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NJ Center for Teaching and Learning ~ 1 ~ www.njctl.org
Graphing Linear Equations Chapter Questions
1. What are the various types of information you can be given to graph a line?
2. What is slope? How is it determined?
3. Explain the four different values of slope and what types of lines you expect to see on the
graph.
4. How can we tell if two lines are parallel, perpendicular, or neither just from their equations?
5. What are the various ways you can use information given to you to determine the equation
of a line?
NJ Center for Teaching and Learning ~ 2 ~ www.njctl.org
Graphing Linear Equations Chapter Problems
Linear Equations
Classwork
For the equations below, make a table with at least 3 ordered pairs, plot the points and connect
them to form the line.
1. y = 3x – 4
2. y = -2x + 4
3. y = x – 3
4. y = 1
2x + 4
5. y = - 2
3x + 1
6. Make a table for the rule y = 3x – 1. Then use the table to graph the rule neatly. Be
sure to include scales and labels for the axes and the graph.
PARCC-type Question:
7. Which points lie on the line whose equation is −4𝑥 + 5𝑦 − 12 = 8? Select all that apply. a. (5, 0) b. (-2, 2.5) c. (8, 5) d. (10, 12) e. (0, 4)
Homework
For the equations below, make a table with at least 3 ordered pairs, plot the points and connect
them to form the line.
8. y = -x – 2
9. y = 2x + 1
10. y = 1
4x
11. y = -2x – 2
12. y = - 1
3x + 4
13. Make a complete graph for y = 3x + 1
14. Using the rule y = –2x + 5,
a. Make a table for the rule.
b. Use your table to graph your rule on graph paper. Make sure your graph is
complete.
NJ Center for Teaching and Learning ~ 3 ~ www.njctl.org
15. Using the tables below, carefully graph these two equations on the same set of axes.
y 12x y 2x 4
x –2 –1 0 1 2 3 4 5
y
x –2 –1 0 1 2 3 4 5
y
a. Estimate where the two graphs intersect.
b. Show how you can check the accuracy of the points you wrote in part (a).
PARCC-type Question:
16. Which points lie on the line whose equation is 8𝑥 − 2𝑦 + 7 = −9? Select all that apply. a. (-2, 0) b. (1, 3) c. (-1, 4) d. (1, -4) e. (0, -1)
Graphing Linear Equations using intercepts
Classwork PARCC-type Questions:
17. Find the x- and y-intercepts in each equation. Then graph the equation.
a. 3x – y = -4
b. 2x + y = 3
c. x – y = -7
d. 2x – 5y = 10
e. 3x + y = -3
f. 2x + 3y = 15
NJ Center for Teaching and Learning ~ 4 ~ www.njctl.org
18. Find the x- and y-intercepts in each graph shown. a) b)
Homework PARCC-type Questions:
19. Find the x- and y-intercepts in each equation. Then graph the equation. a. 6x – y = – 4
b. 3x + y = – 2
c. x – 6y = 3
d. x – y = – 4
e. 2x + 7y = – 14
f. 7x + y = – 7
20. Find the x- and y-intercepts in each graph shown. a) b)
NJ Center for Teaching and Learning ~ 5 ~ www.njctl.org
Horizontal & Vertical Lines
Classwork
21. Determine if the following equations are horizontal, vertical, neither or cannot be
determined.
a. y = -5
b. x = 7
c. 2x + 4y = 8
d. 7x – 21 = 0
e. 3x + 2y = 3x – 4
Homework
22. Determine if the following equations are horizontal, vertical, neither or cannot be
determined.
a. x = -5
b. y = 7
c. 8x + -4y = -2
d. -6x – 3y = -6x + 2
e. -8x = -24
Slope of a Line
Classwork
23. Find the slope of the line in each graph below.
a) b)
NJ Center for Teaching and Learning ~ 6 ~ www.njctl.org
c) d)
24. Find the slope of the line that passes through each of the following two points.
a. (-12, -5), (0, -8)
b. (12, -18), (11, 12)
c. (-18, -20), (-18, -15)
d. (-20, -4), (-12, -10)
e. (8, 10), (0, 14)
f. (6, 9), (3, -9)
g. (1, 2), (5, 7)
h. (3, -3), (12, -2)
i. (-4, -8), (-1, 1)
j. (4, 7), (-3, 7)
NJ Center for Teaching and Learning ~ 7 ~ www.njctl.org
Homework
25. Find the slope of the line in each graph below.
a) b)
c) d)
26. Find the slope of the line through each of the following two points.
a. (3, -9), (1, 1)
b. (7, 4), (3, 8)
c. (-3, 0), (5, 12)
d. (8, -2), (12, -2)
e. (6, -3), (2, 9)
f. (-3, 7), (-4, 8)
NJ Center for Teaching and Learning ~ 8 ~ www.njctl.org
g. (5, 9), (5, -8)
h. (-5, 0.5), (-6, 3)
i. (-7, 1), (7, 8)
j. (-2, 1), (5, 7)
Point-Slope Form
Classwork
27. Graph each equation
a. y – 2 = (x – 3)
b. y + 5 = 2(x + 2)
c. y – 4 = -3(x – 3)
d. y + 7 = 1
2(x – 1)
e. y – 3 = - 3
4(x + 5)
28. Write an equation in point slope form for the line through the given point with the given
slope.
a. (3,4); m = 6
b. (-2,-7); m = - 3
2
c. (7,-4); m = -3
d. (4,0); m = 1
e. (-4,-4); m = 1
3
29. A line passes through the given points. First write an equation for the line in point-slope
form.
a. (-1,0), (1,2)
b. (3,5), (0,0)
c. (6,-2), (9,-8)
d. (-1,-5), (-7,-6)
e. (-3,4), (3,-2)
Homework
30. Graph each equation
a. y – 2 = (x + 4)
b. y + 5 = -2(x – 4)
NJ Center for Teaching and Learning ~ 9 ~ www.njctl.org
c. y – 1 = 4(x – 2)
d. y – 5 = - 1
3(x – 1)
e. y – 0 = 1
5(x – 3)
31. Write an equation in point slope form for the line through the given point with the given
slope.
a. (5,-4); m = -2
b. (-2, -3); m = 4
c. (7,4); m = 1
4
d. (9,-9); m = 3
e. (6,0); m = - 2
3
32. A line passes through the given points. First write an equation for the line in point-slope
form.
a. (-10, -50), (5, 25)
b. (0, 10), (10, -20)
c. (0.5, 9), (50, -90)
d. (8, 9), (5, -6)
e. (-7, 4), (-3, 6)
Slope-Intercept Form
Classwork
33. Use lines A, B, C and D to answers the questions.
a. What is the y-intercept of each line?
b. What is the slope of each line?
c. What is the equation of each line?
NJ Center for Teaching and Learning ~ 10 ~ www.njctl.org
34. Use the lines E, F, G, and H to answer the following questions.
35. Write the equation of the line through the given points in slope-intercept form.
a. (6, 3), (-4, 2)
b. (12, 3), (8, -4)
c. (14, -4), (7, -3)
d. (7, 2), (8, 5)
e. (20, -10), (-30, 0)
Homework
36. Use lines J, K, L, and M to answers the questions.
H
H
a. What is the y-intercept of each line?
b. What is the slope of each line?
c. What is the equation of each line?
a. What is the y-intercept of each line?
b. What is the slope of each line?
c. What is the equation of each line?
NJ Center for Teaching and Learning ~ 11 ~ www.njctl.org
Line 3 Line 5
Line 4
Line 1
Line 2
37. Use lines N, O, P, and Q to answers the questions.
38. Write the equation of the line through the given points in slope-intercept form.
a. (1, 4), (-1, 1)
b. (5, -3), (3, 4)
c. (2, 4), (-3, -6)
d. (5, 3), (4, 5)
e. (0, 0), (-1, -3)
39. Match each clue (a) through (d) below with the line on the graph that fits the description. Then write a clue for the line that has no match.
a. I have a large negative value for m and a small positive value for b.
b. I have a small negative value for m and a value of zero for b.
c. I have a small negative value for m and a large positive value for b.
d. I have a small positive value for m and a small positive value for b.
a. What is the y-intercept of each line?
b. What is the slope of each line?
c. What is the equation of each line?
NJ Center for Teaching and Learning ~ 12 ~ www.njctl.org
40. Growth equations for different types of plants (x = time in days and y = height in inches)
are listed on the left, and descriptions for their growth patterns are listed on the right.
Match the equations with their descriptions. State a reason for each match that you
make. Then write an equation for the plant description left over.
a. y 3x
b. y x 4
c. y 32x 1
d. y 2x 1
e. 𝑦 = 4𝑥 + 1
1) Plant # 1 starts as a seed and grows at a rate
of 4 inches every 2 days. It takes an extra day to
emerge from the ground.
2) Plant # 2 was already 1 inch tall when planted
and grows half of a foot every 4 days.
3) Plant # 3 started as a seed and grows 6
inches every 2 days.
4) Plant # 4 was already 4 inches tall when
planted and grows 1 foot every 12 days.
5) Plant # 5 was already 1 inch tall when
planted and grows 8 inches every 2 days.
NJ Center for Teaching and Learning ~ 13 ~ www.njctl.org
Proportional Relationships
Classwork
41. Determine if a proportional relationship is shown in each graph below. If yes, then
calculate the constant of proportionality. If no, explain why.
a.
b.
42. Determine if a proportional relationship is shown in each table below. If yes, then
calculate the constant of proportionality. If no, explain why.
a. b.
x y x y
1 8 2 -12
12 63 12 -72
-4 -17 -3 18
-15 -72 -14 84
NJ Center for Teaching and Learning ~ 14 ~ www.njctl.org
43. A family drove their car for 288 miles in 4 hours.
a. Write a direct variation equation that relates the distance traveled in respect to
time.
b. Graph the equation.
c. Predict how long it will take the family to drive 612 miles.
PARCC-type questions:
44. A line passes through the points (4, -5), (-2, 13), and (-5, 22).
a. What is the equation of the line?
b. Does this line represent a proportional relationship? Explain your answer.
45. A line passes through the points (16, -12), (24, -18), and (-4, 3).
a. What is the equation of the line?
b. Use the equation of the line to explain why the ratio of the y-coordinate to the x-
coordinate is the same for any point on the line except the y-intercept.
NJ Center for Teaching and Learning ~ 15 ~ www.njctl.org
Homework
46. Determine if a proportional relationship is shown in each graph below. If yes, then
calculate the constant of proportionality. If no, explain why.
a.
b.
47. Determine if a proportional relationship is shown in each table below. If yes, then
calculate the constant of proportionality. If no, explain why.
a. b.
x y x y
3 8 4 5
15 40 16 -13
-6 -16 -5 18.5
-18 -48 -19 39.5
NJ Center for Teaching and Learning ~ 16 ~ www.njctl.org
48. A family drove their car for 408 miles in 6 hours.
a. Write a direct variation equation that relates the distance traveled in respect to
time.
b. Graph the equation.
c. Predict how long it will take the family to drive 765 miles.
PARCC-type questions:
49. A line passes through the points (-3, -2), (12, 8), and (21, 14).
a. What is the equation of the line?
b. Does this line represent a proportional relationship? Explain your answer.
50. A line passes through the points (-5, 2), (-30, 12), and (20, 8).
a. What is the equation of the line?
b. Use the equation of the line to explain why the ratio of the y-coordinate to the x-
coordinate is the same for any point on the line except the y-intercept.
Solving Linear Equations
Classwork
51. A line has an x-intercept of 8 and y-intercept of 12. Write an equation for the line.
NJ Center for Teaching and Learning ~ 17 ~ www.njctl.org
52. Find another point on the line containing the points (–2, –3) and (3, – 4). Write the
coordinates of the point you found.
53. Consider the line containing the points (–1, 3) and (4, 6).
a. Write an equation for the line.
b. Find another point on the line that has integer coordinates. Write the coordinates of the point you found.
54. Write the equations in standard form. a. y – 4 = 2(x – 5)
b. y + 5 = -3(x – 4)
c. 𝑦 = −4
3𝑥 + 9
d. 𝑦 =6
11𝑥 − 5
55. Write the equations in slope-intercept form.
a. y + 6 = 1
4(x – 6)
b. y – 3 = (x + 7)
c. 8x – 7y = 28
d. 9x + 5y = 45
56. What is the slope of the line 12x – 14y = 84?
NJ Center for Teaching and Learning ~ 18 ~ www.njctl.org
PARCC-type Question:
57. The graph shown below represents the function 𝑓(𝑥) =5
2𝑥 − 5. For what value of x does
𝑓(𝑥) = 0?
Homework
58. The points (2, 3) and (6, 11) lie on a line. Show your work when answering the following questions.
a. Write an equation for the line.
b. Find another point on the line that has integer coordinates. Write the coordinates of the point you found.
59. Write the equations in standard form.
a. y – 5 = 2(x – 3)
b. y + 2 = -4(x – 2)
c. 𝑦 =9
2𝑥 + 3
d. 𝑦 = −10
3𝑥 − 7
60. Write the equations in slope-intercept form.
a. y – 3 = 3
7(x – 3)
b. y + 1 = 2(x + 1)
NJ Center for Teaching and Learning ~ 19 ~ www.njctl.org
c. 2x + 9y = 36
d. 3x – 5y = 30
61. A line has an x-intercept of -4 and y-intercept of 16. Write an equation for the line.
62. What is the slope of the line 10x + 12y = 60?
63. Without graphing, find the equation of the line that passes through the points (– 4, 3) and
(6, –2). Show all work in a clear and organized manner.
64. It costs $20 plus $1.50 per hour to rent a golf cart.
a. Find an equation showing the relationship between the cost of renting a cart (y) and the number of hours it was rented (x).
b. Draw a graph to show this relationship.
c. Show how to use your graph to estimate the cost to rent a cart for 5 hours.
d. Show how to use your graph to estimate the number of hours you get the cart if you spend $32.
65. A new student has joined your study group. Write step-by-step explanations telling her
how to use algebra to find the equation of a line given each of the following. Be sure to include examples.
a) A point and the slope.
b) Two points
NJ Center for Teaching and Learning ~ 20 ~ www.njctl.org
PARCC-type Question:
66. The graph shown below represents the function 𝑓(𝑥) = −3
4𝑥 − 6. For what value of x
does 𝑓(𝑥) = 0?
Scatter plots and Line of Best Fit
Classwork
67. Predict the test score of someone who spends 48 minutes studying.
60
65
70
75
80
85
90
95
100
105
110
25 30 35 40 45 50 55 60
Time spent studying
test
sco
res
NJ Center for Teaching and Learning ~ 21 ~ www.njctl.org
68. Predict the test score of someone who spends 34 minutes studying.
69. Draw a scatter plot from the following data:
Size of shoe Height (inches) 5 55 5.5 58 6 62 7 68 6.5 63 7.3 70 8 79 8.7 88
70. Consider the scatter graph to answer the following:
Which two points would give the line of best fit?
a. A and B
b. A and C
c. D and B
d. There is no pattern
60
65
70
75
80
85
90
95
100
105
110
25 30 35 40 45 50 55 60
Time spent studying
test
sco
res
NJ Center for Teaching and Learning ~ 22 ~ www.njctl.org
71. Consider the scatter graph to answer the following:
Which two points would give the line of best fit?
a. A and B
b. B and C
c. C and D
d. There is no pattern
72. Use the two points (7,14) and (15,27) to write an equation for the line of best fit in slope-intercept
form.
73. If the prediction equation is y = 0.5t + 60, where t represents time in minutes, what will the person
get on his test if he studies for 45 minutes?
74. If the prediction equation to determine a test grade is y = 0.5t + 60, and someone received an 80
on the test, how long did they study for?
NJ Center for Teaching and Learning ~ 23 ~ www.njctl.org
75. Consider the scatter graph to answer the following:
What is the equation of the line of best fit in slope-intercept form that passes through (3.4, 7) and
(8, 3)?
76. Consider the scatter graph to answer the following:
The equation for the line of best fit is y = -1.06x + 10.7. Determine the value for x = 15? Is this an
interpolation or extrapolation?
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
NJ Center for Teaching and Learning ~ 24 ~ www.njctl.org
Homework
77. Using the scatter graph, predict the mile time of someone who spends 6 hours a week training.
78. Using the scatter graph, predict the mile time of someone who spends 12 hours a week training.
79. Draw a scatter graph from the following data,
Time spent studying (min) Grade
55 97 31 78 52 90 20 61 42 84 47 90 31 81
5:55
7:07
8:19
9:31
10:43
11:55
0 5 10 15 20 25
Time spent training
Mil
e t
ime
5:55
7:07
8:19
9:31
10:43
11:55
0 5 10 15 20 25
Time spent training
Mil
e t
ime
NJ Center for Teaching and Learning ~ 25 ~ www.njctl.org
80. Consider the scatter graph to answer the following:
Which point would give the line best fit?
a. A
b. B
c. C
d. There is no pattern
81. Consider the scatter graph to answer the following:
Which two points would give the line of best fit?
a. A and D
b. A and C
c. B and D
d. There is no pattern
NJ Center for Teaching and Learning ~ 26 ~ www.njctl.org
82. Using the scatter graph below use the two points (3.4, 7) and (9, 1) to write an equation for the
line of best fit in slope-intercept form.
83. If the prediction equation for a test grade is y = 0.52t + 65, where t represents the time in minutes,
what grade will someone earn if they study for 30 minutes.
84. If the prediction equation for a test grade is y=.52t+65, where t represents the time in minutes,
how long did someone study for if they received an 83 on the exam?
85. Consider the scatter graph to answer the following:
What is the equation of the line of best fit in slope-intercept form passing through (2.7, 11.1) and
(9.4, 3.7)?
9, 1
7.5, 3.75.3, 4.2
4, 5.4
3.4, 7
3, 8.1
2, 9.2
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
NJ Center for Teaching and Learning ~ 27 ~ www.njctl.org
Unit Review
1. The slope of a line is its:
a. Run over rise
b. Rise over run
c. How far it travels
d. Horizontal distance divided by its vertical distance
2. Which of the following equations is slope-intercept?
a. Ax + By= C
b. m(x2-x1) = y2-y1
c. y=mx+b
d. (y2-y1)/(x2-x1)
3. A horizontal line’s slope is:
a. Positive
b. Negative
c. 0
d. Undefined
4. A vertical line’s slope is:
a. Positive
b. Negative
c. 0
d. Undefined
5. For the line -3y = 3x+7, the slope is:
a. Positive
b. Negative
c. 0
d. Undefined
6. The y-intercept of a line will always have the coordinates of:
a. (0, y)
b. (x, 0)
c. (-x, 0)
d. (0, -y)
7. The x-intercept of a line will always have the coordinates of:
a. (0, y)
b. (x, 0)
c. (-x, 0)
d. (0, -y)
NJ Center for Teaching and Learning ~ 28 ~ www.njctl.org
8. The line 12x + 2y = 6 is:
a. Vertical
b. Horizontal
c. Neither
d. Cannot be determined
9. What is the slope of the line that passes through (3, -4) and (4, 9)?
a. 13
b. -13
c. 5
d. -5
10. What is the slope of the line that passes through (-2, -1) and (12, 6)?
a. -1/2
b. -2
c. 2
d. ½
Use the following graph to answer questions 11-16:
11. Which line has a slope of 0?
a. Line A
b. Line B
c. Line C
d. Line D
e. Line E
12. Which line has a negative slope?
a. Line A
b. Line B
c. Line C
d. Line D
e. Line E
13. Which line has an undefined slope?
a. Line A
b. Line B
c. Line C
d. Line D
e. Line E
NJ Center for Teaching and Learning ~ 29 ~ www.njctl.org
14. Which line is represented by y = x?
a. Line A
b. Line B
c. Line C
d. Line D
e. Line E
15. Which line is represented by y = -x?
a. Line A
b. Line B
c. Line C
d. Line D
e. Line E
16. Line D is represented by:
a. y = -4x
b. -4x + y = 0
c. y = 4x + 1
d. y = -4x + 1
17. What are the equations of the line that passes through the point (-4, 5) and has a slope
of 2? Select all that apply.
a. 𝑦 − 5 = 2(𝑥 + 4)
b. 𝑦 + 4 = 2(𝑥 − 5)
c. y = 2x + 13
d. 2x – y = -13
e. y = 2x – 14
f. 2x – y = 14
18. Which points lie on the line whose equation is 2(𝑥 − 3) = 7𝑦 + 8? Select all that apply.
2x – 7y = 14
a. (7, 0)
b. (3.5, 1)
c. (0, -2)
d. (-7, 4)
e. (10.5, -1)
f. (-3.5, -3)
19. Which of the following statements reflect the data on the scatter graph? Select all that apply.
a. The less time spent training,
the greater the mile time
b. The more time spent training,
the lower the mile time
c. The points on the graph
indicate that one can predict
the mile time, given the time
spent training
d. There is no correlation
between the time spent
training and the time it takes
to run a mile
NJ Center for Teaching and Learning ~ 30 ~ www.njctl.org
20. Create a table of values for the equation y + 2x = 5 and graph the line in the coordinate plane below.
21. What is the slope of the line that passes through the points (-4, -3) and (10, 4)?
22. What is the equation of the line for question #20? Graph it in the coordinate plane below.
23. What is the slope of the line 7x – 9y = 63?
NJ Center for Teaching and Learning ~ 31 ~ www.njctl.org
24. Write an equation of a line that passes through (3, 2) and (5, 5). Graph it in the coordinate plane below.
25. Consider the line y = -3x + 2.
a. What is the slope of the line?
b. What is this line in standard form?
26. A line has an x-intercept of -4 and a y-intercept of 8.
a. What is the slope of this line?
b. Write the equation for this line in point-slope form, slope-intercept form, and standard
form.
27. A line that has a slope of 1 and passes through the point (4, 5). What is the equation of
this line in point-slope form, slope-intercept form, and standard form?
NJ Center for Teaching and Learning ~ 32 ~ www.njctl.org
28. The graph below represents the function 𝑓(𝑥) = −3
5𝑥 − 3. For what value of x does
𝑓(𝑥) = 0?
29. A car is driving at 55 miles per hour. They are at mile marker 25. After driving 5 hours,
which mile marker will they be passing?
30. A car is driving at 40 mph. After driving 2 hours, they are at mile marker 120. After
driving 3 hours, which mile marker will they be passing?
31. Marla works at a toy company. She can box 50 toys per hour. How many toys can she
box in 3.5 hours?
32. If Marla needs to box 400 toys, how long will this take her?
33. John tutors at $40 an hour. Write an equation that represents his income with x
representing the number of hours, and y representing his income.
34. Let’s assume John deposits all his money from his tutoring job into his savings account.
If John had $450 in the bank originally, what would the new equation be that represents
the total amount of money in John’s bank account?
NJ Center for Teaching and Learning ~ 33 ~ www.njctl.org
35. Kim sells friendship bracelets. Kim can make 5 an hour. What equation represents how
many Kim can make at x number of hours?
36. If Kim’s friend Shawna wants 40 friendship bracelets made for her birthday party, how
long will Kim spend making friendship bracelets?
37. A study team is having a discussion about how much information a person must have to write an equation for a line.
Ray says that you need both the x and y coordinate of a point on the line.
Miranda says that you have to have at least two points.
Jim says that once you have the both the vertical and horizontal change of the line, then you can write the equation.
Sasha says that if you have the slope, you need at least one point to write the equation. Who is correct? Write a thorough response to one of the students mentioned above. Be sure to comment on the accuracy of the student’s statement and explain your reasoning.
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38. Harper, Ellie, and Raina are having a skateboard race. Harper began at the starting line and rode at a constant rate of 3 feet per second. Ellie got a five-foot head start and rode two feet every 3 seconds. Raina rode 3 feet every five seconds and had a two-foot head start.
a. Graph lines on the axes provided for each of the contestants. Let x represent the time in seconds, and let y represent distance in feet.
b. Write an equation for each line.
c. After how many seconds did Harper catch up to Ellie? How far were they from the starting line what that happened? Confirm your answer algebraically. If the race is 9 feet long, who won? Use both the graph and the rules to justify your answer.
d. Will Raina ever catch up to Ellie? How do you know?
39. Kim has improved with her friendship bracelet making skills. She can now make 10 an
hour. Her friend, Shawna, has been helping Kim and is able to make 4 an hour.
a. What would be the equation for the amount of bracelets Kim can make in x hours?
b. What would be Shawna’s equation for the amount of bracelets she can make in x
hours?
c. If Kim can sell a bracelet at $2.50 each, what would be the equation that shows her
income?
d. How much can Kim make if she sold all the bracelets that she made in 4 hours?
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Answer Key
Note: In problems #1-6, the points may vary 1.
2.
3.
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4.
5.
6. (0, -1), (-2, -7), (1, 2)
7. D and E Note: In problems #8-13, the points may vary
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8.
9.
10.
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11.
12. (0, 4), (12, 0), (3, 3)
13.
14.
a.
x -2 -1 0 1 2
y 9 7 5 3 1
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b.
15.
X -2 -1 0 1 2 3 4 5
Y -1 -½ 0 ½ 1 1 ½ 2 2 ½
X -2 -1 0 1 2 3 4 5
Y 8 6 4 2 0 -2 -4 -6
a. The two lines intersect at (1.6, 0.8) b. Substitute the values into BOTH equations and check the statements are true.
y = ½ x y = -2x + 4 (0.8) = ½ (1.6) (0.8) = -2(1.6) + 4 0.8 = 0.8 0.8 = -3.2 + 4 TRUE 0.8 = 0.8 TRUE
16. A and C 17.
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a. x-int: (-1.333, 0); y-int: (0, 4)
b. x-int: (1.5, 0); y-int: (0, 3)
c. x-int: (-7, 0); y-int: (0, 7)
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d. x-int: (5, 0); y-int: (0, -2)
e. x-int: (-1, 0); y-int: (0, -3)
f. x-int: (7.5, 0); y-int: (0, 5)
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18. a) x-int: (3, 0); y-int: (0, 5) b) x-int: (-4, 0); y-int: (0, 1)
19. a. x-int: (-0.667, 0); y-int: (0, 4)
b. x-int: (-0.667, 0); y-int: (0, -2)
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c. x-int: (3, 0); y-int: (0, -0.5)
d. x-int: (-4, 0); y-int: (0, 4)
e. x-int: (-7, 0); y-int: (0, -2)
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f. x-int: (-1, 0); y-int: (0, -7)
20. a) x-int: (-1, 0); y-int: (0, 3)
b) x-int: (-4, 0); y-int: (0, -2)
21. a. Horizontal b. Vertical c. Neither d. Vertical e. Horizontal
22. a. Vertical b. Horizontal c. Neither d. Horizontal e. Vertical
23.
a. 𝑚 = −5
3
b. 𝑚 =1
4
c. m = undefined d. m = 0
24. a. – ¼
b. -30
c. undefined
d. – ¾
e. – ½
f. 6
g. 5/4
h. 1/9
i. 3
j. 0
25. a. m = undefined b. m = 0 c. m = 3
d. 𝑚 = −1
2
26.
a. 4
b. 1
c. 4/3
d. 0
e. 3
f. -1
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g. undefined
h. -5/2
i. ½
j. -6/7
27.
a.
b.
c.
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d.
e.
28.
a. y – 4 = 6(x – 3)
b. 𝑦 + 7 = −3
2(𝑥 + 2)
c. y + 4 = -3(x – 7)
d. y = x - 4
e. 𝑦 + 4 =1
3(𝑥 + 4)
29.
a. y – 1 = x - 2
b. y – 5 = 5
3(x – 3)
c. y + 8 = -2(x – 9)
d. y + 6 = 1
6(x + 7)
e. y + 2 = -1(x – 3)
30.
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a.
b.
c.
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d.
e.
31.
a. y + 4 = -2(x – 5)
b. y + 3 = 4(x + 2)
c. y – 4 = ¼ (x – 7)
d. y – 9 = 3(x – 3)
e. 𝑦 = 2
3(𝑥 − 6)
32.
a. y = 5(x – 0)
b. y – 10 = -3(x – 0)
c. y + 90 = -2(x – 50)
d. y – 9 = 5(x + 8)
e. y – 4 = ½ (x + 7)
33.
Line A Line B Line C Line D
y-intercept -7 4 -1 -6
slope -3 -2 4 0
equation y = -3x – 7 y = -2x + 4 y = 4x – 1 y = -6
34.
Line E Line F Line G Line H
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y-intercept None 5 -1 -2
slope Undefined -¼ -½ 1/3
equation x = -2 y = -¼x + 5 y = -½x – 1 y = 1/3x – 2
35.
a. 𝑦 =1
2𝑥
b. 𝑦 =7
4𝑥 − 18
c. 𝑦 = −1
7𝑥 − 2
d. 𝑦 = 3𝑥 − 19
e. 𝑦 = −1
5𝑥 − 6
36.
Line J Line K Line L Line M
y-intercept None 5 7 -4
slope Undefined -1 -2 2/5
equation x = -4 y = -x + 5 y = -2x + 7 y = 2/5x – 4
37.
Line N Line O Line P Line Q
y-intercept 0 None 4 -2
slope -1 Undefined 1 0
equation y = -x x = 1.5 y = x + 4 y = -2
38.
a. 𝑦 =3
2𝑥 +
5
2
b. 𝑦 =7
2𝑥 −
13
2
c. 𝑦 = 2𝑥
d. 𝑦 = −2𝑥 + 13
e. 𝑦 = 3𝑥 39.
a. Line 3 b. Line 5 c. Line 2 d. Line 4
40. a. Plant #3 b. Plant #4 c. Plant #2 d. Plant #1 e. Plant #5
41.
a. No, the points do not show a consistent ratio of y-coordinate to x-coordinate; 0
−2≠
2
−1.
b. Yes, the constant of proportionality is −3
5.
42.
a. No, the points do not show a consistent ratio of y-coordinate to x-coordinate; 8
1≠
63
12.
b. Yes, the constant of proportionality is −6 43.
a. y = 72x
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b.
c. 8.5 hours
44. a. y = -3x + 7 or y + 5 = -3(x – 4) b. No, it does not intersect the origin, nor do the points have a proportional relationship.
45.
a. 𝑦 = −3
4𝑥
b. Since the y-intercept is the origin, the ratio of the y-coordinate to the x-coordinate is 0
0, which is
not the same as the constant of proportionality (or slope), which is −3
4.
46.
a. Yes, the constant of proportionality is 3
2.
b. No, the points do not show a consistent ratio of y-coordinate to x-coordinate; 1
0≠
−3
5.
47.
a. Yes, the constant of proportionality is 8
3.
b. No, the points do not show a consistent ratio of y-coordinate to x-coordinate; 5
4≠
−13
16.
48. a. y = 68x b.
c. 11.25 hours
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49.
a. 𝑦 =2
3𝑥
b. Yes, the points have a consistent y-coordinate to x-coordinate ratio of 2
3, and the y-intercept is 0.
50.
a. 𝑦 = −2
5𝑥
b. Since the y-intercept is the origin, the ratio of the y-coordinate to the x-coordinate is 0
0, which is
not the same as the constant of proportionality (or slope), which is −2
5.
51. Any of these equations are acceptable:
𝑦 = −3
2𝑥 + 12, 𝑦 = −
3
2(𝑥 − 8), 𝑦 − 12 = −
3
2𝑥, or 3x + 2y = 24
52. Answers will vary: Sample answer: (-8, 5) 53.
a. Any of these equations are acceptable
𝑦 − 6 =3
5(𝑥 − 4), 𝑦 − 3 =
3
5(𝑥 + 1), 𝑦 =
3
5𝑥 +
18
5, 3x – 5y = –18
b. (-6, 0) 54.
a. 2x – y = 6 b. 3x + y = 7 c. 4x + 3y = 27 d. 6x – 11y = 5
55.
a. 𝑦 =1
4𝑥 +
9
2
b. 𝑦 = 𝑥 + 10
c. 𝑦 =8
7𝑥 − 4
d. 𝑦 = −9
5𝑥 + 9
56. slope: 𝑚 =6
7
57. x = 2; as a point, it’s (2, 0) 58.
a. Any of these equations are acceptable 𝑦 − 3 = 2(𝑥 − 2), 𝑦 − 11 = 2(𝑥 − 6), 𝑦 = 2𝑥 − 1, 2𝑥 − 𝑦 = 1
b. Answers will vary; Sample Answer: (3, 5) 59.
a. 2x – y = 1 b. 4x + y = 6 c. 9x – 2y = -6 d. 10x + 3y = 21
60.
a. 𝑦 =3
7𝑥 +
12
7
b. 𝑦 = 2𝑥 + 1
c. 𝑦 = −2
9𝑥 + 4
d. 𝑦 =3
5𝑥 − 6
61. Any of these equations are acceptable 𝑦 = 4𝑥 + 16, 𝑦 = 4(𝑥 + 4), 𝑦 − 16 = 4𝑥, 4x – y = -16
62. Slope: 𝑚 = −5
6
63. Any of these equations are acceptable
𝑦 − 3 = −1
2(𝑥 + 4), 𝑦 + 2 = −
1
2(𝑥 − 6), 𝑦 = −
1
2𝑥 + 1, x + 2y = 2
64. a. y = 1.5 x + 20
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b.
c. The cost of renting a cart for 5 hours would be $27.50
d. It would take 8 hours for a cost of $32.
65. Multiple explanations are possible.
66. x = 8; as a point, it’s (8, 0)
67. 90
68. 75
69.
70. A and C
71. There is no pattern
72. 𝑦 =13
8𝑥 +
21
8
73. 82.5
60
68
76
84
92
100
15 24 33 42 51 60
Gra
de
Time Spent Studying (minutes)
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74. 40 minutes
75. y = -0.87x + 9.96
76. y= -5.2; extrapolation
77. 8:03
78. 7:07
79.
80. B
81. B and D
82. y= (-6/5.6)x + 59.6/5.6
83. 80.6
84. 34.6 minutes
85. 𝑦 = −1.10𝑥 + 14.08
Unit Review
1. b
2. c
3. c
4. d
5. b
6. a
7. b
8. c
9. a
10. d
11. b
12. c
13. a
14. e
15. c
16. b
17. a, c, d
18. a, c, f
19. a, b
20.
50
58
66
74
82
90
4.5 5.25 6 6.75 7.5 8.25 9
Heig
ht(
inch
es)
Size of Shoe
x y
0 5
1 3
2 1
3 -1
4 -3
5 -5
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21. ½
22. Any of these equations are acceptable:
y = ½ x – 1, x – 2y = 1
𝑦 − 4 =1
2(𝑥 − 10), 𝑦 + 3 =
1
2(𝑥 + 4)
23. 𝑚 =7
9
24. Answer can be any of these equations:
𝑦 =3
2𝑥 −
5
2, 3x – 2y = 5
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𝑦 − 2 =3
2(𝑥 − 3), 𝑦 − 5 =
3
2(𝑥 − 5)
25.
a. m = -3
b. 3x + y = 2
26.
a. m = 2
b. Answer can be any of these equations:
y = 2(x + 4), y – 8 = 2x,
y = 2x – 8, 2x – y = -8
27. Answer can be any of these equations:
y – 5 = x – 4
y = x + 1
x – y = -1
28. x = -5
29. Mile marker 300
30. Mile marker 160
31. 175 toys
32. 8 hours
33. y = 40x
34. y = 40x + 450
35. y = 5x
36. 8 hours
37. Ray and Jim are not correct. Miranda and Sasha are correct. Reasoning may vary.
38.
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a.
b. Harper: y = 3x
Ellie: 𝑦 = 2
3𝑥 + 3
Raina: 𝑦 = 3
5𝑥 + 2
c. Harper caught Ellie just after the two-second mark (2.143 seconds to be exact). They were
approximately 6.5 feet from the start (6.43 feet to be exact). Substitute the values back into the
equations.
d. If the race was 9 feet long, Harper would win.
e. Raina will never catch Ellie, because her rate of change is less.
39.
a. y = 10x
b. y = 4x
c. y = 2.50(10x+4x)
d. Kim would make $140.