Equations With Two Variables - PSAT Math

1. Identify the $x$-intercept of the line given by $4x+3y=12$.

   $3$. Substitute $y=0$: $4x=12$, so $x=3$.

2. Identify whether $(2,3)$ is a solution to $x+y=5$.

   Yes. Check: $2+3=5$ ✓

3. Identify the $y$-intercept of the line $4x+2y=10$.

   $5$. Set $x=0$: $2y=10$, so $y=5$.

4. What is the point-slope form of a linear equation through $(x_1,y_1)$ with slope $m$?

   $y-y_1=m(x-x_1)$. Directly uses the given point and slope to write the equation.

5. What is the slope of the line given by $x=4$?

   Undefined. Vertical lines have no defined slope.

6. What is the equation of a line with slope $2$ and $y$-intercept $-3$?

   $y=2x-3$. Use slope-intercept form with given $m$ and $b$.

7. What is the equation of the line through $(0,-2)$ and $(4,6)$?

   $y=2x-2$. Find slope $m=\frac{6-(-2)}{4-0}=2$, then use $y=mx+b$.

8. Identify the slope of $Ax+By=C$ in terms of $A$ and $B$ (assume $B\ne 0$).

   $m=-\frac{A}{B}$. Solve for $y$ to get $y=-\frac{A}{B}x+\frac{C}{B}$.

9. What condition on slopes shows two lines are parallel?

   Equal slopes: $m_1=m_2$. Parallel lines never intersect, so they have identical slopes.

10. Write the equation in slope-intercept form: $2x+y=7$.

    $y=-2x+7$. Isolate $y$ by subtracting $2x$ from both sides.

11. What does it mean if an ordered pair $(x,y)$ is a solution to an equation in two variables?

    Substitution makes the equation true. The $x$ and $y$ values satisfy the equation.

12. Identify whether $(1,-2)$ is a solution to $2x-y=0$.

    No. Check: $2(1)-(-2)=2+2=4\ne 0$.

13. Find the slope of the line through $(1,3)$ and $(5,1)$.

    $-\frac{1}{2}$. Use slope formula: $m=\frac{1-3}{5-1}=\frac{-2}{4}=-\frac{1}{2}$.

14. Find the missing value $k$ so that $(3,k)$ satisfies $2x+y=11$.

    $5$. Substitute: $2(3)+k=11$, so $6+k=11$, thus $k=5$.

15. What condition on slopes shows two nonvertical lines are parallel?

    $m_1=m_2$. Parallel lines have same steepness, thus equal slopes.

16. What is the slope of the line given by $Ax+By=C$ in terms of $A$ and $B$?

    $m=-\frac{A}{B}$. Derived by solving $Ax+By=C$ for $y$ to get slope-intercept form.

17. Identify the equation of the vertical line through $x=-3$.

    $x=-3$. Vertical lines have constant $x$-value for all $y$.

18. What is the $y$-intercept of $y=-3x+5$, stated as a point?

    $(0,5)$. When $x=0$, $y=5$.

19. Identify the equation of the line with slope $2$ and $y$-intercept $-3$.

    $y=2x-3$. Direct substitution into $y=mx+b$ form.

20. What does it mean for $(x,y)$ to be a solution to an equation in two variables?

    $(x,y)$ makes the equation true when substituted for $x$ and $y$. The point satisfies the equation when its coordinates are plugged in.

21. What is the slope formula using two points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points gives the slope.

22. What is the solution to the system $2x+y=9$ and $x+y=6$?

    $\left(3,3\right)$. Subtract second from first: $x=3$; substitute to get $y=3$.

23. Which option is the standard form of $y=3x-5$ with integer coefficients?

    $3x-y=5$. Move all terms to one side: $3x-y-5=0$ or $3x-y=5$.

24. What is the equation of the line perpendicular to $y=\frac{1}{3}x-4$ passing through $(0,2)$?

    $y=-3x+2$. Perpendicular slope is $-3$, passes through $(0,2)$.

25. What is the slope of the line in standard form $Ax+By=C$ (with $B\ne 0$)?

    $m=-\frac{A}{B}$. Rearrange to slope-intercept form: $y=-\frac{A}{B}x+\frac{C}{B}$.

26. What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$ (with $x_2\ne x_1$)?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.

27. Identify the slope of the line $3x-6y=12$.

    $\frac{1}{2}$. Rewrite as $y=\frac{1}{2}x-2$; coefficient of $x$ is the slope.

28. What condition on slopes shows two nonvertical lines are perpendicular?

    $m_1m_2=-1$. Perpendicular slopes multiply to $-1$ (negative reciprocals).

29. Identify the slope of $2x+5y=10$.

    $m=-\frac{2}{5}$. Use $m=-\frac{A}{B}$ with $A=2$, $B=5$.

30. Identify the $x$-intercept of the line $5x-3y=15$.

    $3$. Set $y=0$: $5x=15$, so $x=3$.

31. What does it mean for an ordered pair $(x,y)$ to be a solution to $2x+3y=12$?

    $(x,y)$ makes $2x+3y=12$ true when substituted. The values satisfy the equation when plugged in.

32. Identify the $y$-intercept of $2x+3y=12$.

    $(0,4)$. Set $x=0$: $3y=12$, so $y=4$.

33. Identify the $y$-intercept of the line $y=-3x+5$.

    $b=5$. Constant term in slope-intercept form.

34. What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

    $y-y_1=m(x-x_1)$. Uses a known point and the slope to define the line.

35. What is the $y$-intercept of the line $Ax+By=C$ when $B\ne 0$?

    $\left(0,\frac{C}{B}\right)$. Set $x=0$ and solve for $y$ to find where line crosses $y$-axis.

36. What is the $x$-intercept of $5x+y=10$?

    $\left(2,0\right)$. Set $y=0$: $5x=10$, so $x=2$.

37. Identify the slope of a horizontal line written as $y=c$.

    $m=0$. Horizontal lines have no change in $y$.

38. What is the condition for two nonvertical lines with slopes $m_1$ and $m_2$ to be perpendicular?

    $m_1m_2=-1$. Perpendicular slopes multiply to $-1$.

39. What is the $x$-intercept of a line found by setting which variable equal to $0$?

    Set $y=0$. The x-intercept occurs where the line crosses the x-axis.

40. What is the equation of the horizontal line through $y=-3$?

    $y=-3$. Horizontal lines have zero slope and constant $y$-value.

41. What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$ with $x_1\ne x_2$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.

42. Identify the equation of the line through $(0,4)$ and $(2,0)$.

    $y=-2x+4$. Slope $m=\frac{0-4}{2-0}=-2$; use point-slope form.

43. What is the equation of the line through $(0,2)$ and $(4,6)$ in slope-intercept form?

    $y=x+2$. Slope is $\frac{6-2}{4-0}=1$, passes through $(0,2)$.

44. What is the $x$-intercept of $Ax+By=C$ (assume $A\ne 0$)?

    $\left(\frac{C}{A},0\right)$. Set $y=0$ and solve for $x$: $Ax=C$, so $x=\frac{C}{A}$.

45. What is the equation of the vertical line through $x=5$?

    $x=5$. Vertical lines have undefined slope and constant $x$-value.

46. What is the slope-intercept form of a linear equation in two variables?

    $y=mx+b$. $m$ is slope and $b$ is y-intercept.

47. What is the equation of the line with slope $-4$ passing through $(0,6)$?

    $y=-4x+6$. Point $(0,6)$ is the y-intercept, so $b=6$.

48. Identify whether $(2,1)$ is a solution to $3x+2y=4$.

    No. Check: $3(2)+2(1)=6+2=8≠4$ ✗

49. What is the $x$-intercept of $Ax+By=C$ when $y=0$?

    $x=\frac{C}{A}$. Set $y=0$ and solve for $x$.

50. What is the slope of the line given by $y=6$?

    $0$. Horizontal lines have zero slope.

51. Find the $x$-intercept of the line $2x+5y=10$.

    $(5,0)$. Set $y=0$: $2x=10$, so $x=5$.

52. What is the slope of the line in $Ax+By=C$ in terms of $A$ and $B$?

    $m=-\frac{A}{B}$. Rearrange to slope-intercept form: $y=-\frac{A}{B}x+\frac{C}{B}$.

53. What condition on slopes $m_1$ and $m_2$ means two lines are parallel?

    $m_1=m_2$. Parallel lines have equal slopes.

54. What is the equation of a vertical line through $x=k$?

    $x=k$. All points have the same $x$-coordinate.

55. Solve for $y$ in terms of $x$: $3x+2y=8$.

    $y=-\frac{3}{2}x+4$. Isolate $y$: $2y=8-3x$, then divide by 2.

56. Identify the slope of the line through $(1,2)$ and $(5,10)$.

    $2$. Use $m=\frac{y_2-y_1}{x_2-x_1}=\frac{10-2}{5-1}=\frac{8}{4}$.

57. What is the condition on slopes for two nonvertical lines to be perpendicular?

    Their slopes satisfy $m_1m_2=-1$. Perpendicular lines meet at 90°; slopes are negative reciprocals.

58. State the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be parallel.

    $m_1=m_2$. Parallel lines have equal slopes.

59. Find the slope of the line $2x+5y=10$.

    $m=-\frac{2}{5}$. Rewrite as $y=-\frac{2}{5}x+2$ to identify slope $m=-\frac{2}{5}$.

60. What is the definition of a solution to an equation in two variables such as $2x+3y=7$?

    An ordered pair $(x,y)$ that makes the equation true. When substituted, both sides of the equation must be equal.

61. Identify the slope of the line given by $3x+6y=12$.

    $-\frac{1}{2}$. Rewrite as $y=-\frac{1}{2}x+2$ to find slope $m=-\frac{1}{2}$.

62. What is the standard form of a linear equation in two variables?

    $Ax+By=C$. $A$, $B$, and $C$ are constants with $A$ and $B$ not both zero.

63. Find the equation in standard form for the line through $(0,4)$ and $(2,0)$.

    $2x+y=4$. Slope $m=\frac{0-4}{2-0}=-2$; use point-slope form.

64. What is the $y$-intercept of $Ax+By=C$ when $x=0$?

    $y=\frac{C}{B}$. Set $x=0$ and solve for $y$.

65. Find the equation of the line with slope $-2$ passing through $(1,3)$ in slope-intercept form.

    $y=-2x+5$. Use $y-3=-2(x-1)$, then simplify to get $y=-2x+5$.

66. What is the equation in standard form for the line $y=-3x+7$?

    $3x+y=7$. Move $x$ term to left side.

67. Identify whether $(2,-1)$ is a solution to $3x+2y=4$.

    Yes. $3(2)+2(-1)=6-2=4$ ✓

68. Identify the $y$-intercept of the line $4x+5y=10$.

    $\left(0,2\right)$. Set $x=0$: $5y=10$, so $y=2$.

69. Determine whether $(2,3)$ is a solution to $x+y=6$.

    No, because $2+3\ne 6$. Check: $2+3=5\ne 6$.

70. What is the slope of the line given by $Ax+By=C$ (with $B\ne 0$)?

    $m=-\frac{A}{B}$. Derived by solving $Ax+By=C$ for $y$ and identifying slope.

71. What is the slope of the line in $Ax+By=C$ (assume $B\ne 0$)?

    $m=-\frac{A}{B}$. Rearrange to slope-intercept form: $y=-\frac{A}{B}x+\frac{C}{B}$.

72. What is the slope of $Ax+By=C$ (assume $B\ne 0$)?

    $m=-\frac{A}{B}$. Rearrange to slope-intercept form to find slope.

73. What is the equation of the line with slope $-2$ passing through $(3,1)$?

    $y=-2x+7$. Use $y-1=-2(x-3)$ and simplify.

74. Identify whether $(1,4)$ is a solution to $2x+y=5$.

    No. Check: $2(1)+4=6≠5$, so not a solution.

75. Write the equation of the line with slope $3$ and $y$-intercept $-2$.

    $y=3x-2$. Direct substitution into $y=mx+b$.

76. What does it mean for an ordered pair $(x,y)$ to be a solution of $ax+by=c$?

    $(x,y)$ makes $ax+by=c$ true when substituted. Substituting the values satisfies the equation.

77. What is the equation in slope-intercept form for $2x+3y=12$?

    $y=-\frac{2}{3}x+4$. Solve for $y$: $3y=12-2x$, then divide by $3$.

78. Identify whether $(2,1)$ is a solution to $3x-2y=4$.

    Yes. Check: $3(2)-2(1)=6-2=4$ ✓

79. What is the equation of the line with slope $3$ and $y$-intercept $-5$?

    $y=3x-5$. Use slope-intercept form with $m=3$ and $b=-5$.

80. What is the $y$-intercept of a line found by setting which variable equal to $0$?

    Set $x=0$. The y-intercept occurs where the line crosses the y-axis.

81. Rewrite $4x+2y=8$ in slope-intercept form.

    $y=-2x+4$. Solve for $y$: $2y=-4x+8$, so $y=-2x+4$.

82. What is the point-slope form of a linear equation through $(x_1,y_1)$ with slope $m$?

    $y-y_1=m(x-x_1)$. Uses a known point and the slope to define the line.

83. What is the slope of the line in standard form $Ax+By=C$ when $B\ne 0$?

    $m=-\frac{A}{B}$. Rearrange to slope-intercept form: $y=-\frac{A}{B}x+\frac{C}{B}$.

84. What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.

85. Find $y$ when $x=-2$ for the equation $y=4x-1$.

    $y=-9$. Substitute: $y=4(-2)-1=-8-1=-9$.

86. What is the point-slope form of a line with slope $m$ through $(x_1,y_1)$?

    $y-y_1=m(x-x_1)$. Uses a known point and slope to define the line.

87. State the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be perpendicular.

    $m_1m_2=-1$. Perpendicular slopes multiply to $-1$.

88. What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

    $y-y_1=m(x-x_1)$. Uses known point $(x_1,y_1)$ and slope $m$ to define the line.

89. Identify the slope of the line $3x-2y=8$.

    $\frac{3}{2}$. Rewrite as $y=\frac{3}{2}x-4$; coefficient of $x$ is the slope.

90. What is the slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points.

91. What is the $x$-intercept of $y=-2x+8$?

    $4$. Set $y=0$: $0=-2x+8$, so $x=4$.

92. Find the $x$-intercept of the line $3x+6y=12$.

    $\left(4,0\right)$. Set $y=0$: $3x=12$, so $x=4$.

93. Find the $y$-intercept of the line $2x+5y=10$.

    $(0,2)$. Set $x=0$: $5y=10$, so $y=2$.

94. What is the condition for two lines with slopes $m_1$ and $m_2$ to be parallel?

    $m_1=m_2$. Parallel lines have equal slopes.

95. What is the equation of the line parallel to $y=-\frac{1}{2}x+3$ passing through $(2,1)$?

    $y=-\frac{1}{2}x+2$. Same slope $-\frac{1}{2}$, use point-slope form.

96. What is the equation of a vertical line through $x=a$?

    $x=a$. All points have the same $x$-coordinate $a$.

97. What is the $y$-intercept of $Ax+By=C$ (assume $B\ne 0$) written as a point?

    $\left(0,\frac{C}{B}\right)$. Set $x=0$ and solve for $y$ to get $y=\frac{C}{B}$.

98. What is the definition of the solution set of an equation in two variables?

    All ordered pairs $(x,y)$ that satisfy the equation. The collection of all points that make the equation true.

99. Find the equation of the line with slope $3$ and $y$-intercept $-2$.

    $y=3x-2$. Use $y=mx+b$ with $m=3$ and $b=-2$.

100. Find the $x$-intercept of $3x-2y=12$.

     $(4,0)$. Set $y=0$: $3x=12$, so $x=4$.

101. What is the definition of a solution to a system of two equations in $x$ and $y$?

     An ordered pair $(x,y)$ that satisfies both equations. The point must make both equations true simultaneously.

102. What does it mean for an ordered pair $(x,y)$ to be a solution to $Ax+By=C$?

     It makes the equation true when $x$ and $y$ are substituted. The ordered pair satisfies the equation when plugged in.

103. What does it mean for an ordered pair $(x,y)$ to be a solution to $ax+by=c$?

     $(x,y)$ is a solution if substituting makes $ax+by=c$ true. The pair satisfies the equation when plugged in.

104. What is the equation of the horizontal line that passes through $(0,7)$?

     $y=7$. Horizontal lines have form $y=$ constant.

105. What is the slope of a line parallel to $y=-4x+9$?

     $-4$. Parallel lines have the same slope.

106. What is the solution to the system $x+y=5$ and $x-y=1$?

     $\left(3,2\right)$. Add equations: $2x=6$, so $x=3$; substitute to get $y=2$.

107. Which ordered pair is a solution to $x+y=7$: $(2,5)$ or $(3,5)$?

     $(2,5)$. Check: $2+5=7$ ✓, but $3+5=8$ ✗.

108. What is the $y$-intercept of $Ax+By=C$ when $x=0$?

     $y=\frac{C}{B}$. Set $x=0$ and solve for $y$ to find where line crosses y-axis.

109. What is the standard form of a linear equation in two variables on the PSAT?

     $Ax+By=C$ with $A,B,C$ constants and $A,B$ not both $0$. General form where coefficients $A$ and $B$ determine the line's direction.

110. Determine whether $(2,-1)$ is a solution to $3x+2y=4$.

     Yes, because $3(2)+2(-1)=4$. Substitute and check if equation holds.

111. What is the $y$-intercept of the line $Ax+By=C$ (assume $B\ne 0$)?

     $\left(0,\frac{C}{B}\right)$. Set $x=0$ and solve for $y$ to find where line crosses $y$-axis.

112. Identify whether $(1,3)$ is a solution to $2x-y=0$.

     No. $2(1)-3=2-3=-1\neq^0$

113. What is the slope of the line through $(1,2)$ and $(5,10)$?

     $2$. $m=\frac{10-2}{5-1}=\frac{8}{4}=2$.

114. Identify the slope and $y$-intercept in $y=-3x+5$.

     $m=-3$, $b=5$. Read directly from $y=mx+b$ form.

115. What is the slope of a line perpendicular to $y=\frac{2}{5}x-1$?

     $-\frac{5}{2}$. Perpendicular slopes multiply to $-1$.

116. Identify the slope of the line passing through $(1,2)$ and $(3,8)$.

     $3$. Apply slope formula: $m=\frac{8-2}{3-1}=\frac{6}{2}=3$.

117. What is the definition of a solution to an equation in two variables?

     An ordered pair $(x,y)$ that makes the equation true. A solution satisfies the equation when substituted.

118. Identify the slope of the line $4x-2y=10$.

     $m=2$. Rewrite as $y=2x-5$; slope is coefficient of $x$.

119. Convert $6x+3y=12$ to slope-intercept form $y=mx+b$.

     $y=-2x+4$. Divide by $3$: $y=-2x+4$.

120. Identify whether $(1,2)$ is a solution to $x+2y=6$.

     No, because $1+2(2)=5$. Substitute $x=1$ and $y=2$: $1+2(2)=1+4=5≠6$ ✗

121. What is the slope of the line given by $2x+5y=10$?

     $-\frac{2}{5}$. Rewrite as $y=-\frac{2}{5}x+2$; coefficient of $x$ is slope.

122. What condition on slopes shows two lines are perpendicular?

     Negative reciprocals: $m_1m_2=-1$. Perpendicular lines meet at 90°; slopes multiply to $-1$.

123. What is the equation of the line with slope $2$ and $y$-intercept $-5$?

     $y=2x-5$. Direct substitution into $y=mx+b$ with $m=2$ and $b=-5$.

124. What are the $x$- and $y$-intercepts of $Ax+By=C$ (assume $A\neq 0$ and $B\neq 0$)?

     $x$-int $=\frac{C}{A}$, $y$-int $=\frac{C}{B}$. Set $y=0$ for x-int, $x=0$ for y-int.

125. What is the equation of the vertical line that passes through $(-3,0)$?

     $x=-3$. Vertical lines have form $x=$ constant.

126. Which point is a solution to $2x+y=7$: $(2,3)$ or $(1,5)$?

     $(2,3)$. Check: $2(2)+3=7$ ✓, but $2(1)+5=7$ ✗.

127. Write the equation of the line with slope $-2$ and $y$-intercept $7$.

     $y=-2x+7$. Direct substitution into $y=mx+b$.

128. What is the slope of the line $4x-2y=10$?

     $2$. Rearrange to $y=2x-5$, so slope is $2$.

129. What condition on slopes $m_1$ and $m_2$ means two nonvertical lines are perpendicular?

     $m_1m_2=-1$. Product of perpendicular slopes is $-1$.

130. What is the equation of a horizontal line passing through $y=5$?

     $y=5$. Horizontal lines have constant $y$-values.

131. Which equation is parallel to $y=\frac{1}{3}x-4$ and passes through $(0,2)$?

     $y=\frac{1}{3}x+2$. Parallel lines share slope $\frac{1}{3}$; through $(0,2)$ gives $b=2$.

132. What condition on slopes $m_1$ and $m_2$ means two lines are perpendicular?

     $m_1m_2=-1$. Perpendicular slopes multiply to $-1$.

133. Solve the system: $y=2x+1$ and $y=2x-3$.

     No solution. Same slope but different y-intercepts means parallel lines.

134. Find the $x$-intercept of the line $3x-2y=12$.

     $\left(4,0\right)$. Substitute $y=0$: $3x-2(0)=12$, so $3x=12$, thus $x=4$.

135. What is the $x$-intercept of the line given by $4x-y=8$?

     $(2,0)$. Set $y=0$ and solve for $x$.

136. What is the $y$-intercept of the line $4x-2y=8$?

     $-4$. Set $x=0$: $-2y=8$, so $y=-4$ is the y-intercept.

137. What is the slope of the line given by $Ax+By=C$ with $B\neq 0$?

     $m=-\frac{A}{B}$. Derived by solving $Ax+By=C$ for $y$ to get slope.

138. Find the equation of the line through $(2,5)$ with slope $-1$.

     $y=-x+7$. Use point-slope: $y-5=-1(x-2)$; simplify.

139. What is the $y$-intercept of $6x+3y=12$?

     $4$. Set $x=0$: $3y=12$, so $y=4$.

140. What is the equation of the line through $(1,4)$ with slope $-3$?

     $y=-3x+7$. Use point-slope form: $y-4=-3(x-1)$, then simplify.

141. What is the $y$-intercept of $4x-2y=8$?

     $\left(0,-4\right)$. Set $x=0$: $-2y=8$, so $y=-4$.

142. Identify whether $y=2x+1$ and $y=2x-5$ intersect, are parallel, or are perpendicular.

     Parallel. Same slope but different y-intercepts.

143. Identify whether $(2,1)$ is a solution to $3x-2y=4$.

     Yes, because $3(2)-2(1)=4$. Substitute $x=2$ and $y=1$: $3(2)-2(1)=6-2=4$ ✓

144. Identify the solution set description for a linear equation in two variables.

     Infinitely many solutions, forming a line in the coordinate plane. Every point on the line satisfies the equation.

145. What is the $x$-intercept of the line $Ax+By=C$ (assume $A\ne 0$)?

     $\left(\frac{C}{A},0\right)$. Set $y=0$ and solve for $x$ to find where line crosses $x$-axis.

146. What is the slope of the line given by $4x-2y=8$?

     $2$. Rewrite as $y=2x-4$; slope is coefficient of $x$.

147. Identify whether $(1,3)$ is a solution to $2x-y=-1$.

     No. $2(1)-3=2-3=-1$ ✓

148. What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

     $y-y_1=m(x-x_1)$. Uses a known point and the slope to define the line.

149. Find the equation in slope-intercept form through $(0,2)$ and $(4,6)$.

     $y=x+2$. Slope $m=\frac{6-2}{4-0}=1$, passes through $(0,2)$ giving $b=2$.

150. Rewrite $3x+6y=12$ in slope-intercept form.

     $y=-\frac{1}{2}x+2$. Solve for $y$: $6y=-3x+12$, then $y=-\frac{1}{2}x+2$.

151. Identify whether $(2,-1)$ is a solution to $y=3x-7$.

     Yes. Check: $-1=3(2)-7=-1$ ✓

152. What type of line is represented by $x=a$ for a constant $a$?

     A vertical line with undefined slope. All points have same $x$-coordinate; slope is undefined.

153. What is the $x$-intercept of the line $Ax+By=C$ when $A\ne 0$?

     $\left(\frac{C}{A},0\right)$. Set $y=0$ and solve for $x$ to find where line crosses $x$-axis.

154. Identify the $y$-intercept of the line given by $2x-5y=10$.

     $-2$. Substitute $x=0$: $-5y=10$, so $y=-2$.

155. Find the $y$-intercept of the line $3x+6y=12$.

     $\left(0,2\right)$. Set $x=0$: $6y=12$, so $y=2$.

156. Which ordered pair is a solution to $x+y=7$: $(2,5)$ or $(3,5)$?

     $(2,5)$. $2+5=7$ is true, but $3+5=8≠7$.

157. What condition shows that $(x,y)$ is a solution to an equation in two variables?

     Substitute; the equation becomes a true statement. When substituted, both sides of the equation are equal.

158. Which option is the slope-intercept form of $2x+y=7$?

     $y=-2x+7$. Solve for $y$: $y=7-2x=-2x+7$.

159. Identify the solution $(x,y)$ to the system $x+y=7$ and $x-y=1$.

     $(4,3)$. Add equations: $2x=8$, so $x=4$; substitute to get $y=3$.

160. What is the $x$-intercept of $Ax+By=C$ when $y=0$?

     $x=\frac{C}{A}$. Set $y=0$ and solve for $x$ to find where line crosses x-axis.

161. What is the slope of the line $2x+3y=12$?

     $-\frac{2}{3}$. Use $m=-\frac{A}{B}=-\frac{2}{3}$.

162. What is the $y$-intercept of the line given by $2x+5y=10$?

     $(0,2)$. Set $x=0$ and solve for $y$.

163. What is the equation of the line with slope $3$ and $y$-intercept $-4$?

     $y=3x-4$. Direct substitution into slope-intercept form.

164. What is the equation of a horizontal line through $y=k$?

     $y=k$. All points have the same $y$-coordinate.

165. What are the $x$- and $y$-intercepts of $Ax+By=C$ in terms of $A,B,C$?

     $x$-int $=\frac{C}{A}$, $y$-int $=\frac{C}{B}$. Set $y=0$ to find $x$-int; set $x=0$ to find $y$-int.

166. What is the $y$-intercept of $Ax+By=C$ (assume $B\ne 0$)?

     $\left(0,\frac{C}{B}\right)$. Set $x=0$ and solve for $y$.

167. What is the slope between two points $(x_1,y_1)$ and $(x_2,y_2)$ with $x_1\ne x_2$?

     $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.

168. Find the slope of the line through $(1,4)$ and $(3,0)$.

     $m=-2$. Use $m=\frac{0-4}{3-1}=\frac{-4}{2}=-2$.

169. Identify whether $(2,3)$ is a solution to $x+y=5$.

     Yes, because $2+3=5$. Substitute: $2+3=5$ is true.

170. How many solutions does the system $y=2x+1$ and $y=2x-3$ have?

     $0$. Parallel lines (same slope) never intersect.

171. What is the condition on slopes for two nonvertical lines to be parallel?

     They have equal slopes: $m_1=m_2$. Parallel lines never intersect, so same steepness.

172. Identify whether $2x-3y=6$ and $4x-6y=12$ are the same line or different lines.

     Same line (equivalent equations). Second equation is first multiplied by $2$.

173. What is the $x$-intercept of the line $3x+6y=12$?

     $4$. Set $y=0$: $3x=12$, so $x=4$ is the x-intercept.

174. Write the equation of the line through $(2,5)$ with slope $-1$.

     $y=-x+7$. Use point-slope: $y-5=-1(x-2)$, then simplify.

175. What is the slope of the line given by $3x-6y=12$?

     $\frac{1}{2}$. Rewrite as $y=\frac{1}{2}x-2$ to find slope.

176. What is the $x$-intercept of $Ax+By=C$ (assume $A\ne 0$) written as a point?

     $\left(\frac{C}{A},0\right)$. Set $y=0$ and solve for $x$ to get $x=\frac{C}{A}$.

177. What is the $x$-intercept of $Ax+By=C$ (assume $A\ne 0$)?

     $\left(\frac{C}{A},0\right)$. Set $y=0$ and solve for $x$.

178. Find the slope between $(1,4)$ and $(3,0)$.

     $-2$. Use slope formula: $\frac{0-4}{3-1}=\frac{-4}{2}=-2$.

179. Find the equation of the line with slope $-2$ passing through $(3,1)$.

     $y=-2x+7$. Use point-slope: $y-1=-2(x-3)$, simplify to slope-intercept form.

180. Find the equation in slope-intercept form for the line through $(0,-1)$ with slope $3$.

     $y=3x-1$. Use $y=mx+b$ with $m=3$ and $b=-1$.

181. Find the slope of the line through $(1,2)$ and $(3,8)$.

     $m=3$. Use $m=\frac{y_2-y_1}{x_2-x_1}=\frac{8-2}{3-1}=\frac{6}{2}=3$.

182. What type of line is represented by $y=b$ for a constant $b$?

     A horizontal line with slope $0$. All points have same $y$-coordinate; no vertical change.

183. Identify the slope and $y$-intercept of $y=-3x+5$.

     $m=-3$, $b=5$. Read directly from $y=mx+b$ form.

184. What is the condition for two non-vertical lines with slopes $m_1$ and $m_2$ to be perpendicular?

     $m_1m_2=-1$. Product of slopes equals negative one.

185. What is the equation of a horizontal line through $y=b$?

     $y=b$. All points have the same $y$-coordinate $b$.

186. Find the equation in standard form for $y=-2x+5$ using integer coefficients and $Ax+By=C$.

     $2x+y=5$. Rearrange: $2x+y=5$ with positive $A$ coefficient.

187. Which line is parallel to $y=\frac{1}{2}x-3$: $y=\frac{1}{2}x+4$ or $y=-2x+4$?

     $y=\frac{1}{2}x+4$. Same slope $\frac{1}{2}$ means parallel.

188. What is the slope of a line parallel to $y=-\frac{3}{4}x+9$?

     $-\frac{3}{4}$. Parallel lines have identical slopes.

189. What is the slope of a line perpendicular to $y=\frac{2}{3}x-1$?

     $-\frac{3}{2}$. Perpendicular slopes multiply to $-1$: $\frac{2}{3}\times(-\frac{3}{2})=-1$.

190. Identify the slope and $y$-intercept in $y=-3x+5$.

     $m=-3$ and $b=5$. Read directly from $y=mx+b$ form: coefficient of $x$ and constant term.

191. Which option is a correct rewrite of $3x+2y=8$ in slope-intercept form?

     $y=-\frac{3}{2}x+4$. Solve for $y$: $2y=-3x+8$, then $y=-\frac{3}{2}x+4$.

192. Identify whether $(2,3)$ is a solution to $x+y=5$.

     Yes, because $2+3=5$. Substitute $(2,3)$: $2+3=5$ is true, confirming it's a solution.

193. What is the $y$-intercept of $Ax+By=C$ (assume $B\ne 0$)?

     $\left(0,\frac{C}{B}\right)$. Set $x=0$ and solve for $y$: $By=C$, so $y=\frac{C}{B}$.

194. Identify the slope of the line $4x-2y=8$.

     $m=2$. Rearrange to $y=2x-4$, so slope is $2$.

195. What is the $x$-intercept of $Ax+By=C$ when $A\ne 0$?

     $\left(\frac{C}{A},0\right)$. Set $y=0$ and solve for $x$: $Ax=C$, so $x=\frac{C}{A}$.

196. What is the slope of the line $4x-2y=6$?

     $2$. Rewrite as $y=2x-3$, so $m=2$.

197. What is the equation of a vertical line passing through $x=-3$?

     $x=-3$. Vertical lines have constant $x$-values.

198. What is the $x$-intercept of $2x+5y=10$, stated as a point?

     $(5,0)$. Set $y=0$: $2x=10$, so $x=5$.

199. Identify the equation of the horizontal line through $y=5$.

     $y=5$. Horizontal lines have constant $y$-value for all $x$.

200. What condition on slopes $m_1$ and $m_2$ means two nonvertical lines are parallel?

     $m_1=m_2$. Parallel lines have equal slopes.

201. What is the slope of the line through $(x_1,y_1)$ and $(x_2,y_2)$?

     $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula calculates rate of change between two points.

202. How many solutions does the system $y=-x+4$ and $2y=-2x+8$ have?

     $\text{Infinitely many}$. Second equation is first multiplied by $2$; same line.

203. Find the $y$-intercept of the line $3x-2y=12$.

     $\left(0,-6\right)$. Substitute $x=0$: $3(0)-2y=12$, so $-2y=12$, thus $y=-6$.

204. What is the $y$-intercept of $Ax+By=C$ when $B\ne 0$?

     $\left(0,\frac{C}{B}\right)$. Set $x=0$ and solve for $y$: $By=C$, so $y=\frac{C}{B}$.

205. Find the slope of the line through $(1,2)$ and $(5,10)$.

     $2$. $m=\frac{10-2}{5-1}=\frac{8}{4}=2$.

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