Exercises: Find the Equation of a Line (3.11)

Exercises: Find an Equation of the Line Given the Slope and [latex]{\color{White}{y}}[/latex]-Intercept

Instructions: For questions 1-16, find the equation of a line with given slope and [latex]y[/latex]-intercept. Write the equation in slope–intercept form.

1. slope [latex]3[/latex] and [latex]y[/latex]-intercept [latex](0,5)[/latex]
2. slope [latex]4[/latex] and [latex]y[/latex]-intercept [latex](0,1)[/latex]
Solution

[latex]y=4x+1[/latex]


3. slope [latex]6[/latex] and [latex]y[/latex]-intercept [latex](0,-4)[/latex]
4. slope [latex]8[/latex] and [latex]y[/latex]-intercept [latex](0,-6)[/latex]
Solution

[latex]y=8x-6[/latex]

5. slope [latex]-1[/latex] and [latex]y[/latex]-intercept [latex](0,3)[/latex]
6. slope [latex]-1[/latex] and [latex]y[/latex]-intercept [latex](0,7)[/latex]
Solution

[latex]y=-x+7[/latex]

7. slope [latex]-2[/latex] and [latex]y[/latex]-intercept [latex](0,-3)[/latex]
8. slope [latex]-3[/latex] and [latex]y[/latex]-intercept [latex](0,-1)[/latex]
Solution

[latex]y=-3x-1[/latex]

9. slope [latex]\frac{3}{5}[/latex] and [latex]y[/latex]-intercept [latex](0,-1)[/latex]
10. slope [latex]\frac{1}{5}[/latex] and [latex]y[/latex]-intercept [latex](0,-5)[/latex]
Solution

[latex]y=\frac{1}{5}x-5[/latex]

11. slope [latex]-\frac{3}{4}[/latex] and [latex]y[/latex]-intercept [latex](0,-2)[/latex]
12. slope [latex]-\frac{2}{3}[/latex] and [latex]y[/latex]-intercept [latex](0,-3)[/latex]
Solution

[latex]y=-\frac{2}{3}x-3[/latex]

13. slope [latex]0[/latex] and [latex]y[/latex]-intercept [latex](0,-1)[/latex]
14. slope [latex]0[/latex] and [latex]y[/latex]-intercept [latex](0,2)[/latex]
Solution

[latex]y=2[/latex]


15. slope [latex]-3[/latex] and [latex]y[/latex]-intercept [latex](0,0)[/latex]

16. slope [latex]-4[/latex] and [latex]y[/latex]-intercept [latex](0,0)[/latex]
Solution

[latex]y=-4x[/latex]

Exercises: Find the Equation of a Line Shown on a Graph

Instructions: For questions 17-24, find the equation of the line shown in each graph. Write the equation in slope–intercept form.

17.

The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (1, negative 2) is plotted. A line intercepts the y-axis at (0, negative 5), passes through the point (1, negative 2), and intercepts the x-axis at (5 thirds, 0).
Figure 3P.11.1

18.

The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, 0) is plotted. A line intercepts the y-axis at (0, 4) and intercepts the x-axis at (2, 0).
Figure 3P.11.2
Solution

[latex]y=-2x+4[/latex]

19.

The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (6, 0) is plotted. A line intercepts the y-axis at (0, negative 3) and intercepts the x-axis at (6, 0).
Figure 3P.11.3

20.

The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (4, 5) is plotted. A line intercepts the x-axis at (negative 8 thirds, 0), intercepts the y-axis at (0, 2), and passes through the point (4, 5).
Figure 3P.11.4
Solution

[latex]y=\frac{3}{4}x+2[/latex]

21.

The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (3, negative 1) is plotted. A line intercepts the y-axis at (0, 2), intercepts the x-axis at (9 fourths, 0), and passes through the point (3, negative 1).
Figure 3P.11.5

22.

The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, negative 4) is plotted. A line intercepts the x-axis at (negative 2 thirds, 0), intercepts the y-axis at (0, negative 1), and passes through the point (2, negative 4).
Figure 3P.11.6
Solution

[latex]y=-\frac{3}{2}x-1[/latex]

23.

The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (2, negative 2) is plotted. A line running parallel to the x-axis intercepts the y-axis at (0, negative 2) and passes through the point (2, negative 2).
Figure 3P.11.7

24.

The graph shows the x y-coordinate plane. The x and y-axes each run from negative 9 to 9. The point (negative 3, 6) is plotted. A line running parallel to the x-axis passes through (negative 3, 6) and intercepts the y-axis at (0, 6).
Figure 3P.11.8
Solution

[latex]y=6[/latex]

Exercises: Find an Equation of the Line Given the Slope and a Point

Instructions: For questions 25-42, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

25. [latex]m=\frac{5}{8}[/latex], point [latex](8,3)[/latex]
26. [latex]m=\frac{3}{8}[/latex], point [latex](8,2)[/latex]
Solution

[latex]y=\frac{3}{8}x-1[/latex]

27. [latex]m=\frac{1}{6}[/latex], point [latex](6,1)[/latex]
28. [latex]m=\frac{5}{6}[/latex], point [latex](6,7)[/latex]
Solution

[latex]y=\frac{5}{6}x+2[/latex]

29. [latex]m=-\frac{3}{4}[/latex], point [latex](8,-5)[/latex]
30. [latex]m=-\frac{3}{5}[/latex], point [latex](10,-5)[/latex]
Solution

[latex]y=-\frac{3}{5}x+1[/latex]

31. [latex]m=-\frac{1}{4}[/latex], point [latex](-12,-6)[/latex]
32. [latex]m=-\frac{1}{3}[/latex], point [latex](-9,-8)[/latex]
Solution

[latex]y=-\frac{1}{3}x-11[/latex]

33. Horizontal line containing [latex](-2,5)[/latex]
34. Horizontal line containing [latex](-1,4)[/latex]
Solution

[latex]y=4[/latex]

35. Horizontal line containing [latex]\left(-2,-3\right)[/latex]
36. Horizontal line containing [latex](-1,-7)[/latex]
Solution

[latex]y=-7[/latex]

37. [latex]m=-\frac{3}{2}[/latex], point [latex](-4,-3)[/latex]
38. [latex]m=-\frac{5}{2}[/latex], point [latex](-8,-2)[/latex]
Solution

[latex]y=-\frac{5}{2}x-22[/latex]

39. [latex]m=-7[/latex], point [latex](-1,-3)[/latex]

40. [latex]m=-4[/latex], point [latex](-2,-3)[/latex]
Solution

[latex]y=-4x-11[/latex]

41. Horizontal line containing [latex](2,-3)[/latex]
42. Horizontal line containing [latex](4,-8)[/latex]
Solution

[latex]y=-8[/latex]

Exercises: Find an Equation of the Line Given Two Points

Instructions: For questions 43-68, find the equation of a line containing the given points. Write the equation in slope–intercept form.

43. [latex]\left(2,6\right)[/latex] and [latex]\left(5,3\right)[/latex]
44. [latex]\left(3,1\right)[/latex] and [latex]\left(2,5\right)[/latex]
Solution

[latex]y=-4x+13[/latex]

45. [latex]\left(4,3\right)[/latex] and [latex]\left(8,1\right)[/latex]
46. [latex]\left(2,7\right)[/latex] and [latex]\left(3,8\right)[/latex]
Solution

[latex]y=x+5[/latex]

47. [latex]\left(-3,-4\right)[/latex] and [latex]\left(5-2\right)[/latex]
48. [latex]\left(-5,-3\right)[/latex] and [latex]\left(4,-6\right)[/latex]
Solution

[latex]y=-\frac{1}{3}x-\frac{14}{3}[/latex]

49. [latex]\left(-1,3\right)[/latex] and [latex]\left(-6,-7\right)[/latex]
50. [latex]\left(-2,8\right)[/latex] and [latex]\left(-4,-6\right)[/latex]
Solution

[latex]y=7x+22[/latex]

51. [latex]\left(6,-4\right)[/latex] and [latex]\left(-2,5\right)[/latex]
52. [latex]\left(3,-2\right)[/latex] and [latex]\left(-4,4\right)[/latex]
Solution

[latex]y=-\frac{6}{7}x+\frac{4}{7}[/latex]

53. [latex]\left(0,4\right)[/latex] and [latex]\left(2,-3\right)[/latex]
54. [latex]\left(0,-2\right)[/latex] and [latex]\left(-5,-3\right)[/latex]
Solution

[latex]y=\frac{1}{5}x-2[/latex]

55. [latex]\left(7,2\right)[/latex] and [latex]\left(7,-2\right)[/latex]
56. [latex]\left(4,2\right)[/latex] and [latex]\left(4,-3\right)[/latex]
Solution

[latex]x=4[/latex]

57. [latex]\left(-7,-1\right)[/latex] and [latex]\left(-7,-4\right)[/latex]
58. [latex]\left(-2,1\right)[/latex] and [latex]\left(-2,-4\right)[/latex]
Solution

[latex]x=-2[/latex]

59. [latex]\left(6,1\right)[/latex] and [latex]\left(0,1\right)[/latex]
60. [latex]\left(6,2\right)[/latex] and [latex]\left(-3,2\right)[/latex]
Solution

[latex]y=2[/latex]

61. [latex]\left(3,-4\right)[/latex] and [latex]\left(5,-4\right)[/latex]
62. [latex]\left(-6,-3\right)[/latex] and [latex]\left(-1,-3\right)[/latex]
Solution

[latex]y=-3[/latex]

63. [latex]\left(4,3\right)[/latex] and [latex]\left(8,0\right)[/latex]
64. [latex]\left(0,0\right)[/latex] and [latex]\left(1,4\right)[/latex]
Solution

[latex]y=4x[/latex]

65. [latex]\left(-2,-3\right)[/latex] and [latex]\left(-5,-6\right)[/latex]
66. [latex]\left(-3,0\right)[/latex] and [latex]\left(-7,-2\right)[/latex]
Solution

[latex]y=\frac{1}{2}x+\frac{3}{2}[/latex]

67. [latex]\left(8,-1\right)[/latex] and [latex]\left(8,-5\right)[/latex]
68. [latex]\left(3,5\right)[/latex] and [latex]\left(-7,5\right)[/latex]
Solution

[latex]y=5[/latex]

Exercises: Find an Equation of a Line Parallel to a Given Line

Instructions: For questions 69-84, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.

69. line [latex]y=4x+2[/latex], point [latex]\left(1,2\right)[/latex]
70. line [latex]y=3x+4[/latex], point [latex]\left(2,5\right)[/latex]
Solution

[latex]y=3x-1[/latex]

71. line [latex]y=-2x-3[/latex], point [latex]\left(-1,3\right)[/latex]
72. line [latex]y=-3x-1[/latex], point [latex]\left(2,-3\right)[/latex]
Solution

[latex]y=-3x+3[/latex]

73. line [latex]3x-y=4[/latex], point [latex]\left(3,1\right)[/latex]
74. line [latex]2x-y=6[/latex], point [latex]\left(3,0\right)[/latex]
Solution

[latex]y=2x-6[/latex]

75. line [latex]4x+3y=6[/latex], point [latex]\left(0,-3\right)[/latex]
76. line [latex]2x+3y=6[/latex], point [latex]\left(0,5\right)[/latex]
Solution

[latex]y=-\frac{2}{3}x+5[/latex]

77. line [latex]x=-3[/latex], point [latex]\left(-2,-1\right)[/latex]
78. line [latex]x=-4[/latex], point [latex]\left(-3,-5\right)[/latex]
Solution

[latex]x=-3[/latex]

79. line [latex]x-2=0[/latex], point [latex]\left(1,-2\right)[/latex]

80. line [latex]x-6=0[/latex], point [latex]\left(4,-3\right)[/latex]
Solution

[latex]x=4[/latex]

81. line [latex]y=5[/latex], point [latex]\left(2,-2\right)[/latex]
82. line [latex]y=1[/latex], point [latex]\left(3,-4\right)[/latex]
Solution

[latex]y=-4[/latex]

83. line [latex]y+2=0[/latex], point [latex]\left(3,-3\right)[/latex]
84. line [latex]y+7=0[/latex], point [latex]\left(1,-1\right)[/latex]
Solution

[latex]y=-1[/latex]

Exercises: Find an Equation of a Line Perpendicular to a Given Line

Instructions: For questions 85-96, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.


85. line [latex]y=-2x+3[/latex], point [latex]\left(2,2\right)[/latex]
86. line [latex]y=-x+5[/latex], point [latex]\left(3,3\right)[/latex]
Solution

[latex]y=x[/latex]

87. line [latex]y=\frac{3}{4}x-2[/latex], point [latex]\left(-3,4\right)[/latex]
88. line [latex]y=\frac{2}{3}x-4[/latex], point [latex]\left(2,-4\right)[/latex]
Solution

[latex]y=-\frac{3}{2}x-1[/latex]

89. line [latex]2x-3y=8[/latex], point [latex]\left(4,-1\right)[/latex]
90. line [latex]4x-3y=5[/latex], point [latex]\left(-3,2\right)[/latex]
Solution

[latex]y=-\frac{3}{4}x-\frac{1}{4}[/latex]

91. line [latex]2x+5y=6[/latex], point [latex]\left(0,0\right)[/latex]
92. line [latex]4x+5y=-3[/latex], point [latex]\left(0,0\right)[/latex]
Solution

[latex]y=\frac{5}{4}x[/latex]

93. line [latex]y-3=0[/latex], point [latex]\left(-2,-4\right)[/latex]
94. line [latex]y-6=0[/latex], point [latex]\left(-5,-3\right)[/latex]
Solution

[latex]x=-5[/latex]

95. line [latex]y[/latex]-axis, point [latex]\left(3,4\right)[/latex]
96. line [latex]y[/latex]-axis, point [latex]\left(2,1\right)[/latex]
Solution

[latex]y=1[/latex]

Exercises: Mixed Practice

Instructions: For questions 97-114, find the equation of each line. Write the equation in slope–intercept form.

97. Containing the points [latex]\left(4,3\right)[/latex] and [latex]\left(8,1\right)[/latex]

98. Containing the points [latex]\left(2,7\right)[/latex] and [latex]\left(3,8\right)[/latex]
Solution

[latex]y=x+5[/latex]


99. [latex]m=\frac{1}{6}[/latex], containing point [latex]\left(6,1\right)[/latex]
100. [latex]m=\frac{5}{6}[/latex], containing point [latex]\left(6,7\right)[/latex]
Solution

[latex]y=\frac{5}{6}x+2[/latex]

101. Parallel to the line [latex]4x+3y=6[/latex], containing point [latex]\left(0,-3\right)[/latex]
102. Parallel to the line [latex]2x+3y=6[/latex], containing point [latex]\left(0,5\right)[/latex]
Solution

[latex]y=-\frac{2}{3}x+5[/latex]

103. [latex]m=-\frac{3}{4}[/latex], containing point [latex]\left(8,-5\right)[/latex]
104. [latex]m=-\frac{3}{5}[/latex], containing point [latex]\left(10,-5\right)[/latex]
Solution

[latex]y=-\frac{3}{5}x+1[/latex]

105. Perpendicular to the line [latex]y-1=0[/latex], point [latex]\left(-2,6\right)[/latex]
106. Perpendicular to the line [latex]y[/latex]-axis, point [latex]\left(-6,2\right)[/latex]
Solution

[latex]y=2[/latex]


107. Containing the points [latex]\left(4,3\right)[/latex] and [latex]\left(8,1\right)[/latex]
108. Containing the points [latex]\left(-2,0\right)[/latex] and [latex]\left(-3,-2\right)[/latex]
Solution

[latex]y=x+2[/latex]

109. Parallel to the line [latex]x=-3[/latex], containing point [latex]\left(-2,-1\right)[/latex]
110. Parallel to the line [latex]x=-4[/latex], containing point [latex]\left(-3,-5\right)[/latex]
Solution

[latex]x=-3[/latex]

111. Containing the points [latex]\left(-3,-4\right)[/latex] and [latex]\left(2,-5\right)[/latex]
112. Containing the points [latex]\left(-5,-3\right)[/latex] and [latex]\left(4,-6\right)[/latex]
Solution

[latex]y=-\frac{1}{3}x-\frac{14}{3}[/latex]

113. Perpendicular to the line [latex]x-2y=5[/latex], containing point [latex]\left(-2,2\right)[/latex]
114. Perpendicular to the line [latex]4x+3y=1[/latex], containing point [latex]\left(0,0\right)[/latex]
Solution

[latex]y=\frac{3}{4}x[/latex]

Exercises: Everyday Math

Instructions: For questions 115-116, answer the given everyday math word problems.
115. Cholesterol. The age, [latex]x[/latex], and LDL cholesterol level, [latex]y[/latex], of two men are given by the points [latex]\left(18,68\right)[/latex] and [latex]\left(27,122\right)[/latex]. Find a linear equation that models the relationship between age and LDL cholesterol level.
116. Fuel consumption. The city mpg, [latex]x[/latex], and highway mpg, [latex]y[/latex], of two cars are given by the points [latex]\left(29,40\right)[/latex] and[latex]\left(19,28\right)[/latex]. Find a linear equation that models the relationship between city mpg and highway mpg.
Solution

[latex]y=1.2x+5.2[/latex]

Exercises: Writing Exercises

Instructions: For questions 117-118, answer the given writing exercises.
117. Why are all horizontal lines parallel?
118. Explain in your own words why the slopes of two perpendicular lines must have opposite signs.
Solution

Answers will vary.

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Fanshawe Pre-Health Sciences Mathematics 1 Copyright © 2022 by Domenic Spilotro, MSc is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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