Study Guide and Review
Choose the term from the list below that best matches each phrase. 1. the ratio of the vertical change to the horizontal change of a line
SOLUTION: The slope of a line is the ratio of the vertical change to the horizontal change.
ANSWER: slope
2. an equation written in the form y = mx + b
SOLUTION: When an equation is written in the form y = mx + b it is in slope-intercept form.
ANSWER: slope-intercept form
3. an ordered list of numbers
SOLUTION: A group of numbers that is written in an ordered list is a sequence.
ANSWER: sequence
4. a description of how one quantity changes in relation to another quantity
SOLUTION: The rate of change is a description of how one quantity changes in relation to another quantity.
ANSWER: rate of change
5. b in the equation y = mx + b
SOLUTION: When an equation is written in the form y = mx + b, b represents the y-intercept.
ANSWER: y-intercept
6. the rate of change between any two data points is the same
SOLUTION: When the rate of change between any two data points is the same, it is a constant rate of change.
ANSWER: constant rate of change
7. in a linear equation, a variable for the input
SOLUTION: In a linear equation with input and output, a variable for the input is the independent variable. In the equation y Page
= 1
mx + b, x represents the independent variable.
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ANSWER: SOLUTION: When the rate of change between any two data points is the same, it is a constant rate of change.
ANSWER: Study
Guide and Review
constant rate of change
7. in a linear equation, a variable for the input
SOLUTION: In a linear equation with input and output, a variable for the input is the independent variable. In the equation y =
mx + b, x represents the independent variable.
ANSWER: independent variable
8. k in the equation y = k x
SOLUTION: For a direct variation equation, y = k x, the letter k in the equation is the constant of variation.
ANSWER: constant of variation
9. a set of equations with the same variables
SOLUTION: A set of equations that have the same variables, such as x and y, are a system of equations.
ANSWER: system of equations
10. a line that is close to most of the data points in a scatter plot
SOLUTION: A line that is close to most of the data points in a scatter plot is called the line of fit.
ANSWER: line of fit
11. Determine whether the relation {(5, 3), (–5, 4), (4, 2), (4, 1)} is a function. Explain.
SOLUTION: The relation is not a function because the domain value 4 is paired with 2 range values, 1 and 2.
ANSWER: No; The domain value 4 is paired with 2 range values, 1 and 2.
12. GASOLINE Use the table that shows the cost of gas in different years. Is the relation a function? Explain.
SOLUTION: The relation is a function because each domain value is paired with only one range value.
ANSWER: Yes; each domain value is paired with only one range value.
Find four solutions of each equation. Write the solution as ordered pairs.
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17. y = –5x
SOLUTION: Page 2
SOLUTION: The relation is a function because each domain value is paired with only one range value.
ANSWER: Study
Guide and Review
Yes; each domain value is paired with only one range value.
Find four solutions of each equation. Write the solution as ordered pairs.
17. y = –5x
SOLUTION: x
–1
0
1
2
y = –5x
y = –5(–1)
y = –5(0)
y = –5(1)
y = –5(2)
y
5
0
–5
–10
(x, y)
(–1, 5)
(0, 0)
(1, –5)
(2, –10)
ANSWER: Sample answer: (–1, 5), (0, 0), (1, –5), (2, –10)
18. y = 4x
SOLUTION: x
–1
0
1
2
y = 4x
y = 4(–1)
y = 4(0)
y = 4(1)
y = 4(2)
y
–4
0
4
8
(x, y)
(–1, –4)
(0, 0)
(1, 4)
(2, 8)
ANSWER: Sample answer: (–1, –4), (0, 0), (1, 4), (2, 8)
19. y = x + 9
SOLUTION: x
–1
0
1
2
y =x+9
y = –1 + 9
y =0+9
y =1+9
y =2+9
y
8
9
10
11
(x, y)
(–1, 8)
(0, 9)
(1, 10)
(2, 11)
ANSWER: Sample answer: (–1, 8), (0, 9), (1, 10), (2, 11)
20. x + y = –1
SOLUTION: Rewrite the equation by solving for y.
x
–1
0
1
2
y = –x – 1
y = –(–1) – 1
y = –0 – 1
y = –1 – 1
y = –2 – 1
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ANSWER: y
0
–1
–2
–3
(x, y)
(–1, 0)
(0, –1)
(1, –2)
(2, –3)
Page 3
1
2
y =1+9
y =2+9
10
11
(1, 10)
(2, 11)
ANSWER: Study
Guide and Review
Sample answer: (–1, 8), (0, 9), (1, 10), (2, 11)
20. x + y = –1
SOLUTION: Rewrite the equation by solving for y.
x
–1
0
1
2
y = –x – 1
y = –(–1) – 1
y = –0 – 1
y = –1 – 1
y = –2 – 1
y
0
–1
–2
–3
(x, y)
(–1, 0)
(0, –1)
(1, –2)
(2, –3)
ANSWER: Sample answer: (–1, 0), (0,–1), (1, –2), (2, –3)
Graph each equation.
21. y = –2x
SOLUTION: Choose four values for x and find the corresponding values for y .
x
y
(x, y)
y = –2x
–1
2
(–1,
2)
y = –2(–1)
0
0
(0, 0)
y = –2(0)
1
–2
(1, –2)
y = –2(1)
2
–4
(2, –4)
y = –2(2)
Graph the ordered pairs on a coordinate plane and draw a line through the points.
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2
y = –2 – 1
–3
(2, –3)
ANSWER: Study
Guide and Review
Sample answer: (–1, 0), (0,–1), (1, –2), (2, –3)
Graph each equation.
21. y = –2x
SOLUTION: Choose four values for x and find the corresponding values for y .
x
y
(x, y)
y = –2x
–1
2
(–1,
2)
y = –2(–1)
0
0
(0, 0)
y = –2(0)
1
–2
(1, –2)
y = –2(1)
2
–4
(2, –4)
y = –2(2)
Graph the ordered pairs on a coordinate plane and draw a line through the points.
ANSWER: 22. y = x + 5
SOLUTION: Choose four values for x and find the corresponding values for y .
x
y
(x, y)
y =x+5
–1
4
(–1,
4)
y = –1 + 5
0
5
(0, 5)
y =0+5
1
6
(1, 6)
y =1+5
2
7
(2, 7)
y =2+5
Graph the ordered pairs on a coordinate plane and draw a line through the points.
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Study Guide and Review
22. y = x + 5
SOLUTION: Choose four values for x and find the corresponding values for y .
x
y
(x, y)
y =x+5
–1
4
(–1, 4)
y = –1 + 5
0
5
(0, 5)
y =0+5
1
6
(1, 6)
y =1+5
2
7
(2, 7)
y =2+5
Graph the ordered pairs on a coordinate plane and draw a line through the points.
ANSWER: 23. SNACKS Each small smoothie x costs $1.50, and each large smoothie y costs $3. Find two solutions of 1.5x + 3y =
12 to determine how many of each type Lisa can buy with $12.
SOLUTION: Solve the equation for y.
x
y
(x, y)
y = –0.5x + 4
Page 6
4
(0, 4)
y = –0.5(0) + 4
6
1
(6, 1)
y = –0.5(6) + 4
The solution (0, 4) means she can buy 0 small smoothies and 4 large smoothies with $12. The solution (6, 1) means
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Study Guide and Review
23. SNACKS Each small smoothie x costs $1.50, and each large smoothie y costs $3. Find two solutions of 1.5x + 3y =
12 to determine how many of each type Lisa can buy with $12.
SOLUTION: Solve the equation for y.
x
y
(x, y)
y = –0.5x + 4
0
4
(0, 4)
y = –0.5(0) + 4
6
1
(6, 1)
y = –0.5(6) + 4
The solution (0, 4) means she can buy 0 small smoothies and 4 large smoothies with $12. The solution (6, 1) means
she can buy 6 small smoothies and 1 large smoothie with $12.
ANSWER: Sample answer: (0, 4) means she can buy 0 small smoothies and 4 large smoothies with $12; (6, 1) means she can
buy 6 small smoothies and 1 large smoothie with $12.
24. Find the rate of change for the linear function shown below.
SOLUTION: The rate of change is an increase of $7.75/h.
ANSWER: increase of $7.75/h
25. ENTERTAINMENT The table shows the total cost of tickets. Compare the rates of change.
SOLUTION: For adults, the rate of change is as follows:
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The rate of change is an increase of $7.75/h.
ANSWER: Study
Guide and Review
increase of $7.75/h
25. ENTERTAINMENT The table shows the total cost of tickets. Compare the rates of change.
SOLUTION: For adults, the rate of change is as follows:
The rate of change for adults is an increase of $18/person.
For children, the rate of change is as follows:
The rate of change for children is an increase of $12.50/person.
The cost for adults increases at a faster rate than the cost for children.
ANSWER: Adults: $18/person; children: $12.50/person; the cost for adults increases at a faster rate than the cost for children.
26. WEATHER The temperature one day is shown in the graph. Find the constant rate of change and interpret its
meaning.
SOLUTION: Use any two points on the graph such as (4, 70) and (2, 66).
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The rate of change for children is an increase of $12.50/person.
The cost for adults increases at a faster rate than the cost for children.
ANSWER: Study
Guide and Review
Adults: $18/person; children: $12.50/person; the cost for adults increases at a faster rate than the cost for children.
26. WEATHER The temperature one day is shown in the graph. Find the constant rate of change and interpret its
meaning.
SOLUTION: Use any two points on the graph such as (4, 70) and (2, 66).
The rate of change is 2° per hour. The temperature increases 2° per hour.
ANSWER: 2° per hour; the temperature increases 2° per hour
Find the slope of the line that passes through each pair of points.
27. F(0, 1), G(6, 4)
SOLUTION: ANSWER: 28. R(–8, –2), S(4, 9)
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28. R(–8, –2), S(4, 9)
SOLUTION: ANSWER: 29. A(–3, 7), G(5, –1)
SOLUTION: ANSWER: –1
30. P(6, –4), S(–1, 10)
SOLUTION: ANSWER: –2
31. ANIMALS A lizard is crawling up a hill that rises 5 feet for every horizontal change of 30 feet. Find the slope.
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ANSWER: Study
Guide and Review
–2
31. ANIMALS A lizard is crawling up a hill that rises 5 feet for every horizontal change of 30 feet. Find the slope.
SOLUTION: The slope of the hill is
.
ANSWER: State the slope and the y-intercept of the graph of each equation.
32. y = 4x + 7
SOLUTION: y = 4x + 7
The slope is 4 and the y-intercept is 7.
ANSWER: 4; 7
33. y =
x
SOLUTION: y=–
x+0
The slope is –
and the y-intercept is 0.
ANSWER: –
;0
34. 5x + y = 0
SOLUTION: First solve for y.
The slope is –5 and the y-intercept is 0.
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35. –x + y = –8
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ANSWER: Study
and Review
;0
– Guide
34. 5x + y = 0
SOLUTION: First solve for y.
The slope is –5 and the y-intercept is 0.
ANSWER: –5; 0
35. –x + y = –8
SOLUTION: First solve for y.
The slope is 1 and the y-intercept is –8.
ANSWER: 1; –8
Graph each equation using the slope and y-intercept.
36. y = –x + 4
SOLUTION: y = –x + 4
slope = –1
y-intercept = 4
Graph the y-intercept point at (0, 4), then use the slope to locate a second point on the line.
Another point on the line is (–1, 5).
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The slope is 1 and the y-intercept is –8.
ANSWER: Study
Guide and Review
1; –8
Graph each equation using the slope and y-intercept.
36. y = –x + 4
SOLUTION: y = –x + 4
slope = –1
y-intercept = 4
Graph the y-intercept point at (0, 4), then use the slope to locate a second point on the line.
Another point on the line is (–1, 5).
ANSWER: 37. y = 2x – 6
SOLUTION: y = 2x – 6
slope = 2
y-intercept = –6
Graph the y-intercept point at (0, –6), then use the slope to locate a second point on the line.
Another point on the line is (1, –4).
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Study Guide and Review
37. y = 2x – 6
SOLUTION: y = 2x – 6
slope = 2
y-intercept = –6
Graph the y-intercept point at (0, –6), then use the slope to locate a second point on the line.
Another point on the line is (1, –4).
ANSWER: 38. y =
x–3
SOLUTION: y=
x–3
slope =
y-intercept = –3
Graph the y-intercept point at (0, –3), then use the slope to locate a second point on the line.
Another point on the line is (2, 0).
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Study Guide and Review
38. y =
x–3
SOLUTION: y=
x–3
slope =
y-intercept = –3
Graph the y-intercept point at (0, –3), then use the slope to locate a second point on the line.
Another point on the line is (2, 0).
ANSWER: 39. y =
x+5
SOLUTION: y=
x+5
slope =
y-intercept = 5
Graph the y-intercept point at (0, 5), then use the slope to locate a second point on the line.
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Another point on the line is (4, 4).
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Study Guide and Review
39. y =
x+5
SOLUTION: y=
x+5
slope =
y-intercept = 5
Graph the y-intercept point at (0, 5), then use the slope to locate a second point on the line.
Another point on the line is (4, 4).
ANSWER: 40. BALLOONS A balloon is rising above the ground. The height in feet y of the balloon can be given by y = 7 + 2x,
where x represents the time in seconds. State the slope and y-intercept of the graph of the equation. Describe what
they represent.
SOLUTION: The slope 2 represents the ascent in ft per second. The y–intercept 7 represents the initial altitude in ft before the
balloon is released.
ANSWER: The slope 2 represents the ascent in ft per second. The y-intercept 7 represents the initial altitude in ft before the
balloon is released.
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Write an equation in slope-intercept form for each line.
41. slope = –2, y-intercept = 5
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Study Guide and Review
40. BALLOONS A balloon is rising above the ground. The height in feet y of the balloon can be given by y = 7 + 2x,
where x represents the time in seconds. State the slope and y-intercept of the graph of the equation. Describe what
they represent.
SOLUTION: The slope 2 represents the ascent in ft per second. The y–intercept 7 represents the initial altitude in ft before the
balloon is released.
ANSWER: The slope 2 represents the ascent in ft per second. The y-intercept 7 represents the initial altitude in ft before the
balloon is released.
Write an equation in slope-intercept form for each line.
41. slope = –2, y-intercept = 5
SOLUTION: y = mx + b
y = –2x + 5
ANSWER: y = –2x + 5
42. slope =
, y-intercept = –1
SOLUTION: y = mx + b
y=
x +(–1)
y=
x–1
ANSWER: y=
x–1
43. slope = 4, y-intercept = 0
SOLUTION: y = mx + b
y = 4x + 0
y = 4x
ANSWER: y = 4x
Write an equation in point-slope form for the line passing through each pair of points.
44. (3, 3), (7, –1)
SOLUTION: Find the slope m.
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y = 4x + 0
y = 4x
ANSWER: Study Guide and Review
y = 4x
Write an equation in point-slope form for the line passing through each pair of points.
44. (3, 3), (7, –1)
SOLUTION: Find the slope m.
Use the slope and the coordinates of either point to write the equation in point-slope form.
ANSWER: y – 3 = –1(x – 3)
45. (1, 5), (2, 8)
SOLUTION: Find the slope m.
Use the slope and the coordinates of either point to write the equation in point-slope form.
ANSWER: y – 5 = 3(x – 1)
46. BIRTHDAYS Diem’s parents wants to rent the local movie theatre for her birthday party. It costs $100 plus $30
per hour to rent the movie theater.
a. Write an equation in slope-intercept form that shows the cost y for renting the theater for x hours.
b. Find the cost of renting the theater for 4 hours.
SOLUTION: a. Let y = the total cost. Let x = the number of hours. The total cost is the $100 rental fee plus $30 per hour. So, y =
30x + 100.
b. To find the cost for 4 hours, substitute 4 for x.
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It will cost $220 to rent the theater for 4 hours.
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ANSWER: Study Guide and Review
y – 5 = 3(x – 1)
46. BIRTHDAYS Diem’s parents wants to rent the local movie theatre for her birthday party. It costs $100 plus $30
per hour to rent the movie theater.
a. Write an equation in slope-intercept form that shows the cost y for renting the theater for x hours.
b. Find the cost of renting the theater for 4 hours.
SOLUTION: a. Let y = the total cost. Let x = the number of hours. The total cost is the $100 rental fee plus $30 per hour. So, y =
30x + 100.
b. To find the cost for 4 hours, substitute 4 for x.
It will cost $220 to rent the theater for 4 hours.
ANSWER: a. y = 30x + 100
b. $220
47. HOUSING The table shows the changes in the median price of existing homes.
a. Make a scatter plot and draw a line of fit for the data.
b. Use the line of fit to predict the median price for an existing home for the year 2015.
SOLUTION: a.
b. . Extend the line to estimate the y-value for an x-value of ‘15. The y-value for ‘15 is about 362. So, predict that
the median home price for an existing home for the year 2015 is $362,00.
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ANSWER: a.
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Guide and Review
b
. . Extend the line to estimate the y-value for an x-value of ‘15. The y-value for ‘15 is about 362. So, predict that
the median home price for an existing home for the year 2015 is $362,00.
ANSWER: a.
b. Sample answer: $362,000
48. MUSIC The table shows the changes in the average concert ticket prices.
a. Make a scatter plot and draw a line of fit for the data.
b. Use the line of fit to predict the average price of a concert ticket in 2015.
SOLUTION: a.
b. . Extend the line to estimate the y-value for an x-value of ‘15. The y-value for ‘15 is about 121. So, predict that
the average concert price in 2015 would be $121.
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b.
. Extend the line to estimate the y-value for an x-value of ‘15. The y-value for ‘15 is about 121. So, predict that
the average concert price in 2015 would be $121.
Use the slope and the coordinates of any point to write the equation in point-slope form.
Substitute 2015 for x.
The average price for a concert ticket in the year 2015 should be about $130.
ANSWER: a.
b. Sample answer: $121
Solve each system of equations by graphing.
49. y = x
y=
x–1
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Study Guide and Review
b. Sample answer: $121
Solve each system of equations by graphing.
49. y = x
y=
x–1
SOLUTION: The solution of the system of equations is (–2, –2).
ANSWER: 50. y = x + 2
y = 3x
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50. y = x + 2
y = 3x
SOLUTION: The solution of the system of equations is (1, 3).
ANSWER: Solve each system of equations by substitution.
51. y = x + 3
x=1
SOLUTION: y=x+3
x=1
Replace x with 1 in the first equation.
The solution of this system of equations is (1, 4).
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(1, 4)
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Study Guide and Review
Solve each system of equations by substitution.
51. y = x + 3
x=1
SOLUTION: y=x+3
x=1
Replace x with 1 in the first equation.
The solution of this system of equations is (1, 4).
ANSWER: (1, 4)
52. y = 2x + 6
y=0
SOLUTION: y = 2x + 6
y=0
Replace y with 0 in the first equation.
The solution of this system of equations is (–3, 0).
ANSWER: (–3, 0)
53. NUMBER SENSE The sum of two numbers is 9, and the difference of the numbers is 1. Write a system of
equations to represent this situation. Then solve the system to find the numbers.
SOLUTION: Sample answer: Let x = the first number and y = the second number.
x +y = 9
x–y =1
Solve the second equation for x.
Substitute y + 1 for x in the first equation.
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The solution of this system of equations is (–3, 0).
ANSWER: Study
Guide and Review
(–3, 0)
53. NUMBER SENSE The sum of two numbers is 9, and the difference of the numbers is 1. Write a system of
equations to represent this situation. Then solve the system to find the numbers.
SOLUTION: Sample answer: Let x = the first number and y = the second number.
x +y = 9
x–y =1
Solve the second equation for x.
Substitute y + 1 for x in the first equation.
Now substitute y = 4 into either equation to find x.
The solution of this system of equations is x = 5 and y = 4.
ANSWER: Sample answer: x + y = 9, x – y = 1; 5 and 4; x = 5; y = 4
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