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

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

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

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

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

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

15. $(−3,3),\;(4,−5)$

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

16. $(−2,4),\;(3,−1)$

17. $(−1,−2),\;(2,5)$

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

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. $(2,5)$; $m=−\frac{1}{3}$

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. $(1,4)$; $m=−\frac{1}{2}$

23. $(−1,−4)$; $m=\frac{4}{3}$

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. $(−3,−5)$; $m=\frac{3}{2}$

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

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: $(−2,0)$; $m=\frac{3}{4}$

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. $y=4x−10$

31. $3x+y=5$

Answer: $m=−3$; $(0,5)$

32. $4x+y=8$

33. $6x+4y=12$

Answer: $m=−\frac{3}{2}$; $(0,3)$

34. $8x+3y=12$

35. $5x−2y=6$

Answer: $m=\frac{5}{2}$; $(0,−3)$

36. $7x−3y=9$

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

37. $y=3x−1$

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. $y=2x−3$

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. $y=−x−4$

41. $y=−\frac{2}{5}x−3$

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

42. $y=−\frac{3}{5}x+2$

43. $3x−2y=4$

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. $3x−4y=8$

Choose the Most Convenient Method to Graph a Line

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

45. $x=2$

Answer: vertical line

46. $y=5$

47. $y=−3x+4$

Answer: slope-intercept

48. $x−y=5$

49. $x−y=1$

Answer: intercepts

50. $y=\frac{2}{3}x−1$

51. $3x−2y=−12$

Answer: 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.

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. $y=\frac{3}{4}x−3$; $3x−4y=−2$

Answer: parallel

62. $3x−4y=−2$; $y=\frac{3}{4}x−3$

63. $2x−4y=6$; $x−2y=3$

Answer: neither

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

65. $x=5$; $x=−6$

Answer: parallel

66. $x=−3$; $x=−2$

67. $4x−2y=5$; $3x+6y=8$

Answer: perpendicular

68. $8x−2y=7$; $3x+12y=9$

69. $3x−6y=12$; $6x−3y=3$

Answer: neither

70. $9x−5y=4$; $5x+9y=−1$

71. $7x−4y=8$; $4x+7y=14$

Answer: perpendicular

72. $5x−2y=11$; $5x−y=7$

73. $3x−2y=8$; $2x+3y=6$

Answer: perpendicular

74. $2x+3y=5$; $3x−2y=7$

75. $3x−2y=1$; $2x−3y=2$

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?

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Practice Makes Perfect

Writing Exercises

Self Check