# Exercises: Graph Linear Equations in Two Variables (3.9)

## Exercises: Plot Points in a Rectangular Coordinate System

Instructions: For questions 1-8, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.

1.

a.$(-4,2)$
b.$(-1,-2)$
c.$(3,-5)$
d.$(-3,5)$
e.$\left(\frac{5}{3},2\right)$

Solution

2.

a.$(-2,-3)$
b.$(3,-3)$
c.$(-4,1)$
d.$(4,-1)$
e.$\left(\frac{3}{2},1\right)$

3.

a.$(3,-1)$
b.$(-3,1)$
c.$(-2,2)$
d.$(-4,-3)$
e.$\left(1,\frac{14}{5}\right)$

Solution

4.

a.$(-1,1)$
b.$(-2,-1)$
c.$(2,1)$
d.$(1,-4)$
e.$\left(3,\frac{7}{2}\right)$

5.

a.$\left(-2,0\right)$
b.$\left(-3,0\right)$
c.$\left(0,0\right)$
d.$\left(0,4\right)$
e.$\left(0,2\right)$

Solution

6.

a.$\left(0,1\right)$
b.$\left(0,-4\right)$
c.$\left(-1,0\right)$
d.$\left(0,0\right)$
e.$\left(5,0\right)$

7.

a.$\left(0,0\right)$
b.$\left(0,-3\right)$
c.$\left(-4,0\right)$
d.$\left(1,0\right)$
e.$\left(0,-2\right)$

Solution

8.

a.$\left(-3,0\right)$
b.$\left(0,5\right)$
c.$\left(0,-2\right)$
d.$\left(2,0\right)$
e.$\left(0,0\right)$

## Exercises: Name Ordered Pairs in a Rectangular Coordinate System

Instructions: For questions 9-12, name the ordered pair of each point shown in the rectangular coordinate system.

9.

Solution

A: $(-4,1)$
B: $(-3,-4)$
C: $(1,-3)$
D: $(4,3)$

10.

11.

Solution

A: $(0,-2)$
B: $(-2,0)$
C: $(0,5)$
D: $(5,0)$

12.

## Exercises: Verify Solutions to an Equation in Two Variables

Instructions: For questions 13-20, which ordered pairs are solutions to the given equations?

13. $2x+y=6$

a.$(1,4)$
b.$(3,0)$
c.$(2,3)$

Solution

a, b

14. $x+3y=9$

a.$(0,3)$
b.$(6,1)$
c.$(-3,-3)$

15. $4x-2y=8$

a.$(3,2)$
b.$(1,4)$
c.$(0,-4)$

Solution

a, c

16. $3x-2y=12$

a.$(4,0)$
b.$(2,-3)$
c.$(1,6)$

17. $y=4x+3$

a.$(4,3)$
b.$(-1,-1)$
c.$\left(\frac{1}{2},5\right)$

Solution

b, c

18. $y=2x-5$

a.$(0,-5)$
b.$(2,1)$
c.$\left(\frac{1}{2},-4\right)$

19. $y=\frac{1}{2}x-1$

a.$(2,0)$
b.$(-6,-4)$
c.$(-4,-1)$

Solution

a, b

20. $y=\frac{1}{3}x+1$

a.$(-3,0)$
b.$(9,4)$
c.$(-6,-1)$

## Exercises: Complete a Table of Solutions to a Linear Equation

Instructions: For questions 21-32, complete the table to find solutions to each linear equation.

21. $y=2x-4$

 $x$ $y$ $(x,y)$ $0$ $2$ $-1$
Solution
 $x$ $y$ $\left(x,y\right)$ $0$ $-4$ $\left(0,-4\right)$ $2$ $0$ $\left(2,0\right)$ $-1$ $-6$ $\left(-1,-6\right)$

22. $y=3x-1$

 $x$ $y$ $\left(x,y\right)$ $0$ $2$ $-1$

23. $y=-x+5$

 $x$ $y$ $\left(x,y\right)$ $0$ $3$ $-2$
Solution
 $x$ $y$ $\left(x,y\right)$ $0$ $5$ $\left(0,5\right)$ $3$ $2$ $\left(3,2\right)$ $-2$ $7$ $\left(-2,7\right)$

24. $y=-x+2$

 $x$ $y$ $\left(x,y\right)$ $0$ $3$ $-2$

25. $y=\frac{1}{3}x+1$

 $x$ $y$ $\left(x,y\right)$ $0$ $3$ $6$
Solution
 $x$ $y$ $\left(x,y\right)$ $0$ $1$ $\left(0,1\right)$ $3$ $2$ $\left(3,2\right)$ $6$ $3$ $\left(6,3\right)$

26. $y=\frac{1}{2}x+4$

 $x$ $y$ $\left(x,y\right)$ $0$ $2$ $4$

27. $y=-\frac{3}{2}x-2$

 $x$ $y$ $\left(x,y\right)$ $0$ $2$ $-2$
Solution
 $x$ $y$ $\left(x,y\right)$ $0$ $-2$ $\left(0,-2\right)$ $2$ $-5$ $\left(2,-5\right)$ $-2$ $1$ $\left(-2,1\right)$

28. $y=-\frac{2}{3}x-1$

 $x$ $y$ $\left(x,y\right)$ $0$ $3$ $-3$

29. $x+3y=6$

 $x$ $y$ $\left(x,y\right)$ $0$ $3$ $0$
Solution
 $x$ $y$ $\left(x,y\right)$ $0$ $2$ $\left(0,2\right)$ $3$ $4$ $\left(3,1\right)$ $6$ $0$ $\left(6,0\right)$

30. $x+2y=8$

 $x$ $y$ $\left(x,y\right)$ $0$ $4$ $0$

31. $2x-5y=10$

 $x$ $y$ $\left(x,y\right)$ $0$ $10$ $0$
Solution
 $x$ $y$ $\left(x,y\right)$ $0$ $-2$ $\left(0,-2\right)$ $10$ $2$ $\left(10,2\right)$ $5$ $0$ $\left(5,0\right)$

32. $3x-4y=12$

 $x$ $y$ $\left(x,y\right)$ $0$ $8$ $0$

## Exercises: Find Solutions to a Linear Equation

Instructions: For questions 33-48, find three solutions to each linear equation.

33. $y=5x-8$

Solution

34. $y=3x-9$

35. $y=-4x+5$
Solution

36. $y=-2x+7$

37. $x+y=8$
Solution

38. $x+y=6$

39. $x+y=-2$
Solution

40. $x+y=-1$

41. $3x+y=5$
Solution

42. $2x+y=3$

43. $4x-y=8$
Solution

44. $5x-y=10$

45. $2x+4y=8$
Solution

46. $3x+2y=6$

47. $5x-2y=10$
Solution

48. $4x-3y=12$

## Exercises: Recognize the Relationship Between the Solutions of an Equation and its Graph

Instructions: For questions 49-52, for each ordered pair, decide:

a. Is the ordered pair a solution to the equation?
b. Is the point on the line?

49. $y=x+2$

a.$\left(0,2\right)$
b.$\left(1,2\right)$
c.$\left(-1,1\right)$
d.$\left(-3,-1\right)$

Solution

a. yes; no
b. no; no
c. yes; yes
d. yes; yes

50. $y=x-4$

a.$\left(0,-4\right)$
b.$\left(3,-1\right)$
c.$\left(2,2\right)$
d.$\left(1,-5\right)$

51. $y=\frac{1}{2}x-3$

a.$\left(0,-3\right)$
b.$\left(2,-2\right)$
c.$\left(-2,-4\right)$
d.$\left(4,1\right)$

Solution

a. yes; yes
b. yes; yes
c. yes; yes
d. no; no

52. $y=\frac{1}{3}x+2$

a.$\left(0,2\right)$
b.$\left(3,3\right)$
c.$\left(-3,2\right)$
d.$\left(-6,0\right)$

## Exercises: Graph a Linear Equation by Plotting Points

Instructions: For questions 53-96, graph by plotting points.

53. $y=3x-1$

Solution

54. $y=2x+3$

55. $y=-2x+2$
Solution

56. $y=-3x+1$

57. $y=x+2$
Solution

58. $y=x-3$

59. $y=-x-3$
Solution

60. $y=-x-2$

61. $y=2x$
Solution

62. $y=3x$

63. $y=-4x$
Solution

64. $y=-2x$

65. $y=\frac{1}{2}x+2$
Solution

66. $y=\frac{1}{3}x-1$

67. $y=\frac{4}{3}x-5$
Solution

68. $y=\frac{3}{2}x-3$

69. $y=-\frac{2}{5}x+1$
Solution

70. $y=-\frac{4}{5}x-1$

71. $y=-\frac{3}{2}x+2$
Solution

72. $y=-\frac{5}{3}x+4$

73. $x+y=6$
Solution

74. $x+y=4$

75. $x+y=-3$
Solution

76. $x+y=-2$

77. $x-y=2$
Solution

78. $x-y=1$

79. $x-y=-1$
Solution

80. $x-y=-3$

81. $3x+y=7$
Solution

82. $5x+y=6$

83. $2x+y=-3$
Solution

84. $4x+y=-5$

85. $\frac{1}{3}x+y=2$
Solution

86. $\frac{1}{2}x+y=3$

87. $\frac{2}{5}x-y=4$
Solution

88. $\frac{3}{4}x-y=6$

89. $2x+3y=12$
Solution

90. $4x+2y=12$

91. $3x-4y=12$
Solution

92. $2x-5y=10$

93. $x-6y=3$
Solution

94. $x-4y=2$

95. $5x+2y=4$
Solution

96. $3x+5y=5$

## Exercises: Graph Vertical and Horizontal Lines

Instructions: For questions 97-108, graph each equation.

97. $x=4$
Solution

98. $x=3$

99. $x=-2$
Solution

100. $x=-5$

101. $y=3$
Solution

102. $y=1$

103. $y=-5$
Solution

104. $y=-2$

105. $x=\frac{7}{3}$
Solution

106. $x=\frac{5}{4}$

107. $y=-\frac{15}{4}$
Solution

108. $y=-\frac{5}{3}$

## Exercises: Graph a Pair of Equations in the Same Rectangular Coordinate System

Instructions: For questions 109-112, graph each pair of equations in the same rectangular coordinate system.

109. $y=2x$ and $y=2$
Solution

110. $y=5x$ and $y=5$

111. $y=-\frac{1}{2}x$ and $y=-\frac{1}{2}$
Solution

112. $y=-\frac{1}{3}x$ and $y=-\frac{1}{3}$

## Exercises: Mixed Practice

Instructions: For questions 113-128, graph each equation.
113. $y=4x$
Solution

114. $y=2x$

115. $y=-\frac{1}{2}x+3$
Solution

116. $y=\frac{1}{4}x-2$

117. $y=-x$
Solution

118. $y=x$

119. $x-y=3$
Solution

120. $x+y=-5$

121. $4x+y=2$
Solution

122. $2x+y=6$

123. $y=-1$
Solution

124. $y=5$

125. $2x+6y=12$
Solution

126. $5x+2y=10$

127. $x=3$
Solution

128. $x=-4$

## Exercises: Identify the $\color{White} x$ and $\color{White} y$-Intercepts on a Graph

Instructions: For questions 129-140, find the $x$ and $y$-intercepts on each graph.

129.

Solution

$\left(3,0\right),\left(0,3\right)$

130.

131.

Solution

$\left(5,0\right),\left(0,-5\right)$

132.

133.

Solution

$\left(-2,0\right),\left(0,-2\right)$

134.

135.

Solution

$\left(-1,0\right),\left(0,1\right)$

136.

137.

Solution

$\left(6,0\right),\left(0,3\right)$

138.

139.

Solution

$\left(0,0\right)$

140.

## Exercises: Find the $\color{White} x$ and $\color{White} y$-Intercepts from an Equation of a Line

Instructions: For questions 141-168, find the intercepts for each equation.

141. $x+y=4$

Solution

$\left(4,0\right),\left(0,4\right)$

142. $x+y=3$

143. $x+y=-2$
Solution

$\left(-2,0\right),\left(0,-2\right)$

144. $x+y=-5$

145. $x–y=5$
Solution

$\left(5,0\right),\left(0,-5\right)$

146. $x–y=1$

147. $x–y=-3$
Solution

$\left(-3,0\right),\text{}\left(0,3\right)$

148. $x–y=-4$

149. $x+2y=8$
Solution

$\left(8,0\right),\left(0,4\right)$

150. $x+2y=10$

151. $3x+y=6$
Solution

$\left(2,0\right),\left(0,6\right)$

152. $3x+y=9$

153. $x–3y=12$
Solution

$\left(12,0\right),\left(0,-4\right)$

154. $x–2y=8$

155. $4x–y=8$
Solution

$\left(2,0\right),\left(0,-8\right)$

156. $5x–y=5$

157. $2x+5y=10$
Solution

$\left(5,0\right),\left(0,2\right)$

158. $2x+3y=6$

159. $3x–2y=12$
Solution

$\left(4,0\right),\left(0,-6\right)$

160. $3x–5y=30$

161. $y=\frac{1}{3}x+1$
Solution

$\left(3,0\right),\left(0,1\right)$

162. $y=\frac{1}{4}x-1$

163. $y=\frac{1}{5}x+2$
Solution

$\left(-10,0\right),\left(0,2\right)$

164. $y=\frac{1}{3}x+4$

165. $y=3x$
Solution

$\left(0,0\right)$

166. $y=-2x$

167. $y=-4x$
Solution

$\left(0,0\right)$

168. $y=5x$

## Exercises: Graph a Line Using the Intercepts

Instructions: For questions 169-194, graph using the intercepts.

169. $–x+5y=10$
Solution

170. $–x+4y=8$

171. $x+2y=4$
Solution

172. $x+2y=6$

173. $x+y=2$
Solution

174. $x+y=5$

175. $x+y=-3$
Solution

176. $x+y=-1$

177. $x–y=1$
Solution

178. $x–y=2$

179. $x–y=-4$
Solution

180. $x–y=-3$

181. $4x+y=4$
Solution

182. $3x+y=3$

183. $2x+4y=12$
Solution

184. $3x+2y=12$

185. $3x–2y=6$
Solution

186. $5x–2y=10$

187. $2x–5y=-20$
Solution

188. $3x–4y=-12$

189. $3x–y=-6$
Solution

190. $2x–y=-8$

191. $y=-2x$
Solution

192. $y=-4x$

193. $y=x$
Solution

194. $y=3x$

## Exercises: Everyday Math

Instructions: For questions 195-200, answer the given everyday math word problems.

195. Weight of a baby. Mackenzie recorded her baby’s weight every two months. The baby’s age, in months, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the third column.

a. Plot the points on a coordinate plane.

b. Why is only Quadrant I needed?

 Age $x$ Weight $y$ $\left(x,y\right)$ $0$ $7$ $(0,7)$ $2$ $11$ $(2,11)$ $4$ $15$ $(4,15)$ $6$ $16$ $(6,16)$ $8$ $19$ $(8,19)$ $10$ $20$ $(10,20)$ $12$ $21$ $(12,21)$

Solution

a.

b. Age and weight are only positive.

196. Weight of a child. Latresha recorded her son’s height and weight every year. His height, in inches, and weight, in pounds, are listed in the table below, and shown as an ordered pair in the third column.

a. Plot the points on a coordinate plane.

b. Why is only Quadrant I needed?

 Height $x$ Weight $y$ $\left(x,y\right)$ $28$ $22$ $(28,22)$ $31$ $27$ $(31,27)$ $33$ $33$ $(33,33)$ $37$ $35$ $(37,35)$ $40$ $41$ $(40,41)$ $42$ $45$ $(42,45)$

197. Motor home cost. The Robinsons rented a motor home for one week to go on vacation. It cost them $594$ plus $0.32$ per mile to rent the motor home, so the linear equation $y=594+0.32x$ gives the cost, $y$, for driving $x$ miles. Calculate the rental cost for driving $400$, $800$, and $1200$ miles, and then graph the line.
Solution

$722$, $850$, $978$

198. Weekly earnings. At the art gallery where he works, Salvador gets paid $200$ per week plus $15\%$ of the sales he makes, so the equation $y=200+0.15x$ gives the amount, $y$, he earns for selling $x$ dollars of artwork. Calculate the amount Salvador earns for selling $900$, $1600$, and $2000$, and then graph the line.

199. Road trip. Damien is driving from Chicago to Denver, a distance of $1000 miles. The [latex]x$-axis on the graph below shows the time in hours since Damien left Chicago. The $y$-axis represents the distance he has left to drive.

a. Find the $x$ and $y$-intercepts.
b. Explain what the $x$ and $y$-intercepts mean for Damien.

Solution

a.$(0,1000),(15,0)$
b. At $(0,1000)$, he has been gone $0$ hours and has $1000$ miles left. At $(15,0)$, he has been gone $15$ hours and has $0$ miles left to go.

200. Road trip. Ozzie filled up the gas tank of his truck and headed out on a road trip. The $x$-axis on the graph below shows the number of miles Ozzie drove since filling up. The $y$-axis represents the number of gallons of gas in the truck’s gas tank.

a. Find the $x$ and $y$-intercepts.
b. Explain what the $x$ and $y$-intercepts mean for Ozzie.

## Exercises: Writing Exercises

Instructions: For questions 201-210,
201. Explain in words how you plot the point $\left(4,-2\right)$ in a rectangular coordinate system.
Solution

202. How do you determine if an ordered pair is a solution to a given equation?

203. Is the point $\left(-3,0\right)$ on the $x$-axis or $y$-axis? How do you know?
Solution

204. Is the point $(0,8)$ on the $x$-axis or $y$-axis? How do you know?

205. Explain how you would choose three $x$-values to make a table to graph the line $y=\frac{1}{5}x-2$.

Solution

206. What is the difference between the equations of a vertical and a horizontal line?

207. How do you find the $x$-intercept of the graph of $3x–2y=6$?
Solution

208. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $4x+y=-4$? Why?

209. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $y=\frac{2}{3}x-2$? Why?
Solution

210. Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation $y=6$? Why?