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
(
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)
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
(
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)
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
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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
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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
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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
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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
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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
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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.
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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.
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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).
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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.
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