Systems of Equations - PSAT Math

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

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

2. Identify the solution type for $2x+4y=6$ and $x+2y=3$.

   Infinitely many solutions. Second equation is first divided by 2; same line.

3. What does it mean if two lines in a linear system are parallel and distinct?

   No solution. Parallel lines never intersect, so no common point exists.

4. What does it mean if a point $(x,y)$ solves a system of equations?

   $(x,y)$ satisfies both equations. The point makes both equations true when substituted.

5. What is the elimination goal when solving a linear system using addition or subtraction?

   Make one variable cancel to $0$ by adding or subtracting equations. Multiply equations if needed to make coefficients opposites.

6. Which point is the solution to the system $y=x+1$ and $y=-2x+7$?

   $(2,3)$. Set equal: $x+1=-2x+7$; solve to get $x=2$, $y=3$.

7. Identify the number of solutions if two lines have slopes $m_1=3$ and $m_2=3$ with different $b$ values.

   No solution. Same slopes mean parallel lines that never intersect.

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

   An ordered pair $(x,y)$ that makes both equations true. Both equations must be satisfied simultaneously by the same values.

9. What is the value of $y$ in the system $3x-2y=4$ and $3x+2y=16$?

   $y=3$. From $x=\frac{10}{3}$, substitute into either equation to get $y=3$.

10. What is the solution to the system $2x+y=9$ and $x-y=1$?

    $(\frac{10}{3},\frac{7}{3})$. Add equations to get $3x=10$, then substitute back.

11. Identify the solution to the system $y=x^2$ and $y=2x$.

    $(0,0)$ and $(2,4)$. Substitute: $x^2=2x$; factor to $x(x-2)=0$, so $x=0$ or $x=2$.

12. What is the quickest way to choose between substitution and elimination?

    Use substitution if a variable is isolated; otherwise use elimination. Isolated variables make substitution faster than elimination.

13. What is the solution type of the system $y=2x+3$ and $y=2x-1$?

    No solution. Same slope (2) but different y-intercepts means parallel lines.

14. What is the substitution method first step for a system of two equations?

    Solve one equation for one variable. Express one variable in terms of the other to substitute.

15. What does it mean if a system of two linear equations has exactly one solution?

    The lines intersect at exactly one point. One solution occurs when lines have different slopes.

16. Identify the solution count for $2x+4y=8$ and $x+2y=5$.

    No solution. Second equation is $2x+4y=10$, parallel to first.

17. What is the elimination goal when solving a system by addition or subtraction?

    Make one variable cancel to get a one-variable equation. Add/subtract equations to eliminate a variable and solve for the other.

18. Which ordered pair is the solution to $x+y=6$ and $2x-y=3$?

    $(3,3)$. Add equations: $3x=9$, so $x=3$; substitute to get $y=3$.

19. Which equation results from substituting $y=3x-2$ into $2x+y=10$?

    $2x+(3x-2)=10$. Direct substitution of the expression for $y$.

20. Identify the value of $k$ so the system $x+ky=6$ and $2x+4y=12$ has infinitely many solutions.

    $k=2$. For same line, need $\frac{1}{2}=\frac{k}{4}=\frac{6}{12}$, so $k=2$.

21. Identify the value of $k$ so the system $x+ky=6$ and $2x+4y=10$ has no solution.

    $k=2$. For parallel lines, need same slope: $\frac{1}{2}=\frac{k}{4}$, so $k=2$.

22. Identify the number of solutions for $2x+4y=8$ and $x+2y=4$.

    Infinitely many solutions. Second equation is half the first; they're the same line.

23. What is the solution to the system $y=2x+1$ and $y=7$?

    $(3,7)$. Substitute $y=7$ into first equation: $7=2x+1$, so $x=3$.

24. Identify the solution to the system: $y=2x+1$ and $y=9$.

    $(4,9)$. Substitute $y=9$ into first equation: $9=2x+1$, so $x=4$.

25. Which classification fits the system $2x+4y=10$ and $x+2y=5$: one, none, or infinitely many solutions?

    Infinitely many solutions. Second equation is first divided by 2; they're the same line.

26. Identify the solution to the system: $x=2y$ and $x+y=9$.

    $(6,3)$. Substitute $x=2y$ into second: $2y+y=9$, so $y=3$.

27. What are the three possible numbers of solutions for a system of two linear equations?

    $0$, $1$, or infinitely many solutions. Linear systems can have unique, no, or infinite solutions.

28. What condition indicates a system of two linear equations has infinitely many solutions?

    Equations are equivalent (same line). Identical lines overlap at every point.

29. Identify the intersection point of $x=2$ and $y=-3$.

    $(2,-3)$. Vertical line $x=2$ meets horizontal line $y=-3$ at $(2,-3)$.

30. What is the solution type for the system: $y=2x+3$ and $y=2x-5$?

    No solution. Same slope $m=2$ but different $y$-intercepts means parallel lines.

31. What conclusion follows if two lines have different slopes, $m_1 \ne m_2$?

    The system has exactly $1$ solution. Different slopes guarantee the lines intersect at one point.

32. What conclusion follows if two lines have the same slope but different intercepts?

    The system has $0$ solutions (parallel lines). Same slope means parallel; different intercepts means they never meet.

33. What is the substitution method for solving a system, stated in one sentence?

    Solve one equation for a variable, substitute into the other. Replace the isolated variable in the second equation.

34. What is the solution to the system $x+y=5$ and $2x+2y=12$?

    No solution. Second simplifies to $x+y=6$, contradicting first equation.

35. What is the solution type of the system $x+y=2$ and $x-y=2$?

    Exactly one solution. Different slopes guarantee the lines intersect once.

36. Identify the solution count for $2x+4y=8$ and $x+2y=4$.

    Infinitely many solutions. Second equation is half the first, same line.

37. What does it mean if elimination produces a true statement like $0=0$?

    Infinitely many solutions. True statement means the equations represent the same line.

38. What is the solution to the system $3x-2y=4$ and $x+2y=8$?

    $(3,\frac{5}{2})$. Add equations to eliminate $y$: $4x=12$, so $x=3$.

39. Identify the solution to the system: $y=-x+2$ and $2x+y=5$.

    $(3,-1)$. Substituting $y=-x+2$ into second equation gives $x=3$, then $y=-1$.

40. What does it mean if two linear equations graph as parallel lines in a system?

    No solution (inconsistent system). Parallel lines never intersect, so no common point exists.

41. What is the slope-intercept form used to compare slopes and intercepts of lines?

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

42. What is the solution to the system $y=3x+2$ and $y=11$?

    $(3,11)$. Substitute $y=11$ into first equation: $11=3x+2$, so $x=3$.

43. Which classification fits the system $y=3x-2$ and $y=3x+5$: one, none, or infinitely many solutions?

    No solution. Same slope (3) but different y-intercepts means parallel lines.

44. Identify the solution to the system: $3x-2y=4$ and $x+2y=8$.

    $(3,\frac{5}{2})$. Add equations to eliminate $y$: $4x=12$, so $x=3$.

45. What does it mean if two equations represent the same line?

    Infinitely many solutions. Same line means every point satisfies both equations.

46. What does it mean if two linear equations have infinitely many solutions?

    They represent the same line (equivalent equations). One equation is a multiple of the other, so they overlap completely.

47. What is the solution type if elimination produces a false statement like $0=5$?

    No solution. All variables cancel, leaving a contradiction.

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

    An ordered pair $(x,y)$ that makes both equations true. Both equations must be satisfied simultaneously by the same point.

49. What is the solution to the system $y=x^2$ and $y=x+2$?

    $(-1,1)$ and $(2,4)$. Set $x^2=x+2$, solve quadratic $x^2-x-2=0$.

50. What is the substitution method used for in systems of equations?

    Replace a variable using one equation, then solve the other. Express one variable in terms of the other to reduce to one equation.

51. Identify the solution to the system $4x+2y=10$ and $2x+y=5$.

    Infinitely many solutions. First equation is second multiplied by 2, so same line.

52. What is the value of $x$ in the system $3x+2y=16$ and $3x-2y=4$?

    $x=\frac{10}{3}$. Add equations to eliminate $y$: $6x=20$, so $x=\frac{10}{3}$.

53. What is the slope-intercept form of a line used to compare slopes in a system?

    $y=mx+b$. Standard form showing slope $m$ and y-intercept $b$.

54. What condition on slopes indicates a system of two lines has exactly one solution?

    Different slopes: $m_1 \ne m_2$. Lines with different slopes must intersect at exactly one point.

55. Which condition guarantees a unique solution for $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$?

    $\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$. Different coefficient ratios mean lines intersect once.

56. What is the solution type if elimination produces a true statement like $0=0$?

    Infinitely many solutions. All variables cancel, leaving a true identity.

57. What condition on slopes and intercepts indicates a system has no solution?

    Same slope, different intercepts: $m_1 = m_2$ and $b_1 \ne b_2$. Parallel lines never intersect.

58. What value of $k$ makes the system have infinitely many solutions? $y=3x+2$ and $y=3x+k$.

    $k=2$. Same slope and intercept means identical lines.

59. Identify the solution to the system: $2x+y=9$ and $x-y=0$.

    $(3,3)$. From $x-y=0$, $x=y$; substitute into first equation.

60. Identify the solution to the system: $x-2y=0$ and $3x+2y=16$.

    $(4,2)$. Adding equations eliminates $y$: $4x=16$, so $x=4$, then $y=2$.

61. What does it mean if a system of two linear equations has exactly one solution?

    The lines intersect at exactly one point $(x,y)$. This occurs when lines have different slopes.

62. What is the standard form of a linear equation commonly used in elimination?

    $Ax+By=C$. Coefficients are integers, making elimination easier.

63. What is the solution $(x,y)$ to the system $y=2x+1$ and $y=9$?

    $(4,9)$. Substitute $y=9$ into first equation: $9=2x+1$.

64. What does a solution $(x,y)$ to a system represent on a coordinate plane?

    The intersection point of the graphs of the equations. Where the two lines cross on the graph.

65. What does it mean for a system of two linear equations to have no solution?

    The lines are parallel: same slope, different intercepts. Parallel lines never meet, so no point satisfies both equations.

66. What does it mean if two equations in a linear system represent the same line?

    Infinitely many solutions. Same line means every point on it satisfies both equations.

67. What is the solution to the system $x^2+y^2=25$ and $y=0$?

    $(5,0)$ and $(-5,0)$. Circle intersects x-axis where $y=0$, so $x^2=25$.

68. What is the meaning of a solution to a system of equations?

    An ordered pair $(x,y)$ that makes both equations true. Values that satisfy both equations simultaneously.

69. What is the meaning of a negative solution like $x=-2$ in a system word problem context?

    It may be extraneous if the context requires $x\ge 0$. Negative values may not make sense for quantities like length.

70. What is the solution set type for $y=3x-2$ and $y=3x+5$?

    No solution. Parallel lines (same slope $m=3$) never intersect.

71. Identify the intersection point of the lines $x=3$ and $y=-2$.

    $(3,-2)$. Vertical line $x=3$ meets horizontal line $y=-2$ at $(3,-2)$.

72. What does it mean for an ordered pair $(x,y)$ to be a solution to a system?

    It makes both equations true when substituted. The point satisfies both equations simultaneously.

73. What is the solution to the system $y=x+4$ and $2x+y=1$?

    $(-1,3)$. Substitute $y=x+4$ into second: $2x+(x+4)=1$, so $x=-1$.

74. Identify the number of solutions: $2x-4y=8$ and $x-2y=4$.

    Infinitely many solutions. Second equation is first divided by 2; they're the same line.

75. What is the substitution method first step when one equation is already solved for a variable?

    Substitute that expression into the other equation. Replace the variable in equation 2 with the expression from equation 1.

76. What is the solution type of the system $2x+4y=6$ and $x+2y=3$?

    Infinitely many solutions. Second equation is first divided by 2, so they're the same line.

77. What is the $x$-value of the solution to $x+y=1$ and $x-y=9$?

    $5$. Adding equations gives $2x=10$, so $x=5$.

78. What is the slope-intercept form of a line used when comparing two equations in a system?

    $y=mx+b$. Standard form showing slope $m$ and y-intercept $b$.

79. Which condition indicates no solution for $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$?

    $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$. Same coefficient ratios but different constant ratio means parallel lines.

80. Identify the solution to the system $x=3y$ and $x+y=12$.

    $(9,3)$. Substitute first into second: $3y+y=12$, so $y=3$ and $x=9$.

81. What does an ordered pair $(x,y)$ represent in a system of two equations?

    A point that satisfies both equations simultaneously. The solution where both lines intersect.

82. What method solves a system by replacing a variable using an equivalent expression?

    Substitution. Solve one equation for a variable, then substitute into the other.

83. What is the solution $(x,y)$ to the system $y=x-2$ and $y=-x+4$?

    $(3,1)$. Set equations equal: $x-2=-x+4$, solve for $x=3$.

84. Identify the solution to the system $2x+y=9$ and $x+y=6$.

    $(3,3)$. Subtracting second from first gives $x=3$; then $3+y=6$ gives $y=3$.

85. What value of $k$ makes the system have no solution? $y=3x+2$ and $y=3x+k$.

    Any $k\ne 2$. Same slope requires no solution; different intercepts needed.

86. What are the three possible solution counts for a system of two linear equations in two variables?

    One solution, no solution, or infinitely many solutions. Linear systems can intersect once, never, or everywhere.

87. What does it mean if two linear equations have exactly one solution?

    The lines intersect at exactly one point. Unique solution occurs when lines have different slopes.

88. Identify the solution to the system: $x=2y$ and $x+y=9$.

    $(6,3)$. Substituting $x=2y$ into $2y+y=9$ gives $y=3$, then $x=6$.

89. What is the solution $(x,y)$ to the system $x+2y=8$ and $y=3$?

    $(2,3)$. Substitute $y=3$ into first equation: $x+2(3)=8$.

90. Identify the solution to the system: $x+y=7$ and $x-y=1$.

    $(4,3)$. Adding equations gives $2x=8$, so $x=4$ and $y=3$.

91. Identify the solution to the system $3x-2y=4$ and $x-2y=0$.

    $(2,1)$. Subtracting second from first gives $2x=4$, so $x=2$; then $y=1$.

92. Identify the solution to the system $2x+3y=12$ and $4x+6y=24$.

    Infinitely many solutions. Second equation is first multiplied by 2; same line.

93. Identify the solution to the system: $y=-x+6$ and $y=x$.

    $(3,3)$. Setting equations equal: $-x+6=x$ gives $x=3$, so $y=3$.

94. Identify the solution to the system $x-3y=-11$ and $2x-3y=-5$.

    $(x,y)=(6,\frac{17}{3})$. Subtract equations to get $-x=-6$, so $x=6$; then $y=\frac{17}{3}$.

95. What is the first step to solve the system $y=3x-2$ and $2x+y=10$ by substitution?

    Substitute $y=3x-2$ into $2x+y=10$. Replace $y$ in the second equation with the expression from the first.

96. What condition on slopes guarantees exactly one solution for $y=m_1x+b_1$ and $y=m_2x+b_2$?

    $m_1\ne m_2$. Different slopes guarantee the lines intersect at one point.

97. What are the three possible numbers of solutions for a system of two linear equations?

    $0$, $1$, or infinitely many solutions. Linear systems can have unique intersection, parallel lines, or identical lines.

98. What condition on slopes indicates a system of two lines has no solution?

    Same slope, different $y$-intercepts. Parallel lines never intersect.

99. What is the slope-intercept form of a line used to compare slopes and intercepts?

    $y=mx+b$. Standard form showing slope $m$ and y-intercept $b$.

100. What is the slope-intercept form of a line used to compare slopes in systems?

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

101. Identify the value of $k$ that makes the system $x+ky=4$ and $2x+6y=8$ have infinitely many solutions.

     $k=3$. For same line, need $\frac{1}{2}=\frac{k}{6}=\frac{4}{8}$, so $k=3$.

102. Identify the solution to the system: $y=2x+1$ and $y=-x+7$.

     $(2,5)$. Set equations equal: $2x+1=-x+7$; solve for $x$.

103. What is the slope-intercept form of a linear equation often used for graphing systems?

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

104. Identify whether the system $x+2y=5$ and $2x+4y=12$ has one, none, or infinitely many solutions.

     No solution. Second equation simplifies to $x+2y=6$, not equivalent to first.

105. Identify the solution to the system: $2x+3y=12$ and $4x+6y=10$.

     No solution. Second is twice first with different constant; parallel lines.

106. What is the solution $(x,y)$ to the system $2x+3y=12$ and $4x+6y=24$?

     Infinitely many solutions. Second equation is $2 \times$ first, representing same line.

107. Identify the solution to the system $x=2y$ and $x+y=9$.

     $(x,y)=(6,3)$. Substitute $x=2y$ into second equation: $2y+y=9$, so $y=3$.

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

     $(2,3)$. Set equations equal: $-x+5=x+1$, so $2x=4$ and $x=2$.

109. Identify the number of solutions if one equation is $2x+4y=6$ and the other is $x+2y=3$.

     Infinitely many solutions. Second equation is first divided by 2, showing same line.

110. 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 $y=3$.

111. What is the solution to the system $y=x^2$ and $y=4$?

     $(2,4)$ and $(-2,4)$. Substitute $y=4$ into $y=x^2$ to get $x^2=4$.

112. Identify the solution $(x,y)$ to the system $3x-2y=4$ and $x+2y=8$.

     $(3,\frac{5}{2})$. Add equations: $4x=12$, so $x=3$; substitute to get $y=\frac{5}{2}$.

113. What is the solution to the system $y=2x+1$ and $y=5$?

     $(2,5)$. Substitute $y=5$ into first equation: $5=2x+1$, so $x=2$.

114. What is the solution $(x,y)$ to the system $x+y=7$ and $x-y=1$?

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

115. Which method is most efficient if one equation is already solved for $y$, as in $y=mx+b$?

     Substitution. Direct substitution avoids solving for a variable first.

116. What is the slope of the line that represents $3x+2y=10$?

     $-\frac{3}{2}$. Rewrite as $y=-\frac{3}{2}x+5$ to identify slope.

117. What is the solution to the system $4x+2y=10$ and $2x+y=5$?

     Infinitely many solutions. First equation is twice the second; they're the same line.

118. What is the substitution method for solving a system of equations?

     Solve for one variable, then substitute into the other equation. Replace one variable with its expression from the other equation.

119. What does it mean if a system of two linear equations has no solution?

     The lines are parallel (same slope, different intercepts). Lines never meet when they have equal slopes but different y-intercepts.

120. Identify the solution to the system $3x-2y=4$ and $x+2y=8$.

     $(3,\frac{5}{2})$. Add equations: $4x=12$, so $x=3$; then $y=\frac{5}{2}$.

121. What is the value of $y$ in the system $3x+2y=16$ and $3x-2y=4$?

     $y=3$. Subtract equations to eliminate $x$: $4y=12$, so $y=3$.

122. What is the value of $x$ in the system $3x-2y=4$ and $3x+2y=16$?

     $x=\frac{10}{3}$. Add equations: $6x=20$, so $x=\frac{20}{6}=\frac{10}{3}$.

123. What is the solution to the system $x+y=7$ and $x-y=1$?

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

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

     $(3,3)$. Subtract second from first: $x=3$; substitute back: $y=3$.

125. What does it mean for a system of two linear equations to have exactly one solution?

     The lines intersect at exactly one point $(x,y)$. This occurs when the lines have different slopes.

126. What is the standard form of a linear equation used in many systems problems?

     $Ax+By=C$. All coefficients and constant on same side.

127. What is the elimination method goal when solving a linear system?

     Add or subtract equations to eliminate one variable. This simplifies the system to one equation with one variable.

128. What condition on slopes and intercepts guarantees infinitely many solutions for $y=m_1x+b_1$ and $y=m_2x+b_2$?

     $m_1=m_2$ and $b_1=b_2$. Both equations describe the exact same line.

129. What is the elimination method for solving a system of equations?

     Add or subtract equations to eliminate one variable. Multiply equations if needed to cancel a variable when combining.

130. What does it mean if two linear equations graph as intersecting lines in a system?

     Exactly one solution (consistent independent system). Lines cross at one point, giving a unique solution.

131. Which value of $k$ makes the system $kx+y=5$ and $2x+y=5$ have no solution?

     $k=2$. For no solution, need same slope: $k=2$ makes both slopes 2.

132. What does it mean if a system of two linear equations has infinitely many solutions?

     The equations represent the same line (equivalent equations). This happens when both equations simplify to the same form.

133. Identify the number of solutions: $y=\frac{1}{2}x+3$ and $y=-2x+3$.

     One solution. Different slopes ($\frac{1}{2}$ and $-2$) guarantee intersection.

134. Identify the solution to the system $x+y=7$ and $x-y=1$.

     $(4,3)$. Adding gives $2x=8$, so $x=4$; subtracting gives $2y=6$, so $y=3$.

135. What is the number of solutions to the system $2x+4y=8$ and $4x+8y=16$?

     Infinitely many solutions. Second equation is exactly twice the first, so same line.

136. Identify the solution count if $a_1b_2-a_2b_1=0$ for two linear equations in $x$ and $y$.

     Either no solution or infinitely many solutions. Zero determinant means parallel or same line.

137. What is the elimination method goal when combining two equations in a linear system?

     Make one variable cancel by adding or subtracting equations. Multiply equations if needed so coefficients match, then add/subtract.

138. Identify the solution to the system: $x+y=5$ and $x-y=1$.

     $(3,2)$. Adding equations gives $2x=6$, so $x=3$; then $y=2$.

139. What is the elimination goal when solving a system by addition/subtraction?

     Make one variable cancel to leave a one-variable equation. Add/subtract equations to eliminate one variable and solve for the other.

140. Identify the solution $(x,y)$ to the system $y=3x-2$ and $2x+y=8$.

     $(2,4)$. Substitute $y=3x-2$ into equation 2: $2x+3x-2=8$, so $x=2$, $y=4$.

141. What is the solution $(x,y)$ to the system $x=2y$ and $x+y=9$?

     $(6,3)$. Substitute $x=2y$ into $x+y=9$: $2y+y=9$, so $y=3$, $x=6$.

142. What is the substitution method first step when solving a linear system?

     Solve one equation for one variable in terms of the other. Then substitute this expression into the other equation.

143. Which system represents a unique solution: $y=x+1$ with $y=-x+3$, or $y=2x$ with $y=2x+5$?

     $y=x+1$ and $y=-x+3$. First pair has different slopes; second has same slope (parallel).

144. What are the three possible numbers of solutions for a linear system of two equations?

     One solution, no solution, or infinitely many solutions. Linear systems can intersect once, never, or everywhere.

145. What is the substitution method first step for solving a system?

     Solve one equation for one variable. Express one variable in terms of the other to substitute.

146. What does the solution $(x,y)$ of a system represent on a graph?

     The intersection point of the graphs. The solution is where both lines meet.

147. What method solves a system by adding equations to eliminate a variable?

     Elimination (linear combination). Add/subtract equations to cancel out one variable.

148. What is the substitution method for solving a system of equations?

     Solve one equation for a variable and substitute into the other. Replace the variable in one equation with its expression from the other.

149. What is the elimination method for solving a system, stated in one sentence?

     Add/subtract equations to eliminate one variable. Combine equations to cancel out one variable.

150. What conclusion follows if two linear equations are equivalent (same line)?

     The system has infinitely many solutions. Same equation means every point on one line is on the other.

151. Which value of $k$ makes the system $kx+2y=8$ and $3x+2y=8$ have infinitely many solutions?

     $k=3$. For same line, need identical equations when $k=3$.

152. Identify the solution to the system $y=2x+1$ and $y=9-x$.

     $\left(\frac{8}{3},\frac{19}{3}\right)$. Set $2x+1=9-x$; solve $3x=8$ to get $x=\frac{8}{3}$.

153. What is the solution to the system $2x+y=9$ and $x-y=3$?

     $(4,1)$. Add equations: $3x=12$, so $x=4$; substitute to find $y=1$.

154. Identify the number of solutions for $y=3x+2$ and $y=3x-5$.

     $0$ solutions. Same slope ($3$) but different y-intercepts means parallel lines.

155. Identify the number of solutions if two lines have slopes $m_1=-2$ and $m_2=5$.

     One solution. Different slopes guarantee lines intersect at one point.

156. What is the elimination method for solving a system of equations?

     Add or subtract equations to eliminate one variable. Multiply equations if needed so coefficients match, then add/subtract.

157. Identify the solution to the system: $x+y=5$ and $2x+2y=10$.

     Infinitely many solutions. Second equation is twice the first, so same line.

158. What is the slope of the line $4x-2y=10$ after rewriting in slope-intercept form?

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

159. What is the solution $(x,y)$ to the system $3x+y=10$ and $3x-y=2$?

     $(2,4)$. Add equations to eliminate $y$: $6x=12$, so $x=2$.

160. What is the meaning of an ordered pair $(x,y)$ that solves a system?

     It makes both equations true at the same time. The values satisfy both equations when substituted.

161. Identify the solution to the system: $2x+y=9$ and $x-y=0$.

     $(3,3)$. From $x-y=0$, we get $x=y$; substituting gives $3x=9$.

162. Identify the number of solutions: $y=-3x+2$ and $y=-3x-5$.

     No solution. Same slope $m=-3$, different $y$-intercepts means parallel lines.

163. What is the slope-intercept form you may graph to solve a linear system?

     $y=mx+b$. Standard form showing slope $m$ and y-intercept $b$.

164. Identify whether the system $y=3x-2$ and $y=3x+5$ has one, none, or infinitely many solutions.

     No solution. Same slope (3) but different y-intercepts means parallel lines.

165. Identify the solution to the system: $2x+y=7$ and $x+y=4$.

     $(3,1)$. Subtracting second from first gives $x=3$, then $y=1$.

166. Identify the solution to the system: $y=\frac{1}{2}x-1$ and $y=\frac{1}{2}x+3$.

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

167. Identify the solution to the system $y=x-4$ and $2x+y=5$.

     $(3,-1)$. Substituting $y=x-4$ into second: $2x+(x-4)=5$ gives $x=3$, $y=-1$.

168. What is the solution $(x,y)$ to the system $x+y=5$ and $2x+2y=12$?

     No solution. Second equation simplifies to $x+y=6$, contradicting first.

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

     $(6,3)$. Substitute $x=2y$ into second: $2y+y=9$, so $y=3$ and $x=6$.

170. What is the value of $k$ that makes the system $x+ky=6$ and $2x+4y=12$ have infinitely many solutions?

     $k=2$. For same line, coefficients must be proportional: $\frac{1}{2}=\frac{k}{4}$.

171. Identify the solution to the system $y=x^2$ and $y=4$.

     $(-2,4)$ and $(2,4)$. Substitute $y=4$ into $y=x^2$: $x^2=4$, so $x=pm 2$.

172. What is the intersection point of $x=2$ and $y=-1$?

     $(2,-1)$. Vertical line $x=2$ meets horizontal line $y=-1$ at $(2,-1)$.

173. Identify the solution to the system: $x+y=7$ and $x-y=1$.

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

174. What is the definition of an ordered-pair solution to a system?

     A pair $(x,y)$ that satisfies both equations simultaneously. The point where both equations are true at the same time.

175. What is the elimination method goal when solving a system of equations?

     Create opposite coefficients to eliminate one variable by adding. Adding equations with opposite coefficients cancels a variable.

176. Identify the solution $(x,y)$ to the system $y=-x+5$ and $y=x+1$.

     $(2,3)$. Set equations equal: $-x+5=x+1$, so $2x=4$, $x=2$, $y=3$.

177. What is the $y$-intercept of the line $2x+3y=12$ after solving for $y$?

     $4$. Rewrite as $y=-\frac{2}{3}x+4$; y-intercept is 4.

178. What does it mean if elimination produces a false statement like $0=5$?

     No solution. False statement means the lines are parallel with no intersection.

179. What is the elimination method goal when combining two linear equations?

     Make one variable coefficient sum to $0$. Adding/subtracting equations to eliminate a variable.

180. What does it mean if two linear equations have no solution?

     The lines are parallel and never intersect. Same slope but different y-intercepts means no intersection.

181. What does it mean if two lines in a system are parallel (different intercepts)?

     No solution. Parallel lines never intersect, so no common point exists.

182. What ordered pair is the solution to a system when you solve it algebraically?

     The $(x,y)$ that makes both equations true. The intersection point satisfies both equations simultaneously.

183. What is the elimination method for solving a system used for?

     Adding or subtracting equations to eliminate a variable. Combine equations to cancel out one variable.

184. What is the ordered-pair solution if two lines intersect at $(a,b)$ on the graph?

     $(a,b)$. The intersection point is the solution to the system.

185. What point represents the solution of a system when you solve by graphing two lines?

     The intersection point $(x,y)$ of the graphs. Where the two lines cross gives the $(x,y)$ values that satisfy both.

186. What does it mean if two linear equations graph as the same line in a system?

     Infinitely many solutions (dependent system). Same line means every point satisfies both equations.

187. What does it mean for a system of two linear equations to have infinitely many solutions?

     The equations are equivalent and represent the same line. Every point on the line satisfies both equations.

188. Identify the solution to the system $2x+y=9$ and $x-y=0$.

     $(3,3)$. From $x-y=0$, we get $x=y$; substitute into first equation.

189. What does it mean if a system of two linear equations has no solution?

     The lines are parallel (same slope, different intercepts). No intersection point exists between parallel lines.

190. What is the solution type for the system: $2x+4y=8$ and $x+2y=4$?

     Infinitely many solutions. Second equation is first divided by 2, so they're the same line.

191. What is the determinant test for a unique solution to $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$?

     Unique solution if $a_1b_2-a_2b_1\ne 0$. Non-zero determinant means lines intersect at one point.

192. Identify the solution to the system $y=x^2$ and $y=4$.

     $(-2,4)$ and $(2,4)$. Substitute $y=4$ into $y=x^2$: $x^2=4$, so $x=\pm 2$.

193. What is the solution to the system $x=4$ and $2x+y=13$?

     $(4,5)$. Substitute $x=4$ into second equation: $8+y=13$, so $y=5$.

194. What are the three possible numbers of solutions for two linear equations in $x$ and $y$?

     $0$, $1$, or infinitely many solutions. Lines can intersect once, never (parallel), or be the same line.

195. What is the slope-intercept form of a line used for graphing systems?

     $y=mx+b$. Shows slope $m$ and y-intercept $b$ for easy graphing.

196. What is the substitution method for solving a system of equations?

     Solve for one variable, substitute into the other equation. Express one variable in terms of the other, then substitute back.

197. Identify whether the system $2x-4y=6$ and $x-2y=3$ has one, none, or infinitely many solutions.

     Infinitely many solutions. Second equation is first multiplied by 2, so same line.

198. Identify the solution to the system $3x-2y=4$ and $x+2y=8$.

     $(3,\frac{5}{2})$. Add equations: $4x=12$, so $x=3$; then $y=\frac{5}{2}$.

199. What condition on slopes and intercepts guarantees no solution for $y=m_1x+b_1$ and $y=m_2x+b_2$?

     $m_1=m_2$ and $b_1\ne b_2$. Same slope means parallel; different intercepts means distinct lines.

200. What are the three possible numbers of solutions for a system of two linear equations?

     One solution, no solution, or infinitely many solutions. Linear systems can have unique, no, or infinite solutions.

201. What does the intersection point of two lines represent in a linear system?

     The solution $(x,y)$ that satisfies both equations. The common point where both lines meet.

202. What is the substitution method for solving a system used for?

     Replacing a variable using one equation to solve the other. Express one variable in terms of the other, then substitute.

203. What condition indicates a system of two lines has infinitely many solutions?

     Same line: equations are equivalent (same $m$ and same $b$). Identical lines overlap completely.

204. What is the substitution method for solving a system of equations?

     Solve one equation for a variable and substitute into the other. Replace the variable in the second equation with its expression.

205. What does it mean if a system of two linear equations has infinitely many solutions?

     The equations represent the same line (equivalent equations). Every point on the line satisfies both equations.

206. What is the solution $(x,y)$ to the system $2x+y=11$ and $x=4$?

     $(4,3)$. Substitute $x=4$ into first equation: $2(4)+y=11$.

207. Identify the number of solutions to the system $2x+3y=12$ and $4x+6y=25$.

     No solution. Left sides proportional but right sides aren't; parallel lines.

208. Identify the solution to the system: $y=2x+1$ and $y=x+4$.

     $(3,7)$. Setting $2x+1=x+4$ gives $x=3$, then $y=7$.

209. Identify the solution to the system $y=2x+1$ and $y=-x+7$.

     $(2,5)$. Setting equal: $2x+1=-x+7$ gives $3x=6$, so $x=2$ and $y=5$.

210. What is the solution set type for $2x-4y=6$ and $x-2y=3$?

     Infinitely many solutions. Second equation is half the first, so they're the same line.

211. Which condition indicates infinitely many solutions for $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$?

     $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$. All ratios equal means the equations are proportional.

212. Identify the value of $k$ that makes the system $x+ky=4$ and $2x+6y=10$ have no solution.

     $k=3$. For parallel lines, need same slope: $\frac{1}{2}=\frac{k}{6}$, so $k=3$.

213. What is the solution to the system $x+y=10$ and $x-y=4$?

     $(7,3)$. Add equations to get $2x=14$, so $x=7$; subtract to get $2y=6$, so $y=3$.

214. What is the slope-intercept form used to compare slopes in a linear system?

     $y=mx+b$. This form makes it easy to identify slope $m$ and y-intercept $b$.

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