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3.2E: Exercises

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Practice Makes Perfect

Find the Slope of a Line

In the following exercises, find the slope of each line shown.

1.
This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 4) and (5, negative 2).

Answer

m=25

2.
This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 5) and (2, negative 2).

3.
This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 1) and (4, 4).

Answer

m=54

4.
This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 2) and (3, 3).

5.
This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 2) and (3, 1).

Answer

m=13

6.
This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, negative 1) and (3, negative 3).

7.
This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 4) and (2, negative 1).

Answer

m=52

8.
This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 2) and (4, negative 1).

In the following exercises, find the slope of each line.

9. y=3

Answer

m=0

10. y=2

11. x=5

Answer

undefined

12. x=4

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

13. (2,5),(4,0)

Answer

m=52

14. (3,6),(8,0)

15. (3,3),(4,5)

Answer

m=87

16. (2,4),(3,1)

17. (1,2),(2,5)

Answer

m=73

18. (2,1),(6,5)

19. (4,5),(1,2)

Answer

m=1

20. (3,6),(2,2)

Graph a Line Given a Point and the Slope

In the following exercises, graph each line with the given point and slope.

21. (2,5); m=13

Answer

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (2, 5) and (5, 4).

22. (1,4); m=12

23. (1,4); m=43

Answer

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (negative 1, negative 4) and (2, 0).

24. (3,5); m=32

25. y-intercept: (0,3); m=25

Answer

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, 3) and (5, 1).

26. x-intercept: (2,0); m=34

27. (4,2); m=4

Answer

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (negative 4, 2) and (negative 3, 6).

28. (1,5); m=3

Graph a Line Using Its Slope and Intercept

In the following exercises, identify the slope and y-intercept of each line.

29. y=7x+3

Answer

m=7; (0,3)

30. y=4x10

31. 3x+y=5

Answer

m=3; (0,5)

32. 4x+y=8

33. 6x+4y=12

Answer

m=32; (0,3)

34. 8x+3y=12

35. 5x2y=6

Answer

m=52; (0,3)

36. 7x3y=9

In the following exercises, graph the line of each equation using its slope and y-intercept.

37. y=3x1

Answer

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 1) and (1, 2).

38. y=2x3

39. y=x+3

Answer

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, 3) and (1, 2).

40. y=x4

41. y=25x3

Answer

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 3) and (5, negative 5).

42. y=35x+2

43. 3x2y=4

Answer

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 2) and (2, 1).

44. 3x4y=8

Choose the Most Convenient Method to Graph a Line

In the following exercises, determine the most convenient method to graph each line.

45. x=2

Answer

vertical line

46. y=5

47. y=3x+4

Answer

slope-intercept

48. xy=5

49. xy=1

Answer

intercepts

50. y=23x1

51. 3x2y=12

Answer

intercepts

52. 2x5y=10

Graph and Interpret Applications of Slope–Intercept

53. The equation P=31+1.75w models the relation between the amount of Tuyet’s monthly water bill payment, P, in dollars, and the number of units of water, w, used.

a. Find Tuyet’s payment for a month when 0 units of water are used.

b. Find Tuyet’s payment for a month when 12 units of water are used.

c. Interpret the slope and P-intercept of the equation.

d. Graph the equation.

Answer

a. $31
b. $52
c. The slope, 1.75, means that the payment, P, increases by $1.75 when the number of units of water used, w, increases by 1. The P-intercept means that when the number units of water Tuyet used is 0, the payment is $31.
d.

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 1 to 21. The y-axis runs from negative 1 to 80. The line goes through the points (0, 31) and (12, 52).

54. The equation P=28+2.54w models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used.

a. Find the payment for a month when Randy used 0 units of water.

b. Find the payment for a month when Randy used 15 units of water.

c. Interpret the slope and P-intercept of the equation.

d. Graph the equation.

55. Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day.

a. Find the amount Bruce is reimbursed on a day when he drives 0 miles.

b. Find the amount Bruce is reimbursed on a day when he drives 220 miles.

c. Interpret the slope and R-intercept of the equation.

d. Graph the equation.

Answer

a. $42
b. $168.50
c. The slope, 0.575 means that the amount he is reimbursed, R, increases by $0.575 when the number of miles driven, m, increases by 1. The R-intercept means that when the number miles driven is 0, the amount reimbursed is $42.
d.

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 50 to 250. The y-axis runs from negative 50 to 300. The line goes through the points (0, 42) and (220, 168.5).

56. Janelle is planning to rent a car while on vacation. The equation C=0.32m+15 models the relation between the cost in dollars, C, per day and the number of miles, m, she drives in one day.

a. Find the cost if Janelle drives the car 0 miles one day.

b. Find the cost on a day when Janelle drives the car 400 miles.

c. Interpret the slope and C-intercept of the equation.

d. Graph the equation.

57. Cherie works in retail and her weekly salary includes commission for the amount she sells. The equation S=400+0.15c models the relation between her weekly salary, S, in dollars and the amount of her sales, c, in dollars.

a. Find Cherie’s salary for a week when her sales were $0.

b. Find Cherie’s salary for a week when her sales were $3,600.

c. Interpret the slope and S-intercept of the equation.

d. Graph the equation.

Answer

a. $400
b. $940
c. The slope, 0.15, means that Cherie’s salary, S, increases by $0.15 for every $1 increase in her sales. The S-intercept means that when her sales are $0, her salary is $400.
d.

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 500 to 3500. The y-axis runs from negative 200 to 1000. The line goes through the points (0, 400) and (3600, 940).

58. Patel’s weekly salary includes a base pay plus commission on his sales. The equation S=750+0.09c models the relation between his weekly salary, S, in dollars and the amount of his sales, c, in dollars.

a. Find Patel’s salary for a week when his sales were 0.

b. Find Patel’s salary for a week when his sales were 18,540.

c. Interpret the slope and S-intercept of the equation.

d. Graph the equation.

59. Costa is planning a lunch banquet. The equation C=450+28g models the relation between the cost in dollars, C, of the banquet and the number of guests, g.

a. Find the cost if the number of guests is 40.

b. Find the cost if the number of guests is 80.

c. Interpret the slope and C-intercept of the equation.

d. Graph the equation.

Answer

a. $1570
b. $5690
c. The slope gives the cost per guest. The slope, 28, means that the cost, C, increases by $28 when the number of guests increases by 1. The C-intercept means that if the number of guests was 0, the cost would be $450.
d.

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 20 to 100. The y-axis runs from negative 1000 to 7000. The line goes through the points (0, 450) and (40, 1570).

60. Margie is planning a dinner banquet. The equation C=750+42g models the relation between the cost in dollars, C, of the banquet and the number of guests, g.

a. Find the cost if the number of guests is 50.

b. Find the cost if the number of guests is 100.

c. Interpret the slope and C-intercept of the equation.

d. Graph the equation.

Use Slopes to Identify Parallel and Perpendicular Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.

61. y=34x3; 3x4y=2

Answer

parallel

62. 3x4y=2; y=34x3

63. 2x4y=6; x2y=3

Answer

neither

64. 8x+6y=6; 12x+9y=12

65. x=5; x=6

Answer

parallel

66. x=3; x=2

67. 4x2y=5; 3x+6y=8

Answer

perpendicular

68. 8x2y=7; 3x+12y=9

69. 3x6y=12; 6x3y=3

Answer

neither

70. 9x5y=4; 5x+9y=1

71. 7x4y=8; 4x+7y=14

Answer

perpendicular

72. 5x2y=11; 5xy=7

73. 3x2y=8; 2x+3y=6

Answer

perpendicular

74. 2x+3y=5; 3x2y=7

75. 3x2y=1; 2x3y=2

Answer

neither

76. 2x+4y=3; 6x+3y=2

77. y=2; y=6

Answer

parallel

78. y=1; y=2

Writing Exercises

79. How does the graph of a line with slope m=12 differ from the graph of a line with slope m=2?

Answer

Answers will vary.

80. Why is the slope of a vertical line “undefined”?

81. Explain how you can graph a line given a point and its slope.

Answer

Answers will vary.

82. Explain in your own words how to decide which method to use to graph a line.

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 7 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “find the slope of a line”, “graph a line given a point and the slope”, “graph a line using its slope and intercept”, “choose the most convenient method to graph a line”, “graph and interpret applications of slope-intercept”, and “use slopes to identify parallel and perpendicular lines”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

b. After reviewing this checklist, what will you do to become confident for all objectives?


This page titled 3.2E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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