# 3.2E: Exercises


## Practice Makes Perfect

Find the Slope of a Line

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

1.

$$m=\frac{2}{5}$$

2.

3.

$$m=\frac{5}{4}$$

4.

5.

$$m = -\frac{1}{3}$$

6.

7.

$$m = -\frac{5}{2}$$

8.

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

9. $$y=3$$

$$m = 0$$

10. $$y=−2$$

11. $$x=−5$$

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12. $$x=4$$

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

13. $$(2,5),\;(4,0)$$

$$m = -\frac{5}{2}$$

14. $$(3,6),\;(8,0)$$

15. $$(−3,3),\;(4,−5)$$

$$m = -\frac{8}{7}$$

16. $$(−2,4),\;(3,−1)$$

17. $$(−1,−2),\;(2,5)$$

$$m = \frac{7}{3}$$

18. $$(−2,−1),\;(6,5)$$

19. $$(4,−5),\;(1,−2)$$

$$m = -1$$

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. $$(2,5)$$; $$m=−\frac{1}{3}$$

22. $$(1,4)$$; $$m=−\frac{1}{2}$$

23. $$(−1,−4)$$; $$m=\frac{4}{3}$$

24. $$(−3,−5)$$; $$m=\frac{3}{2}$$

25. $$y$$-intercept: $$(0, 3)$$; $$m=−\frac{2}{5}$$

26. $$x$$-intercept: $$(−2,0)$$; $$m=\frac{3}{4}$$

27. $$(−4,2)$$; $$m=4$$

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. $$y=−7x+3$$

$$m=−7$$; $$(0,3)$$

30. $$y=4x−10$$

31. $$3x+y=5$$

$$m=−3$$; $$(0,5)$$

32. $$4x+y=8$$

33. $$6x+4y=12$$

$$m=−\frac{3}{2}$$; $$(0,3)$$

34. $$8x+3y=12$$

35. $$5x−2y=6$$

$$m=\frac{5}{2}$$; $$(0,−3)$$

36. $$7x−3y=9$$

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

37. $$y=3x−1$$

38. $$y=2x−3$$

39. $$y=−x+3$$

40. $$y=−x−4$$

41. $$y=−\frac{2}{5}x−3$$

42. $$y=−\frac{3}{5}x+2$$

43. $$3x−2y=4$$

44. $$3x−4y=8$$

Choose the Most Convenient Method to Graph a Line

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

45. $$x=2$$

vertical line

46. $$y=5$$

47. $$y=−3x+4$$

slope-intercept

48. $$x−y=5$$

49. $$x−y=1$$

intercepts

50. $$y=\frac{2}{3}x−1$$

51. $$3x−2y=−12$$

intercepts

52. $$2x−5y=−10$$

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.

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.

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.

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.

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.

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.

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.

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.

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. $$y=\frac{3}{4}x−3$$; $$3x−4y=−2$$

parallel

62. $$3x−4y=−2$$; $$y=\frac{3}{4}x−3$$

63. $$2x−4y=6$$; $$x−2y=3$$

neither

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

65. $$x=5$$; $$x=−6$$

parallel

66. $$x=−3$$; $$x=−2$$

67. $$4x−2y=5$$; $$3x+6y=8$$

perpendicular

68. $$8x−2y=7$$; $$3x+12y=9$$

69. $$3x−6y=12$$; $$6x−3y=3$$

neither

70. $$9x−5y=4$$; $$5x+9y=−1$$

71. $$7x−4y=8$$; $$4x+7y=14$$

perpendicular

72. $$5x−2y=11$$; $$5x−y=7$$

73. $$3x−2y=8$$; $$2x+3y=6$$

perpendicular

74. $$2x+3y=5$$; $$3x−2y=7$$

75. $$3x−2y=1$$; $$2x−3y=2$$

neither

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

77. $$y=2$$; $$y=6$$

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 $$m=2$$?

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

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