0.10e: Exercises - Find the Equation of a Line

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A: Construct a linear equation given a graph

Exercise $\PageIndex{A}$: Construct a linear equation given its graph

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope-intercept form.

1. $This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 5), (1, negative 2), and (2, 1).$	2. $This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 4), (1, 2), and (2, 0).$	3. $This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (2, negative 2), and (6, 0).$	4. $This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (4, 5), and (8, 8).$
5. $This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 3), (3, negative 1), and (6, negative 5).$	6. $This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 1), (2, negative 4), and (4, negative 7).$	7. $This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 2), (1, negative 2), and (2, negative 2).$	8. $This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 6), (1, 6), and (2, 6).$
9. $071A x=1.jpg$	10. $071A x=-4.jpg$

Answers to odd exercises.: 1. $y=3x−5$ 3. $y=\frac{1}{2}x−3$ 5. $y=−\frac{4}{3}x+3$ 7. $y=−2$ 9. $x=1$

B: Construct a linear equation given a point on the line and the slope

Exercise $\PageIndex{B}$: Construct a linear equation given attributes of the line

In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope-intercept form.

13. slope $3$ and $y$-intercept $(0,5)$	14. slope $8$ and $y$-intercept $(0,−6)$	15. slope $−3$ and $y$-intercept $(0,−1)$	16. slope $−1$ and $y$-intercept $(0,3)$
17. slope $\frac{1}{5}$ and $y$-intercept $(0,−5)$	18. slope $−\frac{3}{4}$ and $y$-intercept $(0,−2)$	19. slope $0$ and $y$-intercept $(0,−1)$	20. slope $−4$ and $y$-intercept $(0,0)$

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope-intercept form.

25. $m=\frac{5}{8}$, point $(8,3)$	26. $m=\frac{5}{6}$, point $(6,7)$	27. $m=−\frac{3}{5}$, point $(10,−5)$	28. $m=−\frac{3}{4}$, point $(8,−5)$
29. $m=−\frac{3}{2}$, point $(−4,−3)$	30. $m=−\frac{5}{2}$, point $(−8,−2)$	31. $m=−7$, point $(−1,−3)$	32. $m=−4$, point $(−2,−3)$
33. Horizontal line containing $(−2,5)$	34. Horizontal line containing $(−2,−3)$	35. Vertical line containing $(−1,−7)$	36. Vertical line containing $(4,−8)$

Answers to odd exercises.: 13. $y=3x+5$ 15. $y=−3x−1$ 17. $y=\frac{1}{5}x−5$ 19. $y=−1$
25. $y=\frac{5}{8}x−2$ 27. $y=−\frac{3}{5}x+1$ 29. $y=−\frac{3}{2}x−9$ 31. $y=−7x−10$ 33. $y=5$ 35. $x=−1$

C: Construct a linear equation given two points on the line

Exercise $\PageIndex{C}$: Construct a linear equation given two points on the line

In the following exercises, find the equation of a line containing the given points. Write the equation in slope-intercept form.

41. $(2,6)$ and $(5,3)$	42. $(4,3)$ and $(8,1)$	43. $(−3,−4)$ and $(5,−2)$	44. $(−5,−3)$ and $(4,−6)$
45. $(−1,3)$ and $(−6,−7)$	46. $(−2,8)$ and $(−4,−6)$	47. $(0,4)$ and $(2,−3)$	48. $(0,−2)$ and $(−5,−3)$
49. $(7,2)$ and $(7,−2)$	50. $(−2,1)$ and $(−2,−4)$	51. $(3,−4)$ and $(5,−4)$	52. $(−6,−3)$ and $(−1,−3)$

Answers to odd exercises.: 41. $y=−x+8$ 43. $y=\frac{1}{4}x−\frac{13}{4}$ 45. $y=2x+5$ 47. $y=−\frac{7}{2}x+4$ 49. $x=7$ 51. $y=−4$

D: Construct a linear equation given a point on the line and a parallel or perpendicular line

Exercise $\PageIndex{D}$: Construct a linear equation given attributes of the line

In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form.

55. line $y=4x+2$, point $(1,2)$	56. line $y=−3x−1$, $ \\ \quad \; $ point $(2,−3)$	57. line $2x−y=6$, point $(3,0)$	58. line $2x+3y=6$, point $(0,5)$
59. line $x=−4$, point $(−3,−5)$	60. line $x−2=0$, point $(1,−2)$	61. line $y=5$, point $(2,−2)$	62. line $y+2=0$, point $(3,−3)$

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form.

67. line $y=−2x+3$, $ \\ \quad \; $ point $(2,2)$	68. line $y=−x+5$, $ \\ \quad \; $ point $(3,3)$	69. line $y=\frac{3}{4}x−2$, $ \\ \quad \; $ point $(−3,4)$	70. line $y=\frac{2}{3}x−4$, $ \\ \quad \; $ point $(2,−4)$
71. line $2x−3y=8$, $ \\ \quad \; $ point $(4,−1)$	72. line $4x−3y=5$, $ \\ \quad \; $ point $(−3,2)$	73. line $2x+5y=6$, $ \\ \quad \; $ point $(0,0)$	74. line $4x+5y=−3$, $ \\ \quad \; $ point $(0,0)$
75. line $x=3$, point $(3,4)$	76. line $x=−5$, point $(1,−2)$	77. line $x=7$, point $(−3,−4)$	78. line $x=−1$, point $(−4,0)$
79. line $y−3=0$, $ \\ \quad \; $ point$(−2,−4)$	80. line $y−6=0$, $ \\ \quad \; $ point $(−5,−3)$	81. line $y$-axis, point $(3,4)$	82. line $y$-axis, point $(2,1)$

Answers to odd exercises.: 55. $y=4x−2$ 57. $y=2x−6$        59. $x=−3 \qquad$ 61. $ y=−2$
67. $y=\frac{1}{2}x+1$ 69. $y=−\frac{4}{3}x$ 71. $y=−\frac{3}{2}x+5$ 73. $y=\frac{5}{2}x$
75. $y=4 \qquad $ 77. $ y=−4 \qquad $        79. $ x=−2 \qquad $ 81. $ y=4$

E: Construct a linear equation (Mixed Practice)

Exercise $\PageIndex{E}$: Construct a linear equation given two points on the line

In the following exercises, find the equation of each line. Write the equation in slope-intercept form.

87. Containing the points $(4,3)$ and $(8,1)$	88. Containing the points $(−2,0)$ and $(−3,−2)$	89. $m=\frac{1}{6}$, containing point $(6,1)$	90. $m=\frac{5}{6}$, containing point $(6,7)$
91. Parallel to the line $4x+3y=6$, containing point $(0,−3)$	92. Parallel to the line $2x+3y=6$, containing point $(0,5)$	93. $m=−\frac{3}{4}$, containing point $(8,−5)$	94. $m=−\frac{3}{5}$, containing point $(10,−5)$
95. Perpendicular to the line $y−1=0$, point $(−2,6)$	96. Perpendicular to the line y-axis, point $(−6,2)$	97. Parallel to the line $x=−3$, containing point $(−2,−1)$	98. Parallel to the line $x=−4$, containing point $(−3,−5)$
99. Containing the points $(−3,−4)$ and $(2,−5)$	100. Containing the points $(−5,−3)$ and $(4,−6)$	101. Perpendicular to the line $x−2y=5$, point $(−2,2)$	102. Perpendicular to the line $4x+3y=1$, point $(0,0)$

Answers to odd exercises.: 87. $y=−\frac{1}{2}x+5$ 89. $y=\frac{1}{6}x$ 91. $y=−\frac{4}{3}x−3$ 93. $y=−\frac{3}{4}x+1$
95. $x=−2 \qquad $ 97. $x=−2 \quad $ 99. $y=−\frac{1}{5}x−\frac{23}{5}$ 101. $y=−2x−2$

08/17/2025

13. slope \(3\) and \(y\)-intercept \((0,5)\)	14. slope \(8\) and \(y\)-intercept \((0,−6)\)	15. slope \(−3\) and \(y\)-intercept \((0,−1)\)	16. slope \(−1\) and \(y\)-intercept \((0,3)\)
17. slope \(\frac{1}{5}\) and \(y\)-intercept \((0,−5)\)	18. slope \(−\frac{3}{4}\) and \(y\)-intercept \((0,−2)\)	19. slope \(0\) and \(y\)-intercept \((0,−1)\)	20. slope \(−4\) and \(y\)-intercept \((0,0)\)

25. \(m=\frac{5}{8}\), point \((8,3)\)	26. \(m=\frac{5}{6}\), point \((6,7)\)	27. \(m=−\frac{3}{5}\), point \((10,−5)\)	28. \(m=−\frac{3}{4}\), point \((8,−5)\)
29. \(m=−\frac{3}{2}\), point \((−4,−3)\)	30. \(m=−\frac{5}{2}\), point \((−8,−2)\)	31. \(m=−7\), point \((−1,−3)\)	32. \(m=−4\), point \((−2,−3)\)
33. Horizontal line containing \((−2,5)\)	34. Horizontal line containing \((−2,−3)\)	35. Vertical line containing \((−1,−7)\)	36. Vertical line containing \((4,−8)\)

41. \((2,6)\) and \((5,3)\)	42. \((4,3)\) and \((8,1)\)	43. \((−3,−4)\) and \((5,−2)\)	44. \((−5,−3)\) and \((4,−6)\)
45. \((−1,3)\) and \((−6,−7)\)	46. \((−2,8)\) and \((−4,−6)\)	47. \((0,4)\) and \((2,−3)\)	48. \((0,−2)\) and \((−5,−3)\)
49. \((7,2)\) and \((7,−2)\)	50. \((−2,1)\) and \((−2,−4)\)	51. \((3,−4)\) and \((5,−4)\)	52. \((−6,−3)\) and \((−1,−3)\)

55. line \(y=4x+2\), point \((1,2)\)	56. line \(y=−3x−1\), \( \\ \quad \; \) point \((2,−3)\)	57. line \(2x−y=6\), point \((3,0)\)	58. line \(2x+3y=6\), point \((0,5)\)
59. line \(x=−4\), point \((−3,−5)\)	60. line \(x−2=0\), point \((1,−2)\)	61. line \(y=5\), point \((2,−2)\)	62. line \(y+2=0\), point \((3,−3)\)

67. line \(y=−2x+3\), \( \\ \quad \; \) point \((2,2)\)	68. line \(y=−x+5\), \( \\ \quad \; \) point \((3,3)\)	69. line \(y=\frac{3}{4}x−2\), \( \\ \quad \; \) point \((−3,4)\)	70. line \(y=\frac{2}{3}x−4\), \( \\ \quad \; \) point \((2,−4)\)
71. line \(2x−3y=8\), \( \\ \quad \; \) point \((4,−1)\)	72. line \(4x−3y=5\), \( \\ \quad \; \) point \((−3,2)\)	73. line \(2x+5y=6\), \( \\ \quad \; \) point \((0,0)\)	74. line \(4x+5y=−3\), \( \\ \quad \; \) point \((0,0)\)
75. line \(x=3\), point \((3,4)\)	76. line \(x=−5\), point \((1,−2)\)	77. line \(x=7\), point \((−3,−4)\)	78. line \(x=−1\), point \((−4,0)\)
79. line \(y−3=0\), \( \\ \quad \; \) point\((−2,−4)\)	80. line \(y−6=0\), \( \\ \quad \; \) point \((−5,−3)\)	81. line \(y\)-axis, point \((3,4)\)	82. line \(y\)-axis, point \((2,1)\)

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87. Containing the points \((4,3)\) and \((8,1)\)	88. Containing the points \((−2,0)\) and \((−3,−2)\)	89. \(m=\frac{1}{6}\), containing point \((6,1)\)	90. \(m=\frac{5}{6}\), containing point \((6,7)\)
91. Parallel to the line \(4x+3y=6\), containing point \((0,−3)\)	92. Parallel to the line \(2x+3y=6\), containing point \((0,5)\)	93. \(m=−\frac{3}{4}\), containing point \((8,−5)\)	94. \(m=−\frac{3}{5}\), containing point \((10,−5)\)
95. Perpendicular to the line \(y−1=0\), point \((−2,6)\)	96. Perpendicular to the line y-axis, point \((−6,2)\)	97. Parallel to the line \(x=−3\), containing point \((−2,−1)\)	98. Parallel to the line \(x=−4\), containing point \((−3,−5)\)
99. Containing the points \((−3,−4)\) and \((2,−5)\)	100. Containing the points \((−5,−3)\) and \((4,−6)\)	101. Perpendicular to the line \(x−2y=5\), point \((−2,2)\)	102. Perpendicular to the line \(4x+3y=1\), point \((0,0)\)