’s check to make sure (-4, -1) and (4,5) are on the line too.
Y = 3/4x +2
y = 3/4x +2
-1 = ¾(-4) +2
5 = ¾(4) +2
-1 = -3 +2
5 = 3+2
-1 = -1
5=5
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
They both work, so our equation is
correct.
Algebra 1
Unit 3: Writing Equations
8. Two florists are selling Valentine bouquets for Valentine’s Day. Their advertisements are below:
Judy’s Florist
Florist by the Bay
Valentines Special
Valentines Special
$35 Rose Arrangements
$50 Rose Arrangements
$2 per mile delivery fee
Free delivery
•
Write an equation for each florist that will allow them to determine the price (p) for any number
of miles (m) driven for delivery.
Each of the advertisements tells us the slope (rate per mile for delivery) and the y-intercept which is the
constant (price that does not change, or the cost of the rose arrangement). Therefore, we can write 1 equation
for each company in slope intercept form.
Judy’s Florist: p = 2m +35
•
Florist by the Bay: y = 50
Create a table of values to display your data for the two florists.
To create a table of values, you can substitute values for x into your equation and solve for y. You can also
use the “y=” and “table” functions on your graphing calculator and copy these values into a table of values.
Judy’s Florist
# of miles from
florist (m)
0
1
2
3
4
5
6
7
8
9
10
•
Florist by the Bay
Total price of the
bouquet (p)
35
37
39
41
43
45
47
49
51
53
55
# of miles from
florist (m)
0
1
2
3
4
5
6
7
8
9
10
Total price of the
bouquet (p)
50
50
50
50
50
50
50
50
50
50
50
If you live 10 miles away Florist by the Bay and 9 miles from Judy’s Florist, who would cheaper
to order from?
If you live 10 miles away from Florist by the Bay it would cost $50 and if you lived 9 miles away from Judy’s
Florist it would cost $53. Therefore, it would be cheaper to buy from Florist by the Bay.
•
For what distance do the two florists charge the same amount?
If the florists charge for ½ mile distances, then the two florists would charge the same amount, $50 for 7.5
miles.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
9. Jake is trying to figure out the equation that a new cell phone company is using to determine the total cost
of his bill. This company charges based on the number of minutes used plus a monthly maintenance
fee. The first month, Jake used 210 minutes and his bill was $38.80. The next month, he used 330
minutes and his bill was $48.40.
• Write an equation that can be used to determine the total cost (t) of the bill based on the number of
minutes (m) used monthly. Explain how you determined your answer.
We know that 210 minutes is $38.80, therefore, this is an ordered pair (210, 38.80).
We also know that 330 minutes costs $48.40, therefore this is another ordered pair: (330, 48.40)
We are given 2 ordered pairs in this problem, therefore, we will need to find the slope and then use point-slope
form to write the equation.
=
..
.
= = .08 the slope is .08 Now we can use point-slope form.
Let’s choose the point: (210,38.80). So, m = .08, x1 = 210 y1 = 38.80
m = =.08,
x1 = 210
y1 =38.80
y – y1 = m(x-x1)
y – 38.80 =.08(x-210)
Substitute for m, x1 and y1
y – 38.80 = .08x -16.80
Distribute the .08 throughout the parenthesis
y-38.80+38.80 = .08x-16.80+38.80
Add 38.80 to both sides
y = .08x +22
This is the equation used to determine the total cost of the bill.
t = .08m +22
•
Use the table function on your graphing calculator to determine how much a phone bill would cost for
a total of 550 minutes.
After inputting the equation into the “y=” function, we can use the table to find x = 550. (Go to “table set” to set
the table to start at 550 and this is much easier than scrolling through.
For 550 minutes, the cost would be $66.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
•
If Jakes monthly budget for cell phone use is $50, what is the maximum number of minutes that he
can use to stay in budget?
You can see from the table on the graphing calculator that if his
budget is $50, then the maximum number of minutes that he can
use is 350 minutes. The ordered pair is: (350, 50)
•
What is the monthly maintenance fee for this phone? How do you know?
The monthly maintenance fee is $22. We know this because her plan charges a price per minute which is
the rate of change or slope and the constant (22) is the monthly maintenance fee. This amount never
changes, it is charged every month. Since this is the y-intercept and the y-intercept is always a constant,
I know this is the maintenance fee.
•
How much is Jake charged per minute? How do you know?
Jake is charges $0.08 per minute. This is the slope in the equation and the slope is the rate. His rate is
$0.08 per minute.
10. In the second month, a stock had an average price of $24.50. By month 10, the stock’s average price
was $47.25. What is the average rate of change in the stock’s price?
The question asks for the average rate of change, which means we need to find the slope. We are given
2 points (or 2 pairs of information). We will need to use the slope formula to find the rate of change.
(2, 24.50) (2nd month, price of $24.50)
(10, 47.25) (10th month, price of $47.25)
The average rate of change in the stock’s price is
=
..
=
. = !. "#
$2.84 per month.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
•
What would you need to know in order to make a prediction about month five’s average price?
Estimate the average price for month number 5.
We would need to know an equation in order to estimate month number 5’s average price.
We can use point-slope form since we know the slope and a point.
m = 2.84,
x1 = 2 y1 =24.50
y – y1 = m(x-x1)
y – 24.50 =2.84(x-2)
Substitute for m, x1 and y1
y – 24.50 = 2.84x – 5.68
Distribute the 2.84 throughout the parenthesis
y- 24.50 +24.50 = 2.84x-5.68+24.50
Add 24.50 to both sides
y = 2.84x +18.82
This is the equation used to determine the average price
of the stock for any month.
To find the average price for the fifth month, we can substitute 5 for x into our new equation.
Y = 2.84(5) +18.82
Y = 33.02
The average price of the stock for the fifth month is $33.02
11. Carrie took a cab ride in NYC. The cost for driving 3 miles was $7.75. The taxi driver charges $1.25 per
mile. Write an equation that can be used to determine the total cost (t) for any number of miles (m) in NYC.
Explain how you came up with the equation.
We know that 3miles cost $7.75. These two numbers are related, so it’s an ordered pair: (3, 7.75)
We know that it’s $1.25 per mile. The word “per” indicates that this is slope or a rate. So, m = 1.25.
Since I have slope and one point, I can use point-slope form to write the equation.
m = 1.25,
x1 = 3 y1 =7.75
y – y1 = m(x-x1)
y – 7.75 =1.25(x - 3)
y – 7.75 = 1.25x –3.75
y- 7.75 +7.75 = 1.25x- 3.75 +7.75
y =1.25x +4
t = 1.25m +4
•
Substitute for m, x1 and y1
Distribute the 1.25 throughout the parenthesis
Add 7.75 to both sides
This is the equation that can be used to determine the cost
for any number of miles in a cab in NYC.
How much would a 10 mile taxi ride cost?
To find out how much a 10 mile taxi ride will cost, we need to substitute 10 for m into our equation.
T = 1.25m +4
T = 1.25(10)+4
T = 16.50
It will cost $16.50 for a 10 mile taxi ride.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
•
If you only had $15 on you, what is the farthest distance you could travel in the taxi? Justify your
answer mathematically.
You only have $15, so let’s substitute 15 for T and solve for m.
T = 1.25m +4
15 = 1.25m +4
Substitute 15 for T
15 – 4 = 1.25m +4 – 4
Subtract 4 from both sides
11 = 1.25m
Simplify
11/1.25 = 1.25m/1.25
Divide by 1.25 on both sides
8.8 = m
You could travel 8.8 miles in a tax for $15. Or 8 whole miles is the
farthest
distance you could travel.
Justify: 1.25(8) +4 = $14.
Anna has an online business selling hair bows. She sells each hair
bow for $4.50 plus a fixed shipping and handling price. One customer
paid $27.50
for 5 hair bows. (13 points total)
• How much is Anna charging for shipping and handling? Explain how you determined your answer. (3)
We know the rate is $4.50 per bow, so this is the slope. The shipping and handling would be a constant and
is the y-intercept, which we are not given. Therefore, we would need to find the y-intercept or write an
equation in slope intercept for in order to identify the y-intercept. We know one other point: (5, 27.50).
So, let’s write an equation using point slope form.
m = 4.50,
x1 = 5 y1 =27.50
y – y1 = m(x-x1)
y – 27.50 =4.50(x - 5)
y – 27.50 = 4.50x –22.50
y- 27.50+27.50 = 4.50x- 22.50+27.50
y = 4.50x +5
Substitute for m, x1 and y1
Distribute the 4.50 throughout the parenthesis
Add 27.50 to both sides
Equation for shipping any number of bows.
The shipping and handling price is $5. I used the slope and point and point-slope form to write an
equation in slope intercept form. I knew that the shipping and handling was the y-intercept since it
was a fixed price. In the equation, 5 is the y-intercept; therefore, it represents the shipping and
handling price.
•
How much should Anna charge for 3 hair bows? (2)
For 3 bows, Anna should charge: $18.50
Y = 4.50x +5
Y = 4.50(3) + 5
Y = 18.50
Substitute 3 for x
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
•
Lisa is Anna’s competitor. She sold 5 hair bows for $26.75 and to another customer she sold 7 hair
bows for $34.25. Does Anna charge more or less per hair bow? Explain how you determined your
answer. (3)
Since we need to find how much Anna charges per hair bow, we need to find the slope (or rate). We have 2
points, so we can use the slope formula. (5, 26.75) (7, 34.25)
=
.. =
.
= $. %&
Anna charges less per hair bow. I found my answer by finding
the slope of the two points. The slope represents the rate.
Therefore, Anna charges $3.75 per bow.
•
Use the table on the graphing calculator to determine how many hair bows need to be purchased in
order for the two companies to charge the same amount. (2)
In order to use the graphing calculator, I have to have equations in slope intercept form for both people. I
know the equation for Anna’s business is y = 4.50x +5.
I have to find the equation for Lisa’s business by using the point-slope formula:
m =3.75,
x1 = 5 y1 =26.75
y – y1 = m(x-x1)
y – 26.75 =3.75(x - 5)
y – 26.75 = 3.75 –18.75
y- 26.75+26.75 = 3.75x- 18.75+26.75
y = 3.75x +8
Substitute for m, x1 and y1
Distribute the 4.50 throughout the parenthesis
Add 27.50 to both sides
Lisa’s Equation
We can put both equations into y = at the same time: y1 is Anna’s
y2 is Lisa’s
They both charge the same amount for 4 hair bows. They both charge $23.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
•
If you are buying 15 hair bows, who would be the most cost effective choice? Explain your answer. (3)
If you are buying 15 (x=15) hair bows it would be most cost effective to go with Lisa’s hair bows. We put
Lisa’s equation, (y = 3.75x +8) in for L2. The cost for 15 of Lisa’s hair bows is $64.25 and the cost for 15 of
Anna’s hair bows is $72.50.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3 Writing Equations
Lesson 7: Parallel and Perpendicular Lines
Parallel Lines
Write an equation for each line on the graph:
Red line: ______________________________
Green line: ___________________________
Blue line: _____________________________
Parallel Lines: __________________________
______________________________________
______________________________________
**All vertical lines are parallel.
**All horizontal lines are parallel because
______________________________________
Example 1
Write an equation for a line that is parallel to
the line with equation: x + 3y = 9
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Example 2
Write an equation for a line that passes
through the point (3,-1) and is parallel to the
line with equation: x = 2y +10
Algebra 1
Unit 3 Writing Equations
Perpendicular Lines
This graph represents two lines that are
perpendicular. Two lines that are
perpendicular intersect at 90° angles.
Write an equation for each line on the
graph:
Blue line: ________________________
Red line: ________________________
The slopes of perpendicular lines are
___________________________________
___________________________________
When you multiply the slopes of two
perpendicular lines, the product is always
__________.
Example 1
Write an equation for a line that is
perpendicular to the line with equation:
x – 2y = 8
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Example 2
Write an equation for a line that passes
through the point (2,1) and that is
perpendicular to the line with equation:
1/2y = x + 3
Algebra 1
Unit 3 Writing Equations
Lesson 7: Parallel and Perpendicular Lines
Practice Problems
1. Graph the following two lines on the grid.
y = 3x – 5
A line that is parallel to this line.
•
Write an equation for the parallel line.
2. Graph the following two lines on the grid.
3x +4y = -24
A line that passes through (2,1) and is parallel to this line.
•
Write an equation for the parallel line.
3. Graph the following two lines on the grid.
2x – 2y + 12 = 0
A line that passes through (-5,-2) and is parallel to
this line.
•
•
Explain how you graphed the two equations.
Write an equation for the parallel line.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3 Writing Equations
4. Graph the following two lines on the grid.
y = 1/2x – 2
A line that is perpendicular to this line.
•
Write an equation for the perpendicular line.
5. Graph the following two lines on the grid.
3x +2y = 12
A line that passes through (2,1) and is perpendicular
to this line.
•
Write an equation for the perpendicular line.
6. Graph the following two lines on the grid.
2x – 3y + 9= 0
A line that passes through (-4,2) and is perpendicular
to this line.
•
•
Explain how you graphed the two equations.
Write an equation for the perpendicular line.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3 Writing Equations
7. Write an equation for a line that passes through (3,1) and is parallel to a line whose slope is -1. Explain
how you determined your answer.
8. Write an equation for a line that passes through (6,1) and is perpendicular to a line whose slope is -3/2.
Explain how you determined your answer.
9. Write an equation for a line that passes through (4,2) and is perpendicular to the graph of y = 2.
10. Write an equation for a line that passes through (2,6) and is parallel to the graph of x = 5.
11. Write an equation for a line that is perpendicular to the graph of 3x – 2y = -12 and intersects the graph at
its x-intercept.
12. Write an equation for a line that is perpendicular to the graph of 3x = 2y – 4 and intersects the graph at
its y-intercept.
13. What is the slope of a line that is perpendicular to the line x = 4?
1. Identify the slope of a line that is parallel to the graph of:
4x – 2y = 7 (1 point)
2. Identify the slope of a line that is perpendicular to the graph of:
Y = -3/4x – 10 (1 point)
3.
3. Write an equation that passes through the point (2,2) and is parallel to a line that passes through the
points (5,2) & (6,1) (3 points)
4. Write an equation that passes through the point (7,-2) and is perpendicular to the graph of:
x = 2y +5 (3 points)
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3 Writing Equations
Lesson 7: Parallel and Perpendicular Lines
Practice Problems – Answer Key
1. Graph the following two lines on the grid.
y = 3x – 5
The y-intercept is -5 and the slope is 3. (Red line)
A line that is parallel to this line.
A line that is parallel to this line will have a slope of
3 because two lines that are parallel will have the
same slope. It doesn’t specify any other point, so
we can choose our own y-intercept. Your answers
may vary – but you must have a slope of 3. (Blue
line)
•
Write an equation for the parallel line.
The line that I graphed is: y = 3x + 4.
Your line must have a slope of 3 and any yintercept except for -5.
2. Graph the following two lines on the grid.
3x +4y = -24
Since 24 is divisible by 3 and 4, I will find the x and
y intercepts in order to graph this line.
X intercept:
Y intercept:
3x = -24
4y = -24
3x/3 = -24/3
4y/4 = -24/4
x = -8
y = -6
(This is the red line on the graph)
A line that passes through (2,1) and is parallel to this line.
Since you have the graph, you can graph a point on
(2,1) and use the same slope (-3/4) to find the next
point. **(You can find the slope of the red line by
counting the slope on your graph)
Draw a line through your points and this line should
be parallel to the first because they have the same
slope. (This is the blue line)
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
• Write an equation for the parallel line.
This line looks like it has a y-intercept of 2.5 and
it definitely has a slope of -3/4. The equation is:
Y = -3/4x + 2.5.
Let’s check: y =-3/4x +2.5 x = 2 y = 1
1 = -3/4(2)+2.5
1=1
Algebra 1
Unit 3 Writing Equations
3. Graph the following two lines on the grid.
2x – 2y + 12 = 0
Let’s first rewrite this in correct standard form:
2x – 2y = -12
(Subtract 12 from both sides)
Now I notice that 12 is divisible by 2, so I will find the x
and y intercepts.
X intercept:
Y – intercept:
2x = -12
-2y = -12
2x/2 = -12/2
-2y/-2 =-12/-2
X = -6
y=6
(This is the red line)
A line that passes through (-5,-2) and is parallel to
this line.
•
If I count
the slope of the red line, I notice that it is 1.
Therefore, this line must also have a slope of 1. I will
plot the point (-5, -2) on the grid and then use the
slope of 1 to find the next couple of points. (This is
the blue line.
•
Explain how you graphed the two equations.
Each line is explained above. You should also
have written instructions for how to graph each
line in order to answer this bullet.
Write an equation for the parallel line.
From looking at the graph, I can tell that the yintercept of the blue line is 3 and I know the slope
is 1. Therefore, the equation is: y = x +3
4. Graph the following two lines on the grid.
y = 1/2x – 2
This equation has a y-intercept of -2 and a slope of ½.
Plot the y-intercept of 2 and use the slope to find the
next couple of points. (This is the red line.)
A line that is perpendicular to this line.
A line that is perpendicular with have a negative
reciprocal of the slope. Since the slope is ½, the slope
of a perpendicular line is -2. The equation for my blue
line is: y = -2x +5. **You may have any y-intercept,
therefore answers will vary.
Write an equation for the perpendicular line.
Y = -2x+5
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3 Writing Equations
5. Graph the following two lines on the grid.
3x +2y = 12
Since 12 is divisible by 3 and 2, I will use the x and y
intercepts to graph this line.
X intercept:
3x = 12
3x/3 = 12/3
x=4
This is the red line)
Y intercept:
2y = 12
2y/2 = 12/2
y=6
A line that passes through (2,1) and is perpendicular
to this line.
First you must count the slope of the red line. The
slope is -3/2.
A line that is perpendicular must have a slope that is the
negative reciprocal of -3/2. Therefore, this blue line
must have a slope of 2/3.
Since it passes through (2,1), plot this point on the grid.
Then use the slope of 2/3 to find the next couple of
points. Draw a line through these points.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
•
Write an equation for the
perpendicular line.
It’s difficult to write the equation by reading the
graph because the y-intercept is a fraction. So,
let’s use algebra to figure out this equation.
We know: m = 2/3 x = 2 y = 1
Let’s substitute to find b.
Y = mx+b
1 = 2/3(2) +b
1 = 4/3 +b
1 – 4/3 = 4/3-4/3 +b
-1/3 = b
The equation is: y = 2/3x – 1/3
Algebra 1
Unit 3 Writing Equations
6. Graph the following two lines on the grid.
2x – 3y + 9 = 0
Let’s first rewrite this in correct standard form:
2x – 3y = -9 (Subtract 9 from both sides of the equation)
Since 9 is not divisible by 2, I will rewrite this in slope
intercept form:
2x -2x – 3y = -9 -2x Subtract 2x from both sides
-3y = -2x – 9
-3y/-3 = -2x/-3 – 9/-3
Y = 2/3x +3
m = 2/3 b = 3
A line that passes through (-4,2) and is perpendicular
to this line.
A line that is perpendicular is going to have a slope that is
the negative reciprocal of 2/3. Therefore, the slope of the
blue line will be: -3/2.
Since it passes through (-4,2), we will plot (-4,2) on the
graph and use the slope of -3/2 to find the next couple of
points.
•
Explain how you graphed the two
equations.
Explanations are included with each of
the 2 bullets above. Your answers
should include written explanations too.
•
Write an equation for the
perpendicular line
From looking at the graph, the blue line
crosses the y axis at -4. Therefore, the
y-intercept is -4. We know the slope is
-3/2, so the equation is: y = -3/2x-4
.
7. Write an equation for a line that passes through (3,1) and is parallel to a line whose slope is -1. Explain
how you determined your answer.
Since our line that must be parallel has a slope of -1, we know that our equation must also include a slope of
-1. We have an ordered pair, so we also know x and y coordinates. Let’s substitute to find b and write the
equation.
Y = mx+b
m = -1
x=3
y=1
b=?
1 = -1(3) +b
Substitute values for m, x, and y.
1 = -3+b
Simplify
1+3 = -3+3 +b
Add 3 to both sides
4=b
Now we know that the slope is -1 and the y-intercept is 4. The equation for the line parallel is:
y = -x+4
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3 Writing Equations
8. Write an equation for a line that passes through (6,1) and is perpendicular to a line whose slope is -3/2.
Explain how you determined your answer.
A line that is perpendicular must have a slope that is the negative reciprocal of the other line’s slope.
Therefore, the slope of our perpendicular line is 2/3.
We know the line passes through (6,1) so we know: m = 2/3
x=6
y = 1. We need to find b.
Y = mx+b
1 = 2/3(6) +b
Substitute for m, x, and y.
1 = 4 +b
Simplify
1-4 = 4-4 +b
Subtract 4 from both sides
-3 = b
Since b = -3 our equation is: y = 2/3x - 3
9. Write an equation for a line that passes through (4,2) and is perpendicular to the graph of y = 2.
The graph of y = 2 has a slope of 0 because this is a horizontal line. This means that a perpendicular line
must be a vertical line with an undefined slope. (This equation will be x = …)
We know that the vertical line passes through (4,2). The x coordinate of this ordered pair is 4, therefore, the
line that is perpendicular is x = 4.
X = 4 is the line that is perpendicular to the graph of y = 2 and passes through (4,2)
10. Write an equation for a line that passes through (2,6) and is parallel to the graph of x = 5.
The graph of x = 5 is a vertical line with an undefined slope. A line that is parallel must also be a vertical line
with an undefined slope. Since we know that the parallel line passes through (2,6), we know that the
equation for that line must be x = 2.
The equation for the line that is parallel to x = 5 and passes through (2,6) is x = 2
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3 Writing Equations
11. Write an equation for a line that is perpendicular to the graph of 3x – 2y = -12 and intersects the graph at
its x-intercept.
In order to figure out the equation for a line that is perpendicular, we must first figure out the slope of this
line. Since it’ written in standard form, we need to rewrite this equation in slope intercept form:
3x -3x -2y = -12 – 3x
-2y = -3x – 12
-2y/-2 = -3x/-2 – 12/-2
Y = 3/2x +6
Subtract 3x from both sides
Simplify
Divide by -2
This is the equation written in slope intercept form. We know the slope is 3/2.
A line that is perpendicular will have a slope that is the negative reciprocal of 3/2. Therefore, the slope of
our perpendicular line is -2/3.
Now we know that the perpendicular line intersects at the graphs x-intercept. We have to find the x intercept
for: 3x – 2y = -12.
X intercept:
3x = -12
3x/3 = -12/3
X = -4
its x intercept is -4, which is the point (-4,0)
Now we know the slope and a point, we need to find b in order to write the equation:
Y = mx +b
m = -2/3 x = -4 y = 0
0 = -2/3(-4) +b
Substitute for m, x, and y
0 = 8/3 +b
Simplify
0-8/3 = 8/3-8/3 +b
Subtract 8/3 from both sides
-8/3 = b
The y-intercept is -8/3
The equation for a line perpendicular to 3x-2y = -12 that passes through its’ x intercept is:
y = -2/3x-8/3
12. Write an equation for a line that is perpendicular to the graph of 3x = 2y – 4 and intersects the graph at
its y-intercept.
If we rewrite this equation in slope intercept form, then we will be able to find the slope and the y-intercept.
3x= 2y – 4
Original equation
3x +4 = 2y – 4 + 4
Add 4 to both sides
3x +4 = 2y
Simplify
3/2x + 4/2 = 2y/2
Divide by 2 on both sides
3/2x +2 = y
The equation in slope intercept form
Y = 3/2x +2
We know that the slope is 3/2 and the slope of our perpendicular line must by the opposite reciprocal, so its
slope is -2/3. We know the y-intercept is 2; therefore, our perpendicular line passes through (0,2). Let’s
substitute to find b for the equation of the line that is perpendicular.
Y = mx+b m = -2/3 x = 0 y = 2
2 = -2/3(0) +b
2=b
Since b = 2, the equation of the line perpendicular is: y = -2/3x +2
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3 Writing Equations
13. What is the slope of a line that is perpendicular to the line x = 4?
Since the slope of x = 4 is undefined, this means that the line is a vertical line. A line that is
perpendicular will be a horizontal line. All horizontal lines have a slope of 0. Therefore, the slope of a
line perpendicular to x = 4 is 0.
1. Identify the slope of a line that is parallel to the graph of:
4x – 2y = 7 (1 point)
In order to figure out the slope of a line that is parallel to this line, we must
first find the slope of this line. Since it is written in standard form, we must
rewrite it in slope intercept form.
4x – 4x – 2y = 7-4x
Subtract 4x from both sides
-2y = -4x +7
Simplify
-2y/-2 = -4x/-2 +7/-2
Divide by -2
Y = 2x -7/2
The slope of this line is 2
The slope of a line parallel to this line is also 2 since parallel lines have
the same slope.
2. Identify the slope of a line that is perpendicular to the graph of:
Y = -3/4x – 10 (1 point)
Since the slope of this equation is -3/4, we know that the slope of a perpendicular line will be the negative
reciprocal. Therefore, the slope is 4/3.
3. Write an equation that passes through the point (2,2) and is parallel to a line that passes through the
points (5,2) & (6,1) (3 points)
Since we are given 2 points, we will need to use the slope formula to find the slope of the two points.
= = = −1
The slope of the line is -1
The slope of a line parallel to this line will also be -1. It passes through (2,2), so we must find the y-intercept
in order to write the equation.
Y = mx+b m = -1
2 = -1(2) +b
2= -2 +b
2+2 = -2+2 +b
4=b
x=2
y=2
Substitute for m, x, and y
Simplify
Add 2 to both sides
Since b = 4, the equation for the line parallel and passing through (2,2) is:
Y = -x +4
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3 Writing Equations
4. Write an equation that passes through the point (7,-2) and is perpendicular to the graph of:
x = 2y +5 (3 points)
We first need to find the slope of the line given by rewriting it in slope intercept form.
x = 2y +5
x – 5 = 2y +5-5
x – 5 = 2y
x/2 -5/2 = 2y/2
x/2 -5/2 = y
y = x/2 -5/2
Original equation
Subtract 5 from both sides
Simplify
Divide by 2 on both sides
The slope of this equation is ½.
Since the slope is ½ and we know that the slope of a perpendicular line is the negative reciprocal, we know
that the slope of our perpendicular line is -2.
Now we know the slope is -2 and it passes through (7, -2). Let’s solve for b in order to write the equation.
Y = mx+b
-2 = -2(7) +b
-2 = -14 +b
-2 +14 = -14+14 +b
12 = b
Substitute for m, x, and y.
Simplify
Add 14 to both sides
The y-intercept is 12.
The equation for the line perpendicular to x = 2y +5 and passing through (7, -2) is: y = -2x +12
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
Lesson 8: Making Predictions and Creating Scatter Plots
The table below represents the cost of a car over the
recent years.
Year
2000
2002
2004
2006
2008
2010
2012
Cost of a Car (in US
dollars)
22,500
26,000
32,000
37,500
39,000
45,000
52,000
Create a scatter plot to display the data.
Create a line of best fit that can be used to make
predictions.
Write an equation for the line of best fit.
Making Predictions
1. Based on your line of best fit, what does the slope and y-intercept represent?
2. Based on the line of best fit, what will be the cost of the car in the year 2015? Justify your answer.
3. Predict the cost of the car for the year 2020. Do you think this is a good prediction?
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
Example #2
The table shows the median selling price of a new
single family home in the Washington DC area. (As
per Trulia.com) Make a scatter plot of the data.
Year
2000
2002
2004
2006
2008
2010
Median Price
$130,000
$200,000
$300,000
$400,000
$410,000
$380,000
Use the graphing calculator to write an equation for
the line of best fit.
Draw the line of best fit on your graph.
Making Predictions
1. Based on your line of best fit, what does the slope and y-intercept represent?
2. Predict the median price of a new single family home in DC for the year 2015.
3. Predict the median price of a new single family home in DC for the year 2020.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
Lesson 8: Making Predictions and Creating Scatter Plots
Practice Problems
Directions: For each of the problem below, complete the following:
• Create a scatter plot.
• Write an equation for the line of best fit. (You may either use the graphing calculator, or draw your
own line of best fit and write an equation.)
• Use the line of best fit to predict the missing value in the table.
• Use the line of best fit to answer the remaining questions.
1. The table below represents the number of dogs in the US from the year 2000 to 2008 (in
millions).(Information taken from Statista.com)
Year
2000
2002
2004
2006
2008
2010
# of Dogs
68
65
74
74
77
?
in Millions
Use the line of best fit to predict the number of dogs
in the US in the year 2020.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
2. The table below represents the number of hours a typical person spends watching TV per year from the
year 2002 to 2010 (Information taken from statista.com).
Year
2002
2004
2006
2008
2010
2012
# of Hours
803
875
971
1053
964
?
spend
watching TV
What does the slope and y-intercept represent in the
equation for the line of best fit?
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
3. The table below represents the total amount of computer/video game sales in the US (in billions of
dollars).(Information taken from Statista.com)
Year
2000
2002
2004
2006
2008
2010
2012
Total amount
5.5
6.9
7.3
7.3
11.7
16.9
?
of
computer/video
game sales in
US
Use the line of best fit to predict number of
computer/video game sales in the year 2015.
14. Analyze the following data on your graphing calculator. Determine whether you will be able to accurately
find an equation for the line of best fit for this data to make predictions. Explain your reasoning.
The table below represents the number of fresh water fish in the US (in millions).
Year
2000
2002
2004
2006
Number of
159
185
139
142
Fresh Water
Fish in US (in
millions)
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
2008
171
Algebra 1
Unit 3: Writing Equations
Lisa has $500 to invest in stock. She is deciding between two different
companies to invest her money in for a long term basis. The stock prices
for each company, for the first 6 months of the year are shown in the table
below.
Month
Company 1
Company 2
January
29
35
February
34
39
March
32
40
April
38
44
May
43
47
June
44
48
•
Write an equation for the line of best fit for both companies.
•
Based on the line of best fit, how much will Company 1’s stock cost at the end of the year?
•
Based on your equation for the line of best fit, which company should Lisa invest her money in if she
were going to invest for 13 months (starting with January as the first month)?
•
Based on your equation for the line of best fit, which company should Lisa invest her money in if she
were going to invest for 2 years (starting with January as the first month)?
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
Lesson 8: Making Predictions and Creating Scatter Plots
Practice Problems – Answer Key
Directions: For each of the problem below, complete the following:
• Create a scatter plot.
• Write an equation for the line of best fit. (You may either use the graphing calculator, or draw your
own line of best fit and write an equation.)
• Use the line of best fit to predict the missing value in the table.
• Use the line of best fit to answer the remaining questions.
1. The table below represents the number of dogs in the US from the year 2000 to 2008 (in
millions).(Information taken from Statista.com)
Year
2000
2002
2004
2006
2008
2010
# of Dogs
68
65
74
74
77
?
in Millions
The line of best for this set of data is:
Y = 1.35x +66.2
Use the line of best fit to predict the number of dogs
in the US in the year 2020.
Y = 1.35x +66.2
Y = 1.35(20) +66.2
Y = 93.2
There will be about 93 million dogs
in the US in the year 2020. (I substituted 20 for x
since 2020 is 20 years later than 2000.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Number of Dogs in the US
Algebra 1
Unit 3: Writing Equations
2. The table below represents the number of hours a typical person spends watching TV per year from the
year 2002 to 2010 (Information taken from statista.com).
Year
2002
2004
2006
2008
2010
2012
# of Hours
803
875
971
1053
964
?
spend
watching TV
The line of best for this set of data is:
Y = 25x +783.2
Y = 25x +783.2
Y = 25(12) +783.2
Y =1083.2
People will spend about 1,083
hours watching TV in the year 2012. (I
substituted 12 for x since 2012 is 12 years later
than 2000 which is where I chose to start my
graph.
What does the slope and y-intercept represent in the
equation for the line of best fit?
The slope, 25, represents how many more hours per
year a person will watch TV. So, each year, a
person’s TV viewing increases by 25 hours.
The y-intercept represents how many hours people
watched TV per year in the year 2000. We don’t
have data to represent this, but we can tell from our
line of best fit, that approximately, 783 hours were
spent watching TV per person in the year 2000.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
3. The table below represents the total amount of computer/video game sales in the US (in billions of
dollars).(Information taken from Statista.com)
Year
2000
2002
2004
2006
2008
2010
2012
Total amount
5.5
6.9
7.3
7.3
11.7
16.9
?
of
computer/video
game sales in
US
The line of best for this set of data is:
Y = 1.02x +4.17
Y = 1.02x+4.17
Y = 1.02(12)+4.17
Y = 16.41
About 16.41 billion in sales
were made on computer/video games in the
year 2012. (I substituted 12 for x since the year
2012 is 12 years later than the year 2000.)
Use the line of best fit to predict number of
computer/video game sales in the year 2015.
Y = 1.02x+4.17
Y = 1.02(15)+4.17
Y =19.47
About 19.47 billion in sales were
made on computer/video games in the year 2015.
(I substituted 15 for x since the year 2015 is 15 years
later than the year 2000.)
14. Analyze the following data on your graphing calculator. Determine whether you will be able to accurately
find an equation for the line of best fit for this data to make predictions. Explain your reasoning.
The table below represents the number of fresh water fish in the US (in millions).
Year
2000
2002
2004
2006
Number of
159
185
139
142
Fresh Water
Fish in US (in
millions)
2008
171
This data does not show a linear pattern. The points are more scattered and a straight line would not
accurately describe the data. Therefore, an equation for a line of best fit would not accurately allow
predictions to be made for this set of data.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Algebra 1
Unit 3: Writing Equations
Lisa has $500 to invest in stock. She is deciding between two different
companies to invest her money in for a long term basis. The stock prices
for each company, for the first 6 months of the year are shown in the table
below.
Month
Company 1
Company 2
•
January
29
35
February
34
39
March
32
40
April
38
44
May
43
47
June
44
48
Write an equation for the line of best fit for both companies. (2 points)
(I used the stat and linear regression functions on the graphing calculator to write an equation for the line
of best fit.)
Company #1: y = 3.09x + 25.87
Company #2: y = 2.65x+32.87
•
Based on the line of best fit, how much will Company 1’s stock cost at the end of the year? (2 points)
Based on the line of best fit, Company1’s stock will cost: $62.95 at the end of the year.
Y=3.09x +25.87
Y = 3.09(12)+25.87
Y = 62.95
•
Company 1’s equation.
The end of the year will be the 12th month. Substitute 12 for x.
Company 1’s stock will cost approximately $62.95 at the end of the
year.
Based on your equation for the line of best fit, which company should Lisa invest her money in if she
were going to invest for 13 months (starting with January as the first month)? (2 points)
Company 1: y = 3.09x +25.87
Company 2: y = 2.65x +32.84
Y = 3.09(13)+25.87
y = 2.65(13)+32.84
Y = 66.04
y = 67.32
Company 2 will cost more per share, so she should invest in Company number 2 if she is going to
invest for 13 months.
•
Based on your equation for the line of best fit, which company should Lisa invest her money in if she
were going to invest for 2 years (starting with January as the first month)? (2 points)
Since our equation is based on months, we need to substitute 24 for x since 2 years is 24 months.
Company 1: y = 3.09x +25.87
Company 2: y = 2.65x +32.84
Y = 3.09(24)+25.87
y = 2.65(24)+32.84
Y = 100.03
y = 96.47
Company 1 will cost more per share, so she should invest in Company number 1 if she is going to
invest for 2 years.
Copyright © 2009-2012 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
Unit 3: Writing Equations Chapter Review
Part 1: Writing Equations in Slope Intercept Form. (Lesson 1)
1. Write an equation that represents the line on the graph.
2. Write an equation that has a slope of -3 and a y- intercept of 10.
3. Write an equation that has a slope of ½ and passes through the origin.
4. A day care center charges $9 per hour plus a $3 snack fee for a “Parents Night Out” Special.
•
•
Write an equation that can be used to determine the price of sending a child to the day care
center for x number of hours.
Suppose a parent decides to send their child to the day care center for 5 hours. How much will
this cost? Justify your answer.
Part 2: Writing Equations in Standard Form (Lesson 2)
5. Rewrite the following equations in standard form.
a. y = -4x – 9
Copyright © 2011 Karin Hutchinson – Algebra-class.com
b. y = 6x + 2
Unit 3: Writing Equations
6. Which term in the equation does not have an “integer coefficient”?
1/2x + 3y = 9
7. Write this equation in “correct” standard form: 1/2x + 3y = 9
8. Write the following equation in standard form: 1/3x + 1/2y = 6
9. Write an equation, in standard form, for the line shown on the graph.
a.
b.
Part 3: Standard Form Word Problems (Lesson 3)
10. A new 3-D movie is out and the theatre is charging $10.50 for adults and $7 for children. You
have $50 to spend on the movies. Write an equation in standard form that represents the number of
adult, a and child, c tickets that you can purchase.
•
Suppose you are taking 3 adults, at most, how many children can you take to the movies?
Justify your answer mathematically.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
Part 4: Writing Equations Given Slope and a Point (Lesson 4)
11. Write an equation for the line that has a slope of -3 and passes through the point (5,-7).
12. A local tax service charges $65.25 an hour plus a filing fee. A three hour session costs $208.40.
Write an equation that can be used to find the total cost for any session.
Part 5: Writing Equations Given Two Points (Lesson 5)
13. Write an equation for the line that passes through the points: (6,-2) (-4, 8)
14. In 1991,the cost of tuition for a private school was $5,000 per year. In 2012, the cost of the same
private school is $12,500. Let x = 0 represent the year 1990.
•
•
Write an equation that could be used to predict the cost of tuition for any given year.
Predict what the tuition will be for the year 2015.
15. In 2003, the cost of season football tickets was $2,200. In 2012, the cost of the same season
tickets is $4,550. Let x = 0 represent the year 2000.
•
•
Write an equation that could be used to predict the cost of the tickets for any given year.
Predict what the cost of the tickets will be in the year 2020.
Part 6: Writing Linear Equations in Point-Slope Form (Lesson 6)
16. Write an equation that has a slope of -3 and passes through the point (2,8)
17. Write an equation that passes through the points (1,9) and (3, -8). Explain the steps that you
used to solve this problem.
18. Write an equation that has a slope of 1/2 and an x-intercept of 1. Explain how you solved this
problem.
19. Write an equation that has a y-intercept of 22 and passes through the point (3, -7).
20. Write an equation that has an x-intercept of -5 and a y-intercept of 8.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
21. Mei earned $38 on her investment in 5 months. She earned $62 on the same investment in 8
months.
•
•
•
Write an equation that can be used to find the amount earned (y) on Mei’s invest in x number
of months.
About how much will Mei earn after 15 months?
About how much is Mei earning per month? Explain how you determined your answer.
Part 7: Parallel and Perpendicular Lines (Lesson 7)
Parallel Lines: ________________________________________________________________________
Perpendicular lines: ___________________________________________________________________
____________________________________________________________________________________
22. Write an equation for a line that passes through (2,-8) and is perpendicular to a line whose slope
is 5. Explain how you determined your answer.
23. Write an equation for a line that passes through (-3,6) and is parallel to the graph of x = 5.
24. Write an equation for a line that is perpendicular to the graph of 2x – 2y = 10 and intersects the
graph at its x-intercept.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
Part 8: Scatter Plots and Line of Best Fit (Lesson 8)
25. The following data represents the total U.S. E-commerce sales from 2002-2010. (Statistic from
statista.com) Let x =0 represent the year 2000.
Year
•
•
•
2002
E-commerce Sales
(in billions of dollars)
72
2004
117
2006
171
2008
214
2010
228
Using your graphing calculator, write an equation for the line of best fit for this data.
What does the slope and y-intercept represent in the context of this problem.?
Predict the amount of E-commerce sales in the U.S. for the year 2012. Explain how you
determined your answer.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
Unit 3: Writing Equations Chapter Review – Answer Key
Part 1: Writing Equations in Slope Intercept Form. (Lesson 1)
1. Write an equation that represents the line on the graph.
In order to write this equation, we must identify the
y-intercept and slope from the graph.
In order to write this equation, we must identify the
y-intercept and slope from the graph.
The line crosses the y-axis at y = -6, so this is the yintercept. From this point, to the next identifiable
point on the graph, we count up1 unit and 2 units to
the left, so the slope is -1/2.
The line crosses the y-axis at y = 2, so this is the yintercept. From this point, to the next identifiable
point on the graph, we count up 3 units and 1 unit to
the right, so the slope is 3 .
Y = mx+ b
Y = mx+ b
m = -1/2
b = -6
Y = -1/2x - 6
m=3
b= 2
Y = 3x + 2
2. Write an equation that has a slope of -3 and a y- intercept of 10.
Y = mx + b is the formula for writing an equation in slope intercept form. We need to know the slope (m) and
the y-intercept (b).
Slope (m) = -3
Y = mx+b
Y = -3x+ 10
Y-intercept (b) = 10
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
3. Write an equation that has a slope of ½ and passes through the origin.
The origin is (0,0). Therefore the y-intercept for this equation is 0.
Y = mx+b
Y = 1/2x + 0
m=½
or
b=0
Y = 1/2x
4. A day care center charges $9 per hour plus a $3 snack fee for a “Parents Night Out” Special.
•
Write an equation that can be used to determine the price of sending a child to the day care
center for x number of hours.
Slope is the rate and the key word is usually “per” So, 9 is the slope ($9 per hour)
The y-intercept is a constant. $3 is the fee for snack. This is a constant in the problem.
Y = mx+b
m=9
b=3
Y = 9x + 3 is the equation that represents this problem.
•
Suppose a parent decides to send their child to the day care center for 5 hours. How much will
this cost? Justify your answer.
Since x represents the number of hours, we will substitute 5 for x in the equation.
Y = 9x+3
Y = 9(5)+3
Y = 48
Sending a child to the day care for 5 hours costs $48.
Part 2: Writing Equations in Standard Form (Lesson 2)
5. Rewrite the following equations in standard form.
a. y = -4x – 9
b. y = 6x + 2
Standard Form: Ax + By = C
Standard Form: Ax + By = C
4x + y = -4x+4x – 9
Add 4x to both sides
-6x+y = 6x-6x + 2
Subtract 6x from both sides
4x+y = -9
Simplify
-6x + y = 2
Simplify
The lead coefficient is positive and there are no
fractions or decimals, so this is standard form.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
-1(-6x+y) =2(-1)
Multiply by -1 to make the
lead coefficient positive.
6x – y = -2
Standard form
Unit 3: Writing Equations
6. Which term in the equation does not have an “integer coefficient”?
1/2x + 3y = 9
An integer is a positive or negative whole number: (…-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5…)
The set of integers does not contain fractions or decimals. Therefore, ½ is not an integer.
The term that does not have an “integer coefficient” is 1/2x.
In order for this equation to be written in standard form, we must rewrite with integer coefficients. We must
get rid of the fraction, ½. In order to do this we will multiply ALL terms by 2.
2(1/2x+3y) = 9(2)
x+ 6y = 18
There are no fractions or decimals, and the lead coefficient is positive. This is now written in
standard form.
8. Write the following equation in standard form: 1/3x + 1/2y = 6
This equation has 2 fractions, with different denominators. Therefore, in order to get rid of both fractions, I
must multiply ALL terms by the lowest common multiple (LCM). The LCM is 6.
6(1/3x+ 1/2y) = 6(6)
2x+ 3y = 36 is the proper standard form equation.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
9. Write an equation, in standard form, for the line shown on the graph.
a.
b.
Since the slope and y-intercept are identifiable, it
would be easiest to first write the equation in slope
intercept form, then rewrite it in standard form.
Y = mx+ b
Y = -2x+5
m = -2
b=5
Equation in slope intercept form
Now rewrite in standard form:
Since the slope and y-intercept are identifiable, it
would be easiest to first write the equation in slope
intercept form, then rewrite it in standard form.
Y = mx+ b
Y = 4/5x - 9
m = 4/5
b = -9
Equation in slope intercept form
Now rewrite in standard form:
2x+y = -2x+2x +5
Add 2x
2x+y = 5
Simplify
2x+y = 5 is the equation in standard form.
5y = 5(4/5x-9)
Multiply by 5
5y = 4x – 45
Simplify
-4x + 5y 4x-4x-45
Subtract 4x
-4x +5y =-45
Simplify
-1(-4x+5y) = -45(-1)
Multiply by -1 to make lead
coefficient positive.
4x-5y = 45 is the equation in standard form.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
Part 3: Standard Form Word Problems (Lesson 3)
10. A new 3-D movie is out and the theatre is charging $10.50 for adults and $7 for children. You
have $50 to spend on the movies. Write an equation in standard form that represents the number of
adult, a and child, c tickets that you can purchase.
Price for adults(# of adults) + Price per child(# of children) = Total
10.50a + 7c = 50
•
is the equation that represents the number of tickets that you can purchase.
Suppose you are taking 3 adults, at most, how many children can you take to the movies?
Justify your answer mathematically.
If you are taking 3 adults, then you will substitute 3 for a in your equation from above. Then solve for c.
10.50a + 7c= 50
Original Equation
10.50(3) + 7c = 50
Substitute 3 for a since there are 3 adults
31.50 + 7c = 50
Simplify: 10.50(3) = 31.50
31.50 – 31.50 + 7c = 50-31.50
Subtract 31.50 from both sides
7c = 18.50
Simplify: 50-31.50 = 18.50
7c/7 = 18.50/7
Divide by 7 on both sides
C= 2.64
Since you can’t take “part” of a child to the movies, you will be able to take 2 children to the
movies if you take 3 adults.
Part 4: Writing Equations Given Slope and a Point (Lesson 4)
11. Write an equation for the line that has a slope of -3 and passes through the point (5,-7).
We know: m = -3 x = 5
y = -7 we need to find b in order to write an equation in slope intercept
form. Let’s substitute what we know and solve for b.
Y = mx=b
-7 = -3(5) + b
-7 = -15 + b
-7 + 15 = -15+15+b
8=b
Substitute for m, x, and y
Simplify: -3(5) = -15
Add 15 to both sides
The y-intercept (b) = 8
Now that we know: m = -3 and b = 8, we can write an equation in slope intercept form:
Y = mx+ b
y = -3x+ 8
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
12. A local tax service charges $65.25 an hour plus a filing fee. A three hour session costs $208.40.
Write an equation that can be used to find the total cost for any session.
In this problem, we know the slope is: 65.25 because this is the rate per hour. We also know that a 3 hour
session is 208.40. This is an ordered pair, because 3 hours is directly related to 208.40 (3, 208.40).
So, we know:
m = 65.25
x=3
y = 208.40
Now we need to solve for b.
Y=mx+b
208.40 = 65.25(3) + b
Substitute for m, x, and y.
208.40 = 195.75+ b
Simplify: 65.25(3) = 1953.75
208.40 – 195.75 = 195.75 – 195.75 + b
Subtract 195.75 from both sides
12.65 = b
The y-intercept is 12.65.
Now, write an equation using the slope and y-intercept.
Y =mx+b
Y = 65.25x + 12.65
This is the equation that could be used to find the total cost for any session.
Part 5: Writing Equations Given Two Points (Lesson 5)
13. Write an equation for the line that passes through the points: (6,-2) (-4, 8)
Since we are given two points, we must find the slope and y-intercept. We will first find the slope using the
slope formula. Then we will use the slope and 1 point to find the y-intercept.
y2- y1 = 8-(-2) = 10
x2 – x1 -4 – 6 -10
The slope is -1
Now we will use the slope and 1 point to find the y-intercept. Let’s use (6, -2)
Y= mx+ b
m = -1
2 = -1(6) + b
2 = -6 + b
2+6 = -6+6 + b
8=b
x=6
y = -2
Substitute for m, x, and y
Simplify: (-1)(6) = -6
Add 6 to both sides
The y-intercept = 8
Y = mx+ b
m = -1
b=8
Y = -x + 8 is the equation that passes through the points (6,-2) & (-4,8)
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
14. In 1991,the cost of tuition for a private school was $5,000 per year. In 2012, the cost of the same
private school is $12,500. Let x = 0 represent the year 1990.
•
Write an equation that could be used to predict the cost of tuition for any given year.
In this problem, we know that in the year 1991 the cost was $5000. Since these two are directly related, this is
an ordered pair. (1991, 5000). But, since x = 0 represents 1990, x = 1 represents 1991 since its one year
later, so our actual ordered pair is (1, 5000)
In the year 2012, the cost was $12,500. Again this is a direct relationship, so it’s an ordered pair.
(2012, 12500). Since = 0 represents 1990, x = 22 represents 2012 since its 22 years after 1990. (22, 12500)
Now that we have two ordered pairs: (1, 5000) & (22, 12500) we can find the slope using the slope formula.
y2- y1 = 12500 – 5000 =
x2 – x1
22-1
7500
21
= 357.14
The slope is 357.14 and 1 point is (1, 5000). Let’s us this to find the y-intercept (b).
Y = mx+ b
5000 = 357.14 (1) + b
Substitute for m, x, and y.
5000 = 357.14 + b
Simplify: 357.14(1) = 357.14
5000-357.14 = 357.14 – 357.14 + b
Subtract 35.14 from both sides
4642.86 = b
The y-intercept = 4642.86
Now use the slope and y intercept to write the equation.
Y = mx+ b
Y = 357.14 + 4642.86
•
m = 357.14
b = 4642.86
Predict what the tuition will be for the year 2015.
Now using our new equation, we can predict what the tuition will be for the year 2015.
Y = 357.14x + 4642.86
Y = 357.14(25) + 4642.86
Substitute 25 for x since 2015 is 25 years beyond 1990.
Y = 13571.36
In the year 2015, the tuition for this school will be about $13571.36
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
15. In 2003, the cost of season football tickets was $2,200. In 2012, the cost of the same season
tickets is $4,550. Let x = 0 represent the year 2000.
•
Write an equation that could be used to predict the cost of the tickets for any given year.
In 2003, the cost of tickets was $2200. This is an ordered pair. Since x = 0 represents 2000, our ordered pair
is (3, 2200). 2003 is three years later than 2000.
In 2012 the tickets were $4550. This is also an ordered pair. (12, 4550)
2012 is 12 years later than 2000.
Now we will use our ordered pairs to find the slope for this equation. (3, 2200) (12, 4550)
y2- y1 = 4550 – 2200 = 2350 = 261.1
x2 – x1
12 – 3
9
The slope is 261.1
Now we will use the slope and 1 point (3, 2200) to find the y-intercept.
Y = mx+b
m = 261.1
x=3
y = 2200
2200 = 261.1(3) + b
2200 = 783.3 + b
2200 -783.3 = 783.3-783.3 +b
Substitute for m, x, and y.
Simplify: 261.1*3 = 783.3
Subtract 783.3 from both sides.
1416.70 = b
The y-intercept is 1416.70
Now that we know the slope and y-intercept, we can write our equation.
Y = mx+ b
Y = 261.1x +1416.70
•
is the equation that can be used to predict the cost of the tickets.
Predict what the cost of the tickets will be in the year 2020.
Y = 261.1x + 1416.70
Equation from above
Y = 261.1(20) + 1416.70
Substitute 20 for x since 2020 is 20 years after 2000.
Y = 6638.70
Approximate cost of tickets.
The tickets will cost approximately $6638.70 in the year 2020.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
Part 6: Writing Linear Equations - Point-Slope Form (Lesson 6)
16. Write an equation that has a slope of -3 and passes through the point (2,8)
Since we are given slope and a point, we can use point-slope to write the equation.
m = -3 ,
x1 = 2
y1 =8
y – y1 = m(x-x1)
y – 8 = -3(x-2)
y – 8 = -3x +6
y – 8 +8 = -3x +6 +8
y = -3x +14
Substitute the values into the equation
Distribute -3
Add 8 to both sides
The equation that has a slope of -3 and passes through (2,8)
17. Write an equation that passes through the points (1,9) and (3, -8). Explain the steps that you
used to solve this problem.
Since we are given two points, we must first start by finding the slope of the line that passes through the two
points. We will use the slope formula to find the slope.
− −8 − 9
−17
=
=
− 3−1
2
Now we can use the point-slope form to write the equation. (*You can use either point for x1 and y1)
m = -17/2,
x1 = 1 y1 =9
y – y1 = m(x-x1)
y – 9 = -17/2(x-1)
Substitute the values into the equation.
y-9 = -17/2x +17/2
Distribute the -17/2 throughout the parenthesis
y-9+9 = -17/2x +17/2 +9
Add 9 to both sides
y = -17/2x+ 35/2
18. Write an equation that has a slope of 1/2 and an x-intercept of 1. Explain how you solved this
problem.
We know m = ½ and an x-intercept of 1 means that we have the point (1,0). Use point slope form.
m = 1/2,
x1 = 1
y – y1 = m(x-x1)
y – 0 = 1/2(x-1)
y = 1/2x – 1/2
y1 =0
Substitute the values into the equation.
19. Write an equation that has a y-intercept of 22 and passes through the point (3, -7).
We know the y-intercept is 22, but we don’t know the slope which is essential. If we write the y intercept as a
point: (0,22) and we use the other point (3, -7) you can use the slope formula to find the slope.
=
M = -29/3
=
b = 22
is the slope.
y = -29/3x +22 is the equation.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
20. Write an equation that has an x-intercept of -5 and a y-intercept of 8.
Since we are given the x and y intercept, we can write and equation in standard form pretty easily.
We know that a common multiple of 5 and 8 is 40, so that will be our constant.
Since we know (-5,0) & (0,8) are intercepts, we need to write an equation that works:
A(-5) + B(8) = 40 In order for the x and y intercepts to work, A = -8 and B = 5
-8x + 5y = 40.
8x – 5y = -40 if you want to make the lead coefficient work.
Or
Justify: 8x – 5y = -40
substitute (-5,0) and (0,8) and make sure they work.
8(-5) – 5(0) = -40
8(0) – 5(8) = -40
-40 = -40
-40=-40
It works!
*You can also write an equation in slope intercept form. If you took this route, your equation would be:
Y = 8/5x +8
21. Mei earned $38 on her investment in 5 months. She earned $62 on the same investment in 8
months.
•
Write an equation that can be used to find the amount earned (y) on Mei’s invest in x number
of months.
We are given two points in this problem: (5, 38) (8, 62). We can find the slope and then use point-slope
form to write the equation.
− 62 − 38
24
=
=
=8
− 8−5
3
Now we can use the point-slope form to write the equation. (*You can use either point for x1 and y1)
m = 8,
x1 = 5
y1 =38
y – y1 = m(x-x1)
y – 38 = 8(x-5)
Substitute the slope and point coordinates into the equation.
y – 38 = 8x – 40
Distribute 8 throughout the parenthesis.
Y – 38+38 = 8x -40+38
Add 38 to both sides of the equation.
Y = 8x -2
•
About how much will Mei earn after 15 months?
After 15 months, Mei will have earned $118.
Y = 8x – 2
Y = 8(15) – 2
Y = 118
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
•
About how much is Mei earning per month? Explain how you determined your answer.
Mei is earning about $8 per month on her investment.
Part 7: Parallel and Perpendicular Lines (Lesson 7)
Parallel Lines: Have the same slope and different y-intercepts.
Perpendicular lines: The slopes are the negative reciprocal of each other.
22. Write an equation for a line that passes through (2,-8) and is perpendicular to a line whose slope
is 5. Explain how you determined your answer.
Since a line that is perpendicular will have a slope that is the negative reciprocal, that means that the
slope of this line will be -1/5. Now we can use point slope form to write the equation of the line.
m = -1/5,
x1 = 2
y1 =-8
y – y1 = m(x-x1)
y – -8 = -1/5(x-2)
y+8 = -1/5x +2/5
y + 8 – 8 = -1/5x +2/5 – 8
y = -1/5x – 38/5
Substitute values into the equation.
Distribute -1/5
Subtract 8 from both sides of the equation.
is the line perpendicular.
23. Write an equation for a line that passes through (-3,6) and is parallel to the graph of x = 5.
X = 5 is a vertical line. Therefore, a line that is parallel is also going to be a vertical line. If it passes
through (-3,6), then it must be a vertical line through (-3,6) which means that the equation is: x = -3
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
24. Write an equation for a line that is perpendicular to the graph of 2x – 2y = 10 and intersects the
graph at its x-intercept.
We must first determine the slope of 2x-2y = 10 by rewriting it in slope intercept form.
2x – 2x – 2y = -2x +10
Subtract 2x from both sides
-2y = -2x +10
-2y/-2 = -2x/-2 +10/-2
Divide all terms by -2
Y = x -5
The slope of this line is 1
Now we can use point-slope form. Slope will be -1 for the perpendicular line and passes through the xintercept which is 5 or (5,0).
2x = 10
X=5
m = -1
x1 = 5
y – y1 = m(x-x1)
y – 0 = -1(x-5)
Y = -x +5
y1 =-0
Substitute the given values.
Is the equation for a line perpendicular to 2x-27 = 10 and passes
through its x-intercept.
Part 8: Scatter Plots and Line of Best Fit
(Lesson 8)
25. The following data represents the total U.S. E-commerce sales from 2002-2010. (Statistic from
statista.com) Let x =0 represent the year 2000.
Year
•
2002
E-commerce Sales
(in billions of dollars)
72
2004
117
2006
171
2008
214
2010
228
Using your graphing calculator, write an equation for the line of best fit for this data.
Line of best fit is: y = 20.45x +37.7
•
(all numbers in billions of dollars)
What does the slope and y-intercept represent in the context of this problem.?
In this problem, the slope represents the amount that the E-commerce sales increase per year.
The E-commerce sales increase by 20.45 billion dollars each year.
The y-intercept represents the amount of E-commerce sales in the year 2000 because when x = 0
this represents the year 2000. Therefore, the estimated amount of E-commerce sales in 2000 was
$37.7 billion dollars.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
•
Predict the amount of E-commerce sales in the U.S. for the year 2012. Explain how you
determined your answer.
In order to predict the amount for the year 2012, we must substitute 12 into our line of best fit equation.
y = 20.45x +37.7
y = 20.45(12) +37.7
y = 283.1 billion
E-commerce sales in 2012 will be about $283.1 billion dollars.
Copyright © 2011 Karin Hutchinson – Algebra-class.com
Unit 3: Writing Equations
Writing Equations Chapter Test
1.
A.
B.
C.
D.
Which equation represents the line on the graph?
y = 1/2x -1
y = -2x -1
y = 2x +1
y = -1/2x – 1
2. Write the following equation in standard form: y = -2/3x + 1/3
A.
B.
C.
D.
2x +3y = 1
-2x +3y = 1
2/3x +y = 1/3
2x +y = 1
3. The equation of the line that passes through the points (-1,4) & (3, 8) is:
A. y = x + 3
B. y = x + 5
C. y = -x +5
D. y = 3x +5
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
4. Determine the equation of the line that is perpendicular to y = 4x – 6
A. y = -4x +6
B. y = 4x+ 6
C. 1/4x - 6
D. y = -1/4x +1
5. A moving company charges a fee of $300 plus $25 per hour to provide services for
moving in or out of your home. Which equation could you use to find the cost of moving
out of your home. Let x represent the number of hours and y represent the total cost.
A.
B.
C.
D.
25x +y = 300
300 x +y = 25
y = 300x +25
y = 25x +300
6. Write the equation of a line that has a slope of -5 and passes through the point (2,1).
7. Write the equation that represents the line on the graph.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
8. Which equation represents the line on the graph?
A.
B.
C.
D.
x +2y = 8
1/2x +y =4
x +2y = 4
-x +2y = 8
9. In the following equation, which term does not have an integer coefficient?
9x – 3/4y = 3
10. Write the equation in standard form with integer coefficients.
y = 2/3x +3/4
11.
A.
B.
C.
D.
Which equation has a slope of 4 and passes through the origin?
y=4
y = x +4
y = 4x
4x +y = 0
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
12. Your family has $40 to spend at the movies. An adult ticket costs $4 and a youth ticket
costs, $3. Write an equation in standard form that represents the total number of adult
tickets, x and youth tickets, y, that you can purchase.
13. What is the slope of the line that passes through the points (10,8) (4,7)?
14. Write an equation that is parallel to the graph of 6x + 3y = 6 and passes through the point
(-1,-6).
15. An average 170 pound male burns an average of 10.2 calories per minute when jogging.
He burns an average of 23 calories per minute when sprinting. Let x represent the number of
minutes spent jogging and y represent the number of minutes spent sprinting.
• Write an equation that you could use to find the times that you would need to
sprint and jog in order to burn 800 calories.
• If you plan on jogging for an hour and 10 minutes, how long will you have to sprint
in order to burn 800 calories? Justify your answer mathematically.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
16. You are having a birthday party at McDonalds. McDonald’s charges $25 to rent
their private room plus $5 per child.
• Write an equation that you could use to determine the price of having a birthday
party at McDonald’s.
• Suppose you have 15 children coming to the birthday party. How much will the
birthday party cost? Justify your answer.
17. In 1998, the cost of XYZ’s college tuition was $15000 a year. In 2009, the cost of
XYZ’s college tuition is $22000 a year. Let x = 0 represent 1990.
• Write an equation that could be used to predict the college tuition for any given
year.
• Predict what XYZ’s college tuition with be for the year 2012.
18. The following table displays the data for the U.S. digital music revenue for the years
2006 – 2011. (4 points)
Year
2006
Amount of
Revenue (in
billions of dollars)
1.9
2007
2.8
2008
3.7
2009
4.5
2010
5.2
2011
5.7
Copyright© 2009 Algebra-class.com
•
Write an equation for the line of best that fit that
represents this data. Let x = 0 represent the year
2006 because this is the first year that data was
collected on digital music revenue through this
company.
•
Use the equation for the line of best fit to predict how
much revenue will be produced by digital music in
the year 2014.
What does the slope represent in this equation?
•
Unit 3: Writing Equations
Writing Equations Chapter Test – Answer Key
(32 points Total)
1. Which equation represents the line on the graph? (1 point)
A. y = 1/2x -1
B. y = -2x -1
C. y = 2x +1
D. y = -1/2x – 1
Slope (m) = -1/2
Y-intercept (b) = -1
y = -1/2x -1
2. Write the following equation in standard form: y = -2/3x + 1/3
A.
B.
C.
D.
2x +3y = 1
-2x +3y = 1
2/3x +y = 1/3
2x +y = 1
y= -2/3x +1/3
3[y= -2/3x +1/3]
Multiply by 3 to get rid of the fractions.
3y = -2x +1
2x +3y = -2x +2x +1
2x +3y = 1
Copyright© 2009 Algebra-class.com
(1 point)
Add 2x to both sides.
Unit 3: Writing Equations
3. The equation of the line that passes through the points (-1,4) & (3, 8) is:
A. y = x +3
B. y = x+5
C. y = -x +5
D. y = 3x +5
(1 point)
Find the slope:
Y2 – y1 = 8 – 4 = 4 = 1
X2 – x1 3 – (-1)
4
Find the y-intercept: (I’m going to pick (3,8)
y = mx +b
8 = 1(3) +b
y = x +5
8 = 3 +b
8 – 3 = 3-3 +b b = 5
4. Determine the equation of the line that is perpendicular to y = 4x – 6 (1 point)
A. y = -4x +6
B. y = 4x+ 6
C. 1/4x - 6
D. y = -1/4x +1
A line that is perpendicular to this line will have a slope that is -1/4. The slope of a perpendicular line is
the negative reciprocal. Since the slope of this line is 4, a perpendicular line’s slope is -1/4.
Therefore, the only answer is D, since its slope is -1/4.
5. A moving company charges a fee of $300 plus $25 per hour to provide services for moving
in or out of your home. Which equation could you use to find the cost of moving out of your
home. Let x represent the number of hours and y represent the total cost.
(1 point)
A.
B.
C.
D.
25x +y = 300
300 x +y = 25
y = 300x +25
y = 25x +300
Copyright© 2009 Algebra-class.com
$25 per hour indicates slope
m = 25
$300 is a fee so it is the constant
or y-intercept.
b = 300
y = mx +b
y = 25x + 300
Unit 3: Writing Equations
6. Write the equation of a line that has a slope of -5 and passes through the point (2,1).
(2 points)
Slope (m) = -5
b = 11
Find the y-intercept:
y = mx +b
1 = -5(2) +b
1 = -10 +b
1 +10 = -10 +10 +b
11 = b
x=2
y=1
y = -5x +11
The equation is y = -5x +11.
7. Write the equation that represents the line on the graph.
(2 points)
You need the slope and the y-intercept which you can
find by looking at the graph:
Slope (m) = 1/3
y-intercept (b) = -3
y = mx +b
y = 1/3x -3
The equation that represents the line on the graph is:
y = 1/3x -3
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
8. Which equation represents the line on the graph? (1 point)
A. x +2y = 8
B. 1/2x +y =4
C. x +2y = 4
D. -x +2y = 8
Write the equation in slope intercept form:
Slope (m) = -1/2
Y-intercept (b) = 4
y = -1/2x +4
Rewrite in standard form since the answers are
shown in standard form.
2[y = -1/2x +4]
2y = -x +8
x +2y = -x +x +8
x +2y = 8
Multiply by 2 to get rid of
fractions
Add x to both sides
9. In the following equation, which term does not have an integer coefficient? (1 point)
9x – 3/4y = 3
Your terms are: 9x
9 is an integer
-3/4y -3/4 is not an integer, it is a fraction
3
3 is an integer
Therefore, -3/4y is the term that does not have an integer coefficient.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
10. Write the equation in standard form with integer coefficients.
(2 points)
y = 2/3x +3/4
12[y = 2/3x +3/4]
Multiply by 12 (LCD) in order to get rid of the fractions.
12y = 8x + 9
-8x +12y = 8x -8x +9 Subtract 8x from both sides
-8x +12y = 9
-1[-8x +12y = 9]
Multiply by -1 to make the lead coefficient positive.
8x – 12y = -9
The equation in standard form is: 8x – 12y = -9
11. Which equation has a slope of 4 and passes through the origin? (1 point)
A.
B.
C.
D.
y=4
y = x +4
y = 4x
4x +y = 0
Slope (m) = 4
Origin (0,0) Y-intercept = 0
Since the line passes through the origin, which is (0,0), the y
– intercept is 0.
Y = 4x +0 OR
y = 4x
12. Your family has $40 to spend at the movies. An adult ticket costs $4 and a youth ticket
costs, $3. Write an equation in standard form that represents the total number of adult
tickets, x and youth tickets, y, that you can purchase. (2 points)
You know that your family has a total of $40 to spend. Since you know the total amount, you can most likely write
your equation in standard form. Make sure that you have 2 quantities that you can add together to arrive at this
total. You can add adult tickets and youth tickets to get the total. So we are going to write an equation in standard
form.
Price of adult tickets • # of adult tickets + price of youth tickets • # of youth tickets = Total price
$4
•
x
+
$3
•
y
= $40
4x + 3y = 40 is the equation that represents the number of adult tickets and youth tickets that can be purchased for
$40.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
13. What is the slope of the line that passes through the points (10,8) (4,7)? (1 point)
A. -1/6
y2- y1 = 7 – 8 = -1 = 1/6
x2 – x1 4 – 10
-6
B. 1/6
C. 6
D. -6
14. Write an equation that is parallel to the graph of 6x + 3y = 6 and passes through the point
(-1,-6). ( 2 points)
An equation that is parallel to this equation will have the same slope. Therefore, we need to find the
slope of this equation by rewriting it in slope intercept form.
6x – 6x +3y = -6x +6
3y = -6x +6
3y/3 = -6x/3 +6/3
Y = -2x +2
Now we know the slope is -2 and we have a coordinate, so we can use point-slope form.
y – y1 = m(x-x1)
m = -2 x = -1 y = -6
y – (-6) = -2(x – -1)
Substitute into the equation.
y+6 = -2x – 2
Distribute -2 throughout the parenthesis
y+6 – 6 = -2x – 2 -6
Subtract 6 from both sides
y = -2x – 8
Is the equation that is parallel and passes through (-1, -6)
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
15. An average 170 pound male burns an average of 10.2 calories per minute when jogging.
He burns an average of 23 calories per minute when sprinting. Let x represent the number of
minutes spent jogging and y represent the number of minutes spent sprinting.
• Write an equation that you could use to find the times that you would need to
sprint and jog in order to burn 800 calories.
• If he plans on jogging for an hour and 10 minutes, how long would he have to sprint
in order to burn 800 calories? Justify your answer mathematically.
(3 points)
This problem may seem confusing at first because you are given extra information. For
example, even though 170 is a number that you might think you would need, it is really
irrelevant and is just there to set the problem up as a real world example. (You burn
different amounts of calories based on your weight).
So, we need to see what we are given: 10.2 calories/min when jogging.
23 calories/min when sprinting.
X represents the number of min jogging
Y represents the number of min. sprinting
Burn a total of 800 calories.
Other confusing statements are: 10.2 calories per minute and 23 calories per minute. Since
you see the word “per” you think of rate or slope. Although they are technically rates – we
need to add them together to get our total of 800 calories. So, since we have a designated
total and we can add to arrive at that total, we are going to write an equation in standard
form.
Jogging
+
Sprinting
= Total Calories
Cal/min • # of min
+ Cal/min • # of min
= Total Calories
10.2
+
• x
23 •
y
= 800
• 10.2x +23y = 800 is the equation that you could use to find the times that you would
need to sprint and jog in order to burn 800 calories.
In part 2, you are given: you plan on jogging for an hour and 10 minutes – find how
long you will have to sprint in order to burn 800 calories.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
Since x represents the number minutes jogging, we can substitute for x, but first we
need to rewrite an hour and 10 minutes as the total number of minutes.
An hour is 60 minutes, so 60 +10 = 70 minutes.
Let’s substitute 70 for x:
10.2x +23y = 800
10.2(70) +23y = 800
Substitute 70 for x.
714 +23y = 800
Simplify: 10.2(70) = 714
714 – 714 +23y = 800 – 714
Solve for y: Subtract 714 on both sides
23y = 86
Simplify: 800-714 = 86
23y = 86
23 23
Divide by 23 on both sides
y = 3.74
Simplify: 86/23 = 3.74
• If he plans on jogging for an hour and 10 minutes, then he would need to sprint for 3
minutes and 45 seconds in order to burn 800 calories.
.74 of a minute = .74 • 60 seconds = 44.4 - I rounded up to 45 seconds because if he
only sprints for 44 seconds, he will not burn exactly 800 calories, he will be a tad short!
Justify:
10.2x +23y = 800
10.2(70) +23(3.74) = 800
800.02 = 800
Copyright© 2009 Algebra-class.com
Since we had to round our answer, this is as close as we can
get without being less than 800!
Unit 3: Writing Equations
16. You are having a birthday party at McDonalds. McDonald’s charges $25 to rent their
private room plus $5 per child.
• Write an equation that you could use to determine the price of having a birthday
party at McDonald’s.
• Suppose you have 15 children coming to the birthday party. How much will the
birthday party cost? Justify your answer. (3 points)
Let’s see what we are given:
$25 to rent a room is a constant figure. It is not a total amount, just a constant for renting
the room. Therefore, it is the y-intercept(b).
$5 per child is a rate. The key word per indicates that it is a rate or slope (m).
Slope
Y-intercept
5
25
Point 1
Point 2
Define your variables:
Let x = the number of children
Let y = total cost
Since we have the slope and y-intercept we can write the equation:
y = mx +b
y = 5x + 25
• The equation that we could use to determine the price of having a birthday party at
McDonald’s is: y = 5x +25
• If we have 15 children coming to the birthday party, the birthday party will cost
$100.
Justify: y = 5x +25
Y = 5(15) +25
Substitute 15 for x (the number of children)
100 = 100
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
17. In 1998, the cost of XYZ’s college tuition was $15000 a year. In 2009, the cost of XYZ’s
college tuition is $22000 a year. Let x = 0 represent 1990.
• Write an equation that could be used to predict the college tuition for any given
year.
• Predict what XYZ’s college tuition with be for the year 2012. (3 points)
Let’s see what we are given:
I don’t see any numbers that represent a rate. Therefore, I probably don’t know the slope.
Let’s look for ordered pairs:
The year 1998 and cost $9000 are related; therefore, this will be an ordered pair.
The year 2009 and $20000 are related; therefore, this will be an ordered pair.
Let’s fill in our chart:
Slope
Y-intercept
636.36
Point 1
Point 2
(8,15000)
(19,22000)
Given: Let x = 0 represent 1990
1998 is 8 years after 1990. That’s why I used 8 as the x coordinate.
2009 is 19 years after 1990. That’s why I used 19 as my second x coordinate.
Since I have 2 points and no other information, I will need to find the slope (using the slope
formula) and then use the slope and a point to find the y-intercept. (8,15000) (19, 22000)
y2- y1 = 22000 – 15000 = 7000 = 636.36
x2 – x1
19-8
11
The slope is 636.36, so the tuition rises $636.36 per year.
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
Now we can use point slope form to write the equation.
y –y1 = m(x – x1)
Let’s use the point: (8, 15000) and m = 636.36
y – 15000 = 636.36(x – 8)
y - 15000 = 636.36x – 5090.88
y – 15000 +15000 = 636.36x – 5090.88 +15000
Substitute for y1, x1, and m.
Distribute 636.36 throughout the parenthesis
Add 15000 to each side
y = 636.36x+9909.12
• The equation that can be used to predict the college tuition for any given year is:
y = 636.36x+9909.12
• College tuition for the year 2012 is predicted to be about $23909.04
Justify: y = 636.36x+9909.12
Y = 636.369(22) +9909.12
Substitute 22 since 2012 is 22 years after 1990.
23909.04 = 23909.04
Copyright© 2009 Algebra-class.com
Unit 3: Writing Equations
18. The following table displays the data for the U.S. digital music revenue for the years 2006
– 2011. (4 points)
Year
2006
Amount of
Revenue (in
billions of dollars)
1.9
2007
2.8
2008
3.7
2009
4.5
2010
5.2
2011
5.7
•
•
•
Write an equation for the line of best that fit that
represents this data. Let x = 0 represent the year
2006 because this is the first year that data was
collected on digital music revenue through this
company.
Use the equation for the line of best fit to predict how
much revenue will be produced by digital music in
the year 2014.
What does the slope represent in this equation?
Using the graphing calculator, I found the equation for the line of best fit to be: Y = .77x +2.0
The table under stat would look like this:
L1
0
1
2
3
4
5
L2
1.9
2.8
3.7
4.5
5.2
5.7
Using this equation, the revenue produced in the year 2014 will be approximately, $8.16 billion.
Y = .77x +2
Y = .77(8) +2
Substitute 8 for x because 2014 is 8 years after 2006.
Y = 8.16
In this equation, the slope represents the amount that the digital music revenue increases per year.
The slope is .77 billion which means that the digital music revenue increases by 770 million dollars
per year.
This test is worth a total of 32 points! How did you do?
Copyright© 2009 Algebra-class.com
Cumulative Test: Units 1-3
Algebra Class
Cumulative Test: Units 1-3
Solving Equations, Graphing
Equations & Writing Equations
Cumulative Test: Units 1-3
Algebra 1 Exam – Answer Sheet
Part 1: Multiple Choice - Questions are worth 1 point each.
.
Multiple Choice - Total Correct:
___________________ (out of 15 points total)
Cumulative Test: Units 1-3
Algebra 1 Answer Sheet - Continued
Part 2: Fill in the blank. Answers are worth 2 points each.
16.
17.
18.
Cumulative Test: Units 1-3
19.
20.
21.
Fill In the Blank - Total Correct:
___________________ (out of 12 points total)
Cumulative Test: Units 1-3
Part 3: Short Answer - 3 points each
22.
________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
23.
________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
Cumulative Test: Units 1-3
24.
_______________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
________
Short Answer - Total Correct:
___________________ (out of 9 points total)
Cumulative Test- Total Correct: (Add the total points for all 3 sections)
___________________ (out of 36 points total)
33-36 – A
22-25 - D
29-32 – B
21 and below - E
26-28 -C
Cumulative Test: Units 1-3
Algebra Class Part 1 – Cumulative Test
Solving Equations, Graphing Equations, & Writing Equations
Part 1: Multiple Choice. Choose the best answer for each problem. (1 point each)
1. Which equation is represented on the graph?
A. y = -3/4x + 6
B. y = 4/3x + 7
C. y = -4/3x + 6
D. y = 4/3x + 6
2. Which line on the graph has a slope of ½?
A. Line A
B. Line B
C. Line C
D. Line D
Cumulative Test: Units 1-3
3. Which of the following steps would you use first to solve the following equation?
3(x-7) + 4 = 20
A. Subtract 4 from both sides.
B. Add 4 to both sides.
C. Divide by 3 on both sides.
D. Distribute the 3 throughout the parenthesis.
4. Solve for x:
½(2x-4) + 5 = -1
A. x = -4
B. x = -3/2
C. x = -7/2
D. x = 4
5. Write the following equation in standard form:
A. x – 4y = -3
B. x – 4y = 3
C. ¼x – y = ¾
D. –x + 4y = -3
y = ¼x – ¾
Cumulative Test: Units 1-3
6. Find the slope of the line that passes through the points (5,2) & (-9, 10)
A. 2
B. -2
C. -4/7
D. 4/7
7. Which equation represents the line that passes through the points (-6, 4) & (5,4)
A. y = 4x
B. y = 4
C. y = x + 4
D. y = 11x + 4
8. Which description best represents the graph for the equation:
y = -9
A. A horizontal line through the point (0, -9)
B. A vertical line through the point (0, -9)
C. A vertical line through the point (-9, 0)
D. A line with a rise of -9 and a run of 1 that passes through the origin.
Cumulative Test: Units 1-3
9. Solve for y:
6y + 4 = 4(y-2) + 16
A. y = 10
B. y = -2
C. y = -10
D. y = 2
10. The relation between the sides of a rectangle are shown below. The perimeter of the rectangle
is 32 cm. What is the length of the longest side of the rectangle.
x+2
2x + 2
A. 4cm
B. 6 cm
C. 10 cm
D. 20 cm
11. What is the x intercept for the equation:
A. x-intercept = -5 (-5,0)
B. x-intercept = 5 (5,0)
C. x-intercept = -10
(-10,0)
D. x-intercept = 10 (10,0)
4x – 8y = -40
Cumulative Test: Units 1-3
12. Which equation is equivalent to:
4x + 3y = 6
A. y = -4/3x + 2
B. y = 4/3x + 2
C. y = -4x + 6
D. y = -3/4x + 6
13. A landscaping company charges $7.50 per yard of mulch plus a $15 delivery fee. Which
equation could you use to find the cost of having a yard of mulch delivered?
A. 7.50x + y = 15
B. y = 7.50x + 15
C. 15x + y = 7.50
D. y = 15x + 7.50
14. Theresa is selling candy bars for $1.50 a piece and candles for $5 apiece. She has made a
total of $145.00 in sales. Which equation could be used to determine the amount of candy bars and
candles sold? Let x represent the number of candy bars and y represent the number of candles.
A. y = 1.5x + 5
B.
x + y = 145
C. 5x + 1.50y = 145
D. 1.50x + 5y = 145
15. Adam found a job that offered him an average pay raise of $1500 per year. After 8 years on the
job, Adam’s salary was $72000. What was the starting salary that Adam was offered when he took
the job?
A. $12,000
B. $60,000
C $70,500
D. $65,000
Cumulative Test: Units 1-3
Part 2: Fill in the blank. Solve each problem on your answer sheet. Show all of your work.
(2 points each)
16. Solve for x:
¼x – 8 = ⅔(x – 19.5)
17. Graph the following equation on the grid:
8x – 2y = -16
18. Graph the following equation on the grid:
y = -4/5x + 8
19. In the year 2003, the average cost of a trip to Disney World for a family of four was $2300. In
2010, the average cost of a trip to Disney for a family of four is $4,200. Write an equation that can
be used to predict the cost of a trip to Disney for any year after 2000. Let x = 0 represent the year
2000.
(Round all decimals to the hundredths place)
20. The ticket prices for attending a Yankees baseball game increase by $2.75 per year. In the
year 2009, the ticket price for a premium terrace seat was $80. Write an equation that could be
used to determine the price of a premium terrace seat for any year after 2000. Let x = 0 represent
the year 2000.
21. A local banquet room charges $35 an hour for use of facilities plus a $30 clean up fee. How
many hours can Joseph rent the hall for $240?
Cumulative Test: Units 1-3
Part 3: Short answer. Respond to each problem on your answer sheet. Make sure you
answer all parts of each problem. (3 points each)
22. Laurie knits sweaters for dogs, babies, and children. She sells them at craft shows. She sold 3
times as many baby sweaters than dog sweaters. She sold 5 more children sweaters than dog
sweaters. The prices for each sweater are shown below:
Dog - $7.50
Baby - $10.25
Children - $ 14.75
•
Write an expression to represent the number of baby sweaters sold and an expression to
represent the number of children’s sweaters sold. Let x represent the number of dog
sweaters sold.
•
The total sales for Laurie’s sweaters was $391.75. Write an equation to represent the total
sales of Laurie’s sweaters.
•
How many baby sweaters did Laurie sell? Explain how you determined your answer.
23. The cost of tuition at a private school in the year 2002 was $12,100. In the year 2009 the cost
was $16,900. Let x = 0 represent the year 2000.
•
•
Write an equation that could be used to predict the tuition for any year after 2000.
Predict the tuition for the year 2015.
Cumulative Test: Units 1-3
24. Carl has been tracking the price of round trip airfare between Baltimore and Orlando for 10
weeks. The results are show in the graph below.
•
•
What is the y-intercept in this problem? What does it mean in the context of this problem?
What is the rate of change between weeks 4 and 6?
Unit 3: Writing Equations
Algebra Class Part 1 – Cumulative Test – Answer Key
Solving Equations, Graphing Equations, & Writing Equations
Part 1: Multiple Choice. Choose the best answer for each problem. (1 point each)
1. Which equation is represented on the graph?
The line is falling from left to right,
so you know that the slope is
negative. Therefore, you can
eliminate answers B & D since they
have positive slopes.
A. y = -3/4x + 6
B. y = 4/3x + 7
C. y = -4/3x + 6
D. y = 4/3x + 6
The slope of this line is -4/3 (count
down 4 and right 3). The yintercept is 6. The correct answer
choice is C.
2. Which line on the graph has a slope of ½?
A. Line A
B. Line B
C. Line C
D. Line D
Line A is the only line that has a rise
of 1 and a run of 2. Count up 1
and right 2.
You can eliminate answer choices
B and C since they are falling from
left to right. This means they have
a negative slope.
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Y-intercept = 6
Unit 3: Writing Equations
3. Which of the following steps would you use first to solve the following equation?
3(x-7) + 4 = 20
A. Subtract 4 from both sides.
If the distributive property is present, you must
distribute first to remove the parenthesis.
B. Add 4 to both sides.
C. Divide by 3 on both sides.
D. Distribute the 3 throughout the parenthesis.
4. Solve for x:
½(2x-4) + 5 = -1
A. x = -4
B. x = -3/2
C. x = -7/2
D. x = 4
2[½(2x-4) + 5] = -1(2)
Multiply by 2 to get rid of
the fraction.
1(2x-4) + 10 = -2
2x -4 + 10 = -2
2x + 6 = -2
2x + 6 – 6 = -2 – 6
Result after multiplying by 2
2x = -8
Simplify: (-2 -6= -8)
2x/2 = -8/2
Divide by 2 on both sides.
Combine like terms (-4 + 10)
Subtract 6 from both sides.
x = -4
5. Write the following equation in standard form:
A. x – 4y = -3
B. x – 4y = 3
C. ¼x – y = ¾
D. –x + 4y = -3
(4)y
= 4[¼x – ¾]
y = ¼x – ¾
Multiply by 4 to remove the fraction.
4y = x – 3
Result after multiplying by 4
-x + 4y = x –x – 3
Subtract x on both sides.
-x + 4y = -3
Simplify
-1[-x +4y] = -3(-1)
Multiply by -1 to make the lead coefficient
positive.
x – 4y = 3
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Unit 3: Writing Equations
6. Find the slope of the line that passes through the points (5,2) & (-9, 10)
Use the formula to find the slope of two points:
A. 2
B. -2
C. -4/7
y2- y1 = 10 – 2 = 8 = 4
x2 – x1 -9 – 5 -14 -7
Simplify to lowest terms
D. 4/7
The slope of the line is -4/7
7. Which equation represents the line that passes through the points (-6, 4) & (5,4)
Step 1: First find the slope of the line by using the formula:
A. y = 4x
B. y = 4
y2- y1 = 4 – 4 = 0
x2 – x1 5 –(-6) 11
Slope = 0
C. y = x + 4
Step 2: Find the y-intercept using the slope, and 1 point.
D. y = 11x + 4
y = mx + b
**You may also have realized that both
points given had a y-coordinate of 4.
Therefore this is a horizontal line with a yintercept of 4.
4 = 0(5) + b
4=b
Y = 0x + 4 should be written as: y = 4
8. Which description best represents the graph for the equation:
A. A horizontal line through the point (0, -9)
B. A vertical line through the point (0, -9)
y = -9
All points have a y-coordinate of 9. Therefore, the
result is a horizontal line through the point (0,-9).
**A vertical line would start with x =
C. A vertical line through the point (-9, 0)
D. A line with a rise of -9 and a run of 1 that passes through the origin.
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Unit 3: Writing Equations
9. Solve for y:
6y + 4 = 4(y-2) + 16
6y + 4 = 4y – 8 + 16
Distribute 4 throughout the parenthesis
A. y = 10
6y + 4 = 4y + 8
Combine like terms (-8+16 = 8)
B. y = -2
6y – 4y + 4 = 4y – 4y + 8
Subtract 4y from both sides.
C. y = -10
2y +4 = 8
Combine like terms (6y – 4y = 2y)
D. y = 2
2y + 4 -4 = 8- 4
Subtract 4 from both sides
2y = 4
Simplify: (8 – 4 = 4)
2y/2 = 4/2
Divide by 2 on both sides.
y=2
10. The relation between the sides of a rectangle is shown below. The perimeter of the rectangle is
32 cm. What is the length of the longest side of the rectangle.
x+2
2x + 2
A. 4cm
B. 6 cm
C. 10 cm
D. 20 cm
P = 2L + 2w
The perimeter formula
32 = 2(2x+2) + 2(x+2)
32 = 4x + 4 + 2x + 4
Substitute for P, L & W.
Distribute: (2 sets that involve the distributive
property.)
Combine like terms.
Subtract 8 form both sides.
Simplify: 32-8 = 24
Divide by 6 on both sides
32 = 6x + 8
32 – 8 = 6x + 8 -8
24 = 6x
24/6 = 6x/6
4=x
If x = 4, then the longest side is 2x+2
2(4) +2 = 10 cm
11. What is the x intercept for the equation:
4x – 8y = -40
To find the x-intercept, let y = 0
A. x-intercept = -5 (-5,0)
4x – 8(0) = -40
B. x-intercept = 5 (5,0)
4x = -40
C. x-intercept = -10
4x/4 = -40/4
(-10,0)
D. x-intercept = 10 (10,0)
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Substitute 0 for y.
Divide by 4 on both sides.
x = -10
The x-intercept is -10 or (-10,0)
Unit 3: Writing Equations
12. Which equation is equivalent to:
4x + 3y = 6
Rewrite the equation in slope intercept form:
A. y = -4/3x + 2
4x -4x + 3y = -4x + 6
B. y = 4/3x + 2
3y = -4x + 6
Subtract 4x from both sides.
3y/3 = -4x/3 + 6/3
C. y = -4x + 6
Divide all terms by 3.
y = -4/3x + 2
D. y = -3/4x + 6
13. A landscaping company charges $7.50 per yard of mulch plus a $15 delivery fee. Which
equation could you use to find the cost of having a yard of mulch delivered?
A. 7.50x + y = 15
7.50 per yard is the rate or slope in the problem. “Per” is your keyword for slope.
B. y = 7.50x + 15
15 is a delivery or set fee. This is the y-intercept.
C. 15x + y = 7.50
Since you know the slope and y-intercept, this equation can be written in slope
intercept form:
D. y = 15x + 7.50
Y = mx + b
m = 7.50
b = 15
Y = 7.50x + 15
14. Theresa is selling candy bars for $1.50 a piece and candles for $5 apiece. She has made a
total of $145.00 in sales. Which equation could be used to determine the amount of candy bars and
candles sold? Let x represent the number of candy bars and y represent the number of candles.
A. y = 1.5x + 5
B.
Since we know the total (145) and we can add the sales of candy bars + candles to
get this total, we can write the equation in standard form.
x + y = 145
C. 5x + 1.50y = 145
Price of candy• # of candy + Price of candles • number of candles = total sales
1.50x + 5y = 145
D. 1.50x + 5y = 145
15. Adam found a job that offered him an average pay raise of $1500 per year. After 8 years on the
job, Adam’s salary was $72000. What was the starting salary that Adam was offered when he took
the job?
We know the rate of change or slope (1500 per year). We also know a point (8, 72000)
A. $12,000
B. $60,000
C $70,500
D. $65,000
In order to find the starting salary (when year = 0 or x = 0), we need to find the y-intercept.
Let’s substitute:
Y = mx + b
m = 1500
x=8
y = 72000
72000 = 1500(8) + b
72000 = 12000+ b
72000 – 12000 = 12000-12000 + b
Subtract 12000 from both sides.
60000 = b
Therefore, his starting salary is 600000.
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Unit 3: Writing Equations
Part 2: Fill in the blank. Solve each problem on your answer sheet. Show all of your work.
(2 points each)
16. Solve for x:
12[¼x – 8] = 12[⅔(x –
3x – 96 = 8(x – 19.5)
3x – 96 = 8x – 156
3x -8x – 96 = 8x -8x – 156
-5x -96 = -156
-5x -96 + 96 = -156 + 96
-5x = -60
-5x/-5 = -60/-5
X = 12
¼x – 8 = ⅔(x – 19.5)
19.5)]
Multiply both sides by the LCM, 12.
Result after multiplying by 12.
Distribute the 8 throughout the parenthesis
Subtract 8x from both sides.
Simplify: 3x -8x = -5x
Add 96 to both sides.
Simplify: -156+96 = -60
Divide both sides by -5
Answer: x = 12
17. Graph the following equation on the grid:
8x – 2y = -16
Step 1: Find the x-intercept.
8x -2(0) = -16
Let y = 0
8x = -16
X = -2
x-intercept = -2
Step 2: Find the y-intercept.
8(0) – 2y = -16
-2y = -16
Y=8
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y-intercept = 8
Unit 3: Writing Equations
18. Graph the following equation on the grid:
y = -4/5x + 8
Y = mx + b
Y = -4/5x + 8
Slope
y-intercept
Step 1: Plot the point (0,8) this is the yintercept.
Step 2: From this point count down 4 and right
5. The slope is -4/5. Plot this point and draw a
line through your two points.
19. In the year 2003, the average cost of a trip to Disney World for a family of four was $2300. In
2010, the average cost of a trip to Disney for a family of four is $4,200. Write an equation that can
be used to predict the cost of a trip to Disney for any year after 2000. Let x = 0 represent the year
2000.
(Round all decimals to the hundredths place)
Step 1: We can write two ordered pairs from this problem: (3, 2300) (10, 4200)
Step 2: Use the formula to find the slope.
y2- y1 = 4200 – 2300 = 1900 = 271.43
x2 – x1
10 -3
7
The slope is 271.43
Step 3: Choose 1 point to substitute into the slope intercept form equation. I chose (3, 2300)
Y = mx + b
2300 = 271.43(3) + b
2300 = 814.29 + b
Simplify: 271.43(3) = 814.29
2300 – 814.29 = 814.29 – 814.29 + b
Subtract 814.29 from both sides
1485.71 = b
Simplify: 2300- 814.29 = 1485.71
Now that we know the slope and y-intercept we can write the equation.
Y = 271.43x + 1485.71
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Unit 3: Writing Equations
20. The ticket prices for attending a Yankees baseball game increase by $2.75 per year. In the
year 2009, the ticket price for a premium terrace seat was $80. Write an equation that could be
used to determine the price of a premium terrace seat for any year after 2000. Let x = 0 represent
the year 2000.
In this problem, we know the rate (slope) is 2.75 per year. (Per year are your key words for slope)
We also know a point (9, 80) (In the year 2009, the price was 80)
We can use the slope and a point and substitute into y = mx + b to find b (the y-intercept)
Y = mx + b
m = 2.75
(9, 80)
80 = 2.75 (9) + b
Substitute
80 = 24.75 + b
Simplify: 2.75 • 9 = 24.75
80 – 24.75 = 24.75 – 24.75 + b
Subtract 24.75 from both sides.
55.25 = b
Simplify: 80-24.75 = 55.25
Y = mx + b
m = 2.75
b = 55.25
Y = 2.75x + 55.25
The equation written in slope intercept form.
21. A local banquet room charges $35 an hour for use of facilities plus a $30 clean up fee. How
many hours can Joseph rent the hall for $240?
Step 1: In this problem we know the rate (slope) is $35 an hour. The fee is $30 flat, so this is the yintercept.
Since we know the slope and y-intercept we can write the equation in slope intercept form.
Y = mx + b
m = 35
b = 30
Y = 35x + 30
y = total cost & x= the number of hours
Step 2: We know the total amount spent is $240, therefore this represents y in the equation.
Y = 35x + 30
240 = 35x + 30
Substitute 240 for y.
240 – 30 = 35x + 30 -30
Subtract 30 from both sides.
210 = 35x
Simplify: 240-30 = 210
210/35 = 35x/35
Divide by 35 on both sides.
6=x
x = 6 Joseph could rent the hall for 6 hours for $240.
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Unit 3: Writing Equations
Part 3: Short answer. Respond to each problem on your answer sheet. Make sure you
answer all parts of each problem. (3 points each)
22. Laurie knits sweaters for dogs, babies, and children. She sells them at craft shows. She sold 3
times as many baby sweaters than dog sweaters. She sold 5 more children sweaters than dog
sweaters. The prices for each sweater are shown below:
Dog - $7.50
•
Baby - $10.25
Children - $ 14.75
Write an expression to represent the number of baby sweaters sold and an expression to
represent the number of children’s sweaters sold. Let x represent the number of dog
sweaters sold.
x = dog sweaters
3x = baby sweaters
x + 5 = children’s sweaters
•
The total sales for Laurie’s sweaters was $391.75. Write an equation to represent the total
sales of Laurie’s sweaters.
Price (dog) • # of dog + Price (baby) • # of baby + Price (children’s) • # of children’s = total sales
7.50x + 10.25(3x) + 14.75(x+5) = 391.75
•
How many baby sweaters did Laurie sell? Explain how you determined your answer.
7.50x + 10.25(3x) + 14.75(x+5) = 391.75
Equation
7.50x + 30.75x + 14.75x + 73.75 = 391.75
Distribute
53x + 73.75 = 391.75
Combine like terms (x terms)
53x + 73.75 – 73.75 = 391.75 – 73.75
Subtract 73.75
53x = 318
Simplify (391.75-73.75 = 318)
53x/53 = 318/53
Divide by 53 on both sides
X=6
Baby sweaters = 3x
3 • 6 = 18
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Laurie sold 18 baby sweaters. I solved the equation
for x. I found that x = 6 which meant that she had
sold 6 dog sweaters. Since she sold 3 times as many
baby sweaters as dog sweaters, I multiplied 3 times 6
to get 18.
Unit 3: Writing Equations
23. The cost of tuition at a private school in the year 2002 was $12,100. In the year 2009 the cost
was $16,900. Let x = 0 represent the year 2000.
•
•
Write an equation that could be used to predict the tuition for any year after 2000.
Predict the tuition for the year 2015.
I can write two ordered pairs: (2, 12100) & (9, 16900)
Step 1: Find the slope using the formula.
y2- y1 = 16900 – 12100 = 4800 = 685.71
x2 – x1
9-2
7
The slope (m) is 685.71
Step 2: Substitute the slope and 1 point into the slope intercept form equation to find b.
Y = mx + b
m = 685.71
x=2
y = 12100
12100 = 685.71(2) + b
Substitute
12100 = 1371.42 + b
Simplify: 685.71(2) = 1371.42
12100 – 1371.42 = 1371.42 -1371.42 + b
Subtract 1371.42 from both sides
10728.58 = b
M = 685.71
•
b = 10728.58
Y = 685.71x + 10728.58
after 2000.
is the equation that can be used to predict the tuition for any year
Step 3: To predict the tuition for the year 2015, substitute 15 for x into the equation.
Y = 685.71x + 10728.58
Y = 685.71(15) + 10728.58
Y = 21014.23
•
The cost of tuition for the year 2015 is predicted to be 21014.23.
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Unit 3: Writing Equations
Umm
24. Carl has been tracking the price of round trip airfare between Baltimore and Orlando for 10
weeks. The results are show in the graph below.
•
•
•
What is the y-intercept in this problem? What does it mean in the context of this problem?
What is the rate of change between weeks 4 and 6?
The y-intercept in this problem is 250. This means that when Carl first started tracking
the airfares, the cost of roundtrip airfare to Orlando was $250.
In order to find the rate of change between weeks 4 and 6, we will need to write two points and
use the slope formula.
(4, 350) (6, 150)
y2- y1 = 150 – 350 = -200 = -100
x2 – x1
6-4
2
The rate of change between weeks 4 and 6 is -100. This means that the cost of the airfare
dropped about $100 per week during this 2 week time span.
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Unit 3: Writing Equations
Cumulative Test Part 1: Analysis Sheet
Directions: For any problems, that you got wrong on the answer sheet, circle the
number of the problem in the first column. When you are finished, you will be able to
see which Algebra units you need to review before moving on. (If you have more
than 2 circles for any unit, you should go back and review the examples and practice
problems for that particular unit.)
Problem Number
Algebra Unit
3, 4, 9, 10, 16, 21, 22
Solving Equations Unit
2, 6, 8, 11, 12, 17, 18, 24
Graphing Equations Unit
1, 5, 7, 13, 14, 15, 19, 20, 23
Writing Equations Unit
Please take the time to go back and review the problems that you got incorrect. All of
the skills that you learned in these three units, will be needed to solve problems in the
upcoming units.
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