Systems of Polynomial Equations - PSAT Math

1. What is the elimination step when two equations are equal to $y$, such as $y=f(x)$ and $y=g(x)$?

   Subtract to get $f(x)-g(x)=0$. This creates a single equation to find where curves intersect.

2. What should you do after finding possible $x$-values from substitution in a system?

   Substitute back to find $y$ and verify. Find corresponding $y$-values and check both equations are satisfied.

3. Identify the best first step to solve $y=2x+1$ and $x^2+y^2=5$ using substitution.

   Substitute $y=2x+1$ into $x^2+y^2=5$. Replace $y$ in the circle equation to get a quadratic in $x$.

4. Solve the system $y=x^2-1$ and $y=0$; what are the solution points?

   $(1,0)$ and $(-1,0)$. Set $x^2-1=0$: $(x-1)(x+1)=0$, so $x=pm 1$.

5. Solve the system $y=x^2$ and $x+y=6$ for $(x,y)$.

   $(2,4)$ and $(-3,9)$. Substitute $y=6-x$ into $y=x^2$ to get $x^2+x-6=0$.

6. What are the intersection points of $y=x^2+1$ and $y=2x+1$?

   $(0,1)$ and $(2,5)$. Set $x^2+1=2x+1$, which gives same $x$-values as $x^2=2x$.

7. Solve the system $x^2+y^2=4$ and $x=0$ for all solutions $(x,y)$.

   $(0,-2)$ and $(0,2)$. Substitute $x=0$ into circle equation: $y^2=4$, so $y=\pm 2$.

8. What substitution step is used if a system includes $y=f(x)$ and another equation in $x$ and $y$?

   Replace $y$ with $f(x)$ in the other equation. This substitution eliminates $y$ to solve for $x$ first.

9. What is the substitution method for a system like $y=f(x)$ and $y=g(x)$?

   Set $f(x)=g(x)$, solve for $x$, then find $y$. Since both equal $y$, set right sides equal to eliminate $y$.

10. Identify the intersection points of $y=x^2$ and $y=x+2$.

    $(-1,1)$ and $(2,4)$. Substitute to get $x^2=x+2$; solve $x^2-x-2=0$.

11. Solve the system $y=x^2+1$ and $y=2x$; what are the solutions?

    $(1,2)$. $x^2+1=2x$ gives $x^2-2x+1=0$, so $(x-1)^2=0$.

12. Find $k$ so the system $y=x^2$ and $y=2x+k$ has exactly one real solution.

    $k=-1$. Line tangent to parabola when discriminant of $x^2-2x-k=0$ equals zero.

13. Solve the system $y=x^2$ and $y=-x$; what are the solutions?

    $(0,0)$ and $(-1,1)$. $x^2=-x$ gives $x^2+x=0$, so $x(x+1)=0$.

14. What does it mean if substitution gives a contradiction such as $0=5$?

    No solution. The equations contradict each other, so no point satisfies both.

15. What is the elimination method for a polynomial system when terms can be canceled by adding equations?

    Multiply as needed, then add or subtract equations to eliminate a variable. Scale equations so matching terms cancel when combined.

16. What method solves a system by adding or subtracting equations to eliminate a variable?

    Elimination. Add/subtract equations to cancel out a variable.

17. Find the solutions to the system $y=x+1$ and $y^2=4$.

    $(-3,-2)$ and $(1,2)$. From $y^2=4$, $y=pm^2$, then find $x$ values.

18. Solve the system $y=x^2-4x+3$ and $y=0$; what are the solutions?

    $(1,0)$ and $(3,0)$. Factor: $(x-1)(x-3)=0$ gives $x=1,3$ with $y=0$.

19. Identify the intersection points of $y=x^2$ and $y=4$.

    $(2,4)$ and $(-2,4)$. Set $x^2=4$, so $x=pm 2$ with $y=4$ for both.

20. 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 $x^2+0^2=25$.

21. What equation represents the $x$-coordinates of intersection points of $y=f(x)$ and $y=g(x)$?

    Solve $f(x)-g(x)=0$ for $x$. Rearranging gives the equation whose roots are x-coordinates.

22. What does it mean if two polynomial equations have no real solution as a system?

    Their graphs have no real intersection points. The curves don't touch or cross in the real coordinate plane.

23. What is the standard elimination step if you have $p(x,y)=0$ and $q(x,y)=0$ and one equation is linear in $y$?

    Solve the linear equation for $y$ and substitute into the other. This reduces the system to a single-variable equation.

24. Solve the system $x^2+y^2=25$ and $y=0$; what are the solution points?

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

25. What are the $x$-values of intersection for $y=x^2$ and $y=4$?

    $x=-2$ and $x=2$. Solve $x^2=4$ to get $x=pm 2$.

26. What does each solution $(x,y)$ of a polynomial system represent on a graph?

    An intersection point of the graphs. Solutions occur where the curves meet.

27. What is the degree of the equation formed by substituting a linear equation into a cubic?

    Degree $3$ (a cubic in one variable). Linear substituted into cubic yields cubic.

28. What is the first step to solve $y=x^2+1$ and $y=2x+1$ by substitution?

    Set $x^2+1=2x+1$. Since both equal $y$, set the right sides equal.

29. Identify the solution set of $\begin{cases}y=2x+1\\y=x^2+1\end{cases}$.

    $\{(0,1),(2,5)\}$. Set $2x+1=x^2+1$: $x^2-2x=0$, factor to $x(x-2)=0$.

30. What is the standard substitution step after solving one equation for $y$ in a system?

    Replace $y$ in the other equation with that expression, then solve for $x$. Substitution eliminates one variable by replacement.

31. What is the result of substituting $y=2x-3$ into $x^2+y^2=13$ (single equation in $x$)?

    $x^2+(2x-3)^2=13$. Direct substitution expands to $x^2+4x^2-12x+9=13$.

32. What is the discriminant condition for exactly one intersection of $y=ax^2+bx+c$ and $y=mx+n$?

    After setting equal, discriminant $\Delta=0$. One solution means the resulting quadratic has discriminant zero.

33. What is the solution to the system $y=x^2-4$ and $y=0$?

    $(-2,0)$ and $(2,0)$. Find where parabola crosses $x$-axis by solving $x^2-4=0$.

34. Identify the solution set of $\begin{cases}y=x^2\\y=x+2\end{cases}$.

    $\{(-1,1),(2,4)\}$. Set $x^2=x+2$: $x^2-x-2=0$, factor to $(x-2)(x+1)=0$.

35. Find the solutions to the system $x+y=5$ and $xy=6$.

    $(2,3)$ and $(3,2)$. Solve quadratic $x^2-5x+6=0$ from substitution.

36. What is the degree of the equation formed by eliminating $y$ between a line and a parabola?

    Degree $2$. Line has degree 1, parabola has degree 2; product is 2.

37. What are the intersection points of $y=x^2$ and $y=4$?

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

38. What are the intersection points of $y=x^2$ and $y=x^2+1$?

    No solution (it would require $0=1$). Parallel parabolas never intersect.

39. What is the maximum possible number of real intersections of a line and a quadratic?

    At most $2$ real intersection points. A line crosses a parabola at most twice.

40. What do you do after finding an $x$-value from $f(x)=g(x)$ in a system $y=f(x)$, $y=g(x)$?

    Substitute $x$ into either equation to find the matching $y$. Use the $x$-value in either original equation to get $y$.

41. Solve the system $x^2+y^2=1$ and $y=0$ for all solutions $(x,y)$.

    $(-1,0)$ and $(1,0)$. Substitute $y=0$ into circle equation: $x^2=1$, so $x=\pm 1$.

42. What method solves a system by adding multiples of equations to eliminate a variable?

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

43. Identify the maximum number of real intersection points of a line and a parabola.

    $2$. A line can intersect a parabola at most twice.

44. What are the solution points to $y=x^2$ and $y=-1$?

    No real solutions. Since $x^2 \geq 0$, it can never equal $-1$.

45. Identify the solution set of $\begin{cases}y=x^2-4\\y=0\end{cases}$.

    $\{(-2,0),(2,0)\}$. Set $x^2-4=0$: $x^2=4$, so $x=\pm 2$.

46. Solve the system $y=x^2$ and $y=x^2+1$; how many solutions are there?

    $0$ solutions. Parallel parabolas (same shape, shifted vertically) never intersect.

47. What is the first step to solve the system $y=x^2$ and $y=2x+3$ algebraically?

    Set $x^2=2x+3$. Since both equal $y$, equate the right-hand sides.

48. What are the solutions to the system $y=x^2$ and $y=2x$?

    $(0,0)$ and $(2,4)$. Substitute $x=0$ and $x=2$ into either equation for $y$.

49. What are the solution points to $y=x^2+1$ and $y=1$?

    $(0,1)$. Solve $x^2+1=1$: only $x=0$ works.

50. What does it mean if $f(x)-g(x)$ has no real roots when solving $f(x)=g(x)$?

    No real intersection points; the system has no real solutions. The curves never meet in the real plane.

51. What are the intersection points of $y=x^2$ and $y=-x$?

    $(0,0)$ and $(-1,1)$. Solve $x^2=-x$ to get $x^2+x=0$, so $x(x+1)=0$.

52. What method solves a system by replacing one variable using an equation like $y=x^2$?

    Substitution. Replace one variable with its expression from another equation.

53. What are the intersection points of $y=x^2$ and $y=2x$?

    $(0,0)$ and $(2,4)$. Substitute $x=0$ and $x=2$ into either equation to find $y$.

54. Solve the system $y=(x-1)^2$ and $y=x+1$ for $(x,y)$.

    $(1,2)$ and $(4,5)$. Expand and set $(x-1)^2=x+1$; solve $x^2-3x=0$.

55. Solve the system $y=x^2-1$ and $y=0$; what are the solutions?

    $(1,0)$ and $(-1,0)$. $x^2-1=0$ gives $x=\pm 1$, then $y=0$ for both.

56. Solve the system $y=x^2$ and $y=x$ for all solutions $(x,y)$.

    $(0,0)$ and $(1,1)$. Set $x^2=x$ to get $x(x-1)=0$, so $x=0$ or $x=1$.

57. Solve the system $y=2x$ and $y=x^2$ for all real solutions.

    $(0,0)$ and $(2,4)$. Set $2x=x^2$: $x^2-2x=x(x-2)=0$, so $x=0$ or $x=2$.

58. Find the solution points to $y=x^2$ and $y=2x$.

    $(0,0)$ and $(2,4)$. Solve $x^2=2x$: $x(x-2)=0$, giving $x=0,2$.

59. What does it mean if $f(x)-g(x)$ is the zero polynomial when solving $f(x)=g(x)$?

    Infinitely many solutions; the equations represent the same curve. The two equations describe identical curves.

60. Find the solutions to the system $x-y=1$ and $x^2-y^2=5$.

    $(3,2)$. Factor as $(x+y)(x-y)=5$ with $x-y=1$.

61. Solve the system $y=x^2+2$ and $y=2x+2$ for all solutions $(x,y)$.

    $(0,2)$ and $(2,6)$. Set $x^2+2=2x+2$ to get $x^2-2x=0$, so $x=0$ or $x=2$.

62. What does it mean if $f(x)-g(x)$ factors as $(x-a)^2$ when solving $f(x)=g(x)$?

    $x=a$ is a double root; the graphs are tangent there. The curves touch but don't cross at $x=a$.

63. What do the real solutions of a system $f(x)=g(x)$ represent on a graph?

    The $x$-coordinates of intersection points. Where the graphs intersect, their $x$-values are the solutions.

64. Solve the system $y=x^2$ and $x+y=0$; what are the solution points?

    $(0,0)$ and $(-1,1)$. Substitute $y=-x$ into $y=x^2$: $-x=x^2$, so $x^2+x=0$.

65. Solve the system $y=x^2-2$ and $y=x$; what are the solutions?

    $(-1,-1)$ and $(2,2)$. $x^2-2=x$ gives $x^2-x-2=0$, factoring to $(x+1)(x-2)=0$.

66. Solve the system $y=x^2$ and $y=2x$; what are the solution points?

    $(0,0)$ and $(2,4)$. From $x^2=2x$: $x(x-2)=0$, so $x=0$ or $x=2$.

67. What does a solution $(x,y)$ to a system of two polynomial equations represent on a graph?

    An intersection point of the two graphs. Solutions occur where both curves meet.

68. Solve the system $x+y=5$ and $x^2+y^2=13$; what are the solution points?

    $(2,3)$ and $(3,2)$. Substitute $y=5-x$ into circle equation and solve quadratic.

69. What is the maximum possible number of real solutions to a system of two quadratic equations?

    $4$. Two parabolas can intersect at most 4 times.

70. Solve the system $y=x^2+2$ and $y=x$; how many real solutions are there?

    $0$ real solutions. $x^2+2=x$ gives $x^2-x+2=0$; discriminant is negative.

71. What is the elimination step if you have $y=f(x)$ and $y=g(x)$ in a system?

    Set $f(x)=g(x)$. Since both equal $y$, set the expressions equal to each other.

72. Identify the first step to solve $y=x^2$ and $y=2x$ using substitution.

    Set $x^2=2x$. Since both equal $y$, equate the right sides.

73. Identify the number of real solutions to $x^2+y^2=1$ and $y=2$.

    $0$. Line $y=2$ is outside unit circle centered at origin.

74. What does it mean if a system has exactly one real solution?

    The graphs intersect at exactly one real point. This is the unique solution to the system.

75. Identify the polynomial equation in $x$ after eliminating $y$: $y=x^2+1$ and $y=3x-2$.

    $x^2+1=3x-2$. Set the two $y$-expressions equal to eliminate $y$.

76. What are the solution points to $y=x^2$ and $y=x^2$?

    Infinitely many solutions: all points on $y=x^2$. Identical equations mean every point on the curve satisfies both.

77. What is the maximum number of real intersection points between a line and a parabola?

    At most $2$. A quadratic equation has at most 2 real roots.

78. What are the solutions to the system $x^2+y^2=25$ and $y=0$?

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

79. What method solves a system by setting $y=f(x)$ from both equations and equating them?

    Substitution: set $f(x)=g(x)$ after isolating the same variable. Both equations express y in terms of x, so equate the expressions.

80. What is the maximum number of real intersection points of $y=x^2$ and $y=ax+b$?

    $2$. A parabola and line can intersect at most twice.

81. What is the maximum possible number of real intersection points of $y=$ quadratic and $y=$ cubic?

    At most $3$ real intersection points. A cubic minus a quadratic yields a cubic with at most 3 roots.

82. What is the maximum possible number of real intersections of $y=x^3$ and $y=mx+b$?

    $3$. A cubic and line can intersect at most three times.

83. What is the substitution step if you have $y=x^2$ and $y=2x+3$?

    Set $x^2=2x+3$. Substitute to eliminate $y$ since both expressions equal $y$.

84. What are the $x$-values solving the system $y=x^2$ and $y=2x$?

    $x=0$ or $x=2$. Solve $x^2=2x$: $x(x-2)=0$.

85. Solve the system $x^2+y^2=25$ and $y=4$ for $(x,y)$.

    $(3,4)$ and $(-3,4)$. Substitute $y=4$ into the circle equation: $x^2+16=25$.

86. What are the $x$-values of solutions to $y=x^2$ and $y=x$?

    $x=0$ or $x=1$. Solve $x^2=x$: $x(x-1)=0$.

87. What does it mean if two polynomial equations have infinitely many solutions as a system?

    The equations define the same curve (are equivalent). One equation is a multiple of the other, so they're the same curve.

88. Identify the solutions to $x^2+y^2=25$ and $y=0$.

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

89. Solve the system $y=x^2+1$ and $y=x^2+1$; how many solutions are there?

    Infinitely many solutions. Identical equations represent the same curve.

90. Identify the number of real intersections if $x^2+1=x$.

    $0$ real intersections. Rearranging gives $x^2-x+1=0$ with negative discriminant.

91. What is the solution set to the system $y=x^2$ and $y=x^2$?

    Infinitely many solutions: all points on $y=x^2$. Same equation twice means every point on curve satisfies both.

92. What are the solutions to the system $x^2+y^2=1$ and $y=0$?

    $(1,0)$ and $(-1,0)$. Substitute $y=0$ into circle equation: $x^2=1$.

93. Identify the system's solutions in the coordinate plane when $y=f(x)$ and $y=g(x)$ are given.

    All intersection points $(x,y)$ of the two graphs. Solutions occur where the curves meet in the plane.

94. Identify the best first step to solve the system $y=x^2+1$ and $y=3x+1$.

    Set $x^2+1=3x+1$. Since both equal $y$, set the expressions equal.

95. Solve the system $x^2+y^2=25$ and $y=3$; what are the solution points?

    $(4,3)$ and $(-4,3)$. Substitute $y=3$: $x^2+9=25$, so $x^2=16$, $x=±4$.

96. What are the solution points to $y=x^2$ and $y=-x$?

    $(0,0)$ and $(-1,1)$. Solve $x^2=-x$: $x(x+1)=0$, so $x=0$ or $x=-1$.

97. Identify the number of solutions if the system is $y=x^2+2x+1$ and $y=(x+1)^2$.

    Infinitely many solutions. Both equations represent the same parabola: $(x+1)^2=x^2+2x+1$.

98. Identify the degree of the equation formed after substitution: $y=x^2$ into $y=3x+1$.

    $2$ (it becomes $x^2=3x+1$). Substituting gives $x^2=3x+1$, which has degree 2.

99. Which equation in $x$ results from eliminating $y$ in $x+y=5$ and $xy=6$?

    $x(5-x)=6$. Substitute $y=5-x$ from first equation into second equation $xy=6$.

100. Identify the number of real solutions to $y=x^2$ and $y=1$.

     $2$. Solve $x^2=1$ to get $x=pm 1$.

101. Solve the system $y=x^2$ and $y=-x$; what are the solution points?

     $(0,0)$ and $(-1,1)$. Set $x^2=-x$: $x^2+x=0$, $x(x+1)=0$, so $x=0$ or $x=-1$.

102. What does it mean for $(a,b)$ to be a solution to a system of polynomial equations?

     $(a,b)$ makes every equation in the system true. The point must satisfy all equations simultaneously.

103. Identify the number of real solutions to $y=x^2+1$ and $y=0$.

     $0$. Parabola $y=x^2+1$ has vertex at $(0,1)$, never touches $x$-axis.

104. What does it mean if substitution produces a contradiction such as $0=5$?

     No solution (no intersection). The curves never meet.

105. Identify the best next step to solve $y=x^2$ and $y=2x+3$ by substitution.

     Set $x^2=2x+3$. Substitute the second equation into the first.

106. What are the intersection points of $y=x^2+2x$ and $y=0$?

     $(-2,0)$ and $(0,0)$. Factor $x^2+2x=0$ as $x(x+2)=0$ to find $x$-intercepts.

107. What is the maximum possible number of real intersection points of two parabolas?

     $4$. Two parabolas (degree 2) can intersect at most $2 \times 2 = 4$ points.

108. What method solves a system by setting two expressions for $y$ equal, such as $y=f(x)$ and $y=g(x)$?

     Substitution: solve $f(x)=g(x)$, then back-substitute for $y$. This method works when both equations are already solved for the same variable.

109. What does it mean if a system has infinitely many solutions?

     The equations represent the same curve (equivalent equations). Every point on one curve is also on the other.

110. What is the maximum possible number of real intersection points of $y=ax^2+bx+c$ and $y=mx+b$?

     $2$. A parabola and line can intersect at most twice.

111. What are the solutions to the system $x^2+y^2=4$ and $y=x$?

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

112. Identify the solution set of $\begin{cases}y=x^2\\2y=2x^2\end{cases}$.

     Infinitely many solutions (same equation). Second equation is first multiplied by 2.

113. Identify the maximum number of real intersection points of two parabolas.

     $4$. Two parabolas can intersect at up to 4 points.

114. What method solves a system by substituting one polynomial expression into the other equation?

     Substitution. Replace one variable with its expression from the other equation.

115. What does a real solution $(x,y)$ to a system of two polynomial equations represent on a graph?

     An intersection point of the two graphs. Solutions occur where the curves meet on the coordinate plane.

116. What method solves a system by adding or subtracting equations to eliminate a variable?

     Elimination: combine equations to cancel one variable. Multiply equations by constants if needed to make coefficients opposites.

117. What is the solution set of the system $y=x^2$ and $y=2x$?

     $(0,0)$ and $(2,4)$. Solve $x^2=2x$ to get $x(x-2)=0$, so $x=0$ or $x=2$.

118. What is the solution set of the system $y=x^2$ and $y=4$?

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

119. What are the $x$-values solving the system $y=x^2$ and $y=2x$?

     $x=0$ or $x=2$. Solve $x^2=2x$ to get $x(x-2)=0$.

120. Solve the system $y=x^2+1$ and $y=x+1$ for all solutions $(x,y)$.

     $(0,1)$ and $(1,2)$. Set $x^2+1=x+1$ to get $x^2-x=0$, so $x=0$ or $x=1$.

121. What does it mean for $(x,y)$ to be a solution to a system of polynomial equations?

     $(x,y)$ makes every equation in the system true at the same time. A solution satisfies all equations simultaneously.

122. Solve the system $y=x^2$ and $x+y=2$ for $(x,y)$.

     $(1,1)$ and $(-2,4)$. Substitute $y=2-x$ into $y=x^2$ and solve.

123. What is the solution set of the system $y=x^2-2$ and $y=x$?

     $( -1,-1)$ and $(2,2)$. Solve $x^2-2=x$ to get $x^2-x-2=(x+1)(x-2)=0$.

124. What are the solutions to the system $y=x^2$ and $y=2x$?

     $(0,0)$ and $(2,4)$. Substitute $x$-values back to find $y$-values.

125. Identify the number of real solutions to $y=x^2$ and $y=-2$.

     $0$ real solutions. A parabola opening upward can't intersect a line below its vertex.

126. What elimination step is used if a system has two equations both equal to $0$?

     Add or subtract equations to eliminate a variable term. Combining equations strategically simplifies the system.

127. What are the solutions to the system $y=x^2$ and $x+y=2$?

     $(1,1)$ and $(-2,4)$. Substitute $y=2-x$ into $y=x^2$ and solve quadratic.

128. What is the solution set of the system $y=x^2+1$ and $y-x^2=1$?

     Infinitely many solutions. Both equations represent the same parabola.

129. What type of equation in one variable often results after substitution in a polynomial system?

     A polynomial equation in one variable. Substitution eliminates one variable, leaving a single-variable equation.

130. What are the $x$-values where $y=x^2$ and $y=2x$ intersect?

     $x=0$ and $x=2$. Solve $x^2=2x$ to get $x^2-2x=0$, so $x(x-2)=0$.

131. Identify the number of intersection points possible for two distinct lines.

     $0$, $1$, or infinitely many. Lines can be parallel (0), intersect at one point (1), or coincide (∞).

132. What are the solution points to $y=x^2$ and $y=x^2+1$?

     No solutions. The equations $x^2=x^2+1$ gives $0=1$, impossible.

133. Solve the system $y=x^2$ and $y=4$; what are the solutions?

     $(2,4)$ and $(-2,4)$. $x^2=4$ gives $x=\pm 2$, then $y=4$ for both.

134. What is the resulting equation after simplifying $x^2+9x^2=25$?

     $10x^2=25$. Combine like terms: $x^2+9x^2=10x^2$.

135. What does each real solution $(x,y)$ of a polynomial system represent on a graph?

     An intersection point of the graphs. Solutions occur where the curves meet.

136. What are the solution points to $y=x^2$ and $y=4$?

     $(-2,4)$ and $(2,4)$. Solve $x^2=4$: $x=\pm 2$, both give $y=4$.

137. What is the discriminant test for the number of real solutions of $ax^2+bx+c=0$?

     $b^2-4ac>0:2, =0:1, <0:0$ real solutions. Discriminant sign determines if roots are real, repeated, or complex.

138. Solve the system $y=x^2-3x$ and $y=0$; what are the solutions?

     $(0,0)$ and $(3,0)$. Factor: $x(x-3)=0$ gives $x=0,3$ with $y=0$.

139. Solve the system $y=x^2$ and $y=x$; what are the solution points?

     $(0,0)$ and $(1,1)$. Set $x^2=x$: $x^2-x=0$, $x(x-1)=0$, so $x=0$ or $x=1$.

140. What does a solution $(x,y)$ to a system of polynomial equations represent on a graph?

     An intersection point of the graphs. Where both polynomial curves meet on the coordinate plane.

141. What is the solution set of the system $y=x^2-4x+3$ and $y=0$?

     $(1,0)$ and $(3,0)$. Factor $x^2-4x+3=(x-1)(x-3)=0$ to get $x=1$ or $x=3$.

142. What is the solution set of the system $y=x^2$ and $y=x+2$?

     $( -1,1)$ and $(2,4)$. Solve $x^2=x+2$ to get $x^2-x-2=0$, factoring as $(x+1)(x-2)=0$.

143. Identify the solution set of $\begin{cases}x^2+y^2=25\\y=0\end{cases}$.

     $\{(-5,0),(5,0)\}$. Circle intersects x-axis where $y=0$: $x^2=25$, so $x=\pm 5$.

144. Solve the system $y=x^2-4$ and $y=0$; what are the solution points?

     $(-2,0)$ and $(2,0)$. Where parabola crosses $x$-axis: $x^2-4=0$ gives $x=±2$.

145. What is the elimination result when you subtract $y=x^2+3$ from $y=2x+1$?

     $2x+1=x^2+3$. Subtracting eliminates $y$ to solve for $x$.

146. Find the solution points to $y=x^2-1$ and $y=0$.

     $(-1,0)$ and $(1,0)$. Solve $x^2-1=0$: $(x-1)(x+1)=0$.

147. Identify the number of real solutions to $y=x^2$ and $y=0$.

     $1$. Parabola $y=x^2$ touches origin only.

148. What are the solutions to the system $y=x^2$ and $x=3$?

     $(3,9)$. Substitute $x=3$ into $y=x^2$.

149. Identify the number of solutions if $y=x^2+1$ and $y=x^2-3$ are in the same system.

     $0$ solutions. Parallel parabolas (same shape, different vertical shifts) never intersect.

150. What do the solutions of a system $y=f(x)$ and $y=g(x)$ represent on a graph?

     The intersection points of the graphs of $f$ and $g$. Where the curves cross, both equations are satisfied.

151. Solve the system $y=x^2+1$ and $y=2x+1$; what are the solution points?

     $(0,1)$ and $(2,5)$. Same $x$-values as $y=x^2$ and $y=2x$ since adding 1 shifts both up.

152. What method eliminates a variable by forming a new equation like $f(x)-g(x)=0$?

     Elimination by subtraction (equate-and-subtract). Subtracting equations eliminates $y$ when both equal $y$.

153. What are the intersection points of $y=x^2$ and $y=x+2$?

     $(-1,1)$ and $(2,4)$. Solve $x^2=x+2$ to get $x^2-x-2=0$, factor as $(x-2)(x+1)=0$.

154. Solve the system $y=x^2+2x$ and $y=0$; what are the solution points?

     $(0,0)$ and $(-2,0)$. Set $x^2+2x=0$: $x(x+2)=0$, so $x=0$ or $x=-2$.

155. What is the solution set of the system $y=x^2+1$ and $y=2$?

     $( -1,2)$ and $(1,2)$. Set $x^2+1=2$ to get $x^2=1$, so $x=pm 1$.

156. Identify the solution set of $\begin{cases}y=x^2\\y=4\end{cases}$.

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

157. What is the maximum possible number of real intersection points of a line and a parabola?

     $2$. A line can cross a parabola at most twice.

158. Identify the number of real intersections if $x^2=x$.

     $2$ real intersections. Factor as $x(x-1)=0$ giving two distinct solutions.

159. Solve the system $y=x^2-4x+3$ and $y=0$; what are the solution points?

     $(1,0)$ and $(3,0)$. Set $x^2-4x+3=0$: $(x-1)(x-3)=0$, so $x=1$ or $x=3$.

160. What method solves a system by setting two expressions for the same variable equal, such as $y=f(x)$ and $y=g(x)$?

     Substitution: set $f(x)=g(x)$ and solve. This eliminates one variable by equating the expressions.

161. What is the maximum possible number of real solutions to a system of a cubic and a line?

     $3$. A cubic and line can intersect at most 3 times.

162. Solve for $x$ in the intersection equation $x^2+1=2x+1$.

     $x=0$ or $x=2$. Simplify to $x^2-2x=0$, factor as $x(x-2)=0$.

163. Identify the intersection point of $y=x^2$ and $y=0$.

     $(0,0)$. The parabola touches the x-axis only at the origin.

164. What equation do you set up to solve the system $y=f(x)$ and $y=g(x)$ using substitution?

     Set $f(x)=g(x)$, then solve for $x$. Since both equal $y$, set the expressions equal to each other.

165. What does an intersection point of $y=f(x)$ and $y=g(x)$ represent in a system?

     A solution $(x,y)$ satisfying $f(x)=g(x)$. At intersections, both functions have the same $y$-value.

166. Solve the system $y=x^2$ and $y=x$; what are the solutions?

     $(0,0)$ and $(1,1)$. $x^2=x$ gives $x(x-1)=0$, so $x=0$ or $x=1$.

167. What method solves a system by setting two expressions for $y$ equal: $y=f(x)$ and $y=g(x)$?

     Substitution: solve $f(x)=g(x)$, then find $y$. Setting equal eliminates $y$ to solve for $x$ first.

168. What does it mean if substituting gives an identity such as $0=0$?

     Infinitely many solutions. The equations are equivalent, so every point satisfying one satisfies both.

169. What is the solution set of the system $y=x^2+2x$ and $y=0$?

     $( -2,0)$ and $(0,0)$. Factor $x^2+2x=x(x+2)=0$ to get $x=0$ or $x=-2$.

170. Solve the system $x^2+y^2=25$ and $y=3$ for $(x,y)$.

     $(4,3)$ and $(-4,3)$. Substitute $y=3$ into the circle equation: $x^2+9=25$.

171. What are the intersection points of $y=x^2$ and $y=x$?

     $(0,0)$ and $(1,1)$. Solve $x^2=x$ to get $x=0$ or $x=1$.

172. Solve the system $y=x^2$ and $y=x+2$; what are the solution points?

     $(-1,1)$ and $(2,4)$. From $x^2=x+2$: $x^2-x-2=0$, factor to $(x+1)(x-2)=0$.

173. Identify the number of solutions if $y=x^2$ and $y=x^2+1$ are in a system.

     $0$ solutions. Parallel parabolas never intersect.

174. What is the maximum possible number of real intersections of two parabolas (two quadratics)?

     $4$. Two quadratics can intersect at most four times.

175. What does it mean if substitution produces a contradiction such as $0=5$?

     No solution (the graphs do not intersect). The equations represent parallel or non-intersecting curves.

176. Identify the intersection points of $y=x^2$ and $y=2x$.

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

177. Solve the system $y=x^2-1$ and $y=0$ for all solutions $(x,y)$.

     $(-1,0)$ and $(1,0)$. Set $x^2-1=0$ to get $x=\pm 1$, both with $y=0$.

178. What does it mean if $f(x)-g(x)=0$ has no real solutions for $y=f(x)$ and $y=g(x)$?

     The graphs have no real intersection points. No real solutions means the curves don't intersect.

179. What is the elimination step for systems: how do you remove a variable using addition/subtraction?

     Multiply if needed, then add or subtract equations to cancel one variable. Elimination combines equations to remove a variable.

180. What is the substitution method for a system where one equation gives $y$ in terms of $x$?

     Substitute $y=f(x)$ into the other equation to get one equation in $x$. Replace $y$ with its expression to eliminate one variable.

181. What method solves a system by adding or subtracting equations to eliminate a variable?

     Elimination: combine equations to cancel one variable. Add/subtract equations strategically to make a variable disappear.

182. Solve the system $y=x^2$ and $y=-x$ for all solutions $(x,y)$.

     $(0,0)$ and $(-1,1)$. Set $x^2=-x$ to get $x(x+1)=0$, so $x=0$ or $x=-1$.

183. Solve the system $y=x^2$ and $y=4$ for all solutions $(x,y)$.

     $(-2,4)$ and $(2,4)$. Set $x^2=4$ to get $x=\pm 2$, both with $y=4$.

184. Solve the system $y=x^2$ and $y=4$; what are the solution points?

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

185. Find the solution points to $y=x^2+2x$ and $y=0$.

     $(-2,0)$ and $(0,0)$. Factor: $x(x+2)=0$, so $x=0$ or $x=-2$.

186. What is the solution set to the system $y=x^2$ and $y=x^2+1$?

     No solution. Parallel parabolas never intersect.

187. Solve the system $y=x^2+1$ and $y=2x$; what are the solution points?

     $(1,2)$. Set $x^2+1=2x$: $x^2-2x+1=0$, $(x-1)^2=0$, so $x=1$.

188. Find the intersection points of $y=x^2+1$ and $y=2x+1$.

     $(0,1)$ and $(2,5)$. Substitute $x=0$ and $x=2$ back to find $y$-values.

189. What is the maximum possible number of real intersection points of a line and a parabola?

     $2$ real intersection points. A line can cross a parabola at most twice.

190. What is the solution to the system $y=x^2+1$ and $y=x^2+3$?

     No solution. Parallel parabolas never intersect.

191. What equation results from setting the system $y=f(x)$ and $y=g(x)$ equal?

     $f(x)=g(x)$. Since both equal $y$, set the expressions equal to each other.

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

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

193. Which equation is formed by equating $y=2x+5$ and $y=-x^2+1$ to solve the system?

     $2x+5=-x^2+1$. Set equal since both expressions equal $y$.

194. What is the maximum number of real intersection points between two quadratics?

     At most $4$. Two degree-2 curves can intersect at most 4 times.

195. What is the solution set of the system $y=x^2+4$ and $y=4$?

     $(0,4)$ only. Since $x^2+4=4$ gives $x^2=0$, only $x=0$ works.

196. What is the degree of the equation formed by substituting a linear equation into a quadratic?

     Degree $2$ (a quadratic in one variable). Linear substituted into quadratic yields quadratic.

197. Identify the solution set of $\begin{cases}y=x^2+1\\y=x^2-3\end{cases}$.

     $\varnothing$ (no solution). Parallel parabolas $y=x^2+1$ and $y=x^2-3$ never intersect.

198. What does the graph of a system of two polynomial equations represent geometrically?

     The intersection points of the two polynomial graphs. Solutions occur where the curves meet on the coordinate plane.

199. Solve the system $x+y=0$ and $y=x^2$; what are the solutions?

     $(0,0)$ and $(-1,1)$. Substitute $y=-x$ into $y=x^2$: $-x=x^2$.

200. What does the intersection of $y=f(x)$ and $y=g(x)$ represent in a polynomial system?

     All solutions $(x,y)$ where $f(x)=g(x)$ and $y=f(x)$. Intersection points occur where both functions equal each other.

201. What is the maximum possible number of real intersection points of two distinct parabolas?

     $2$. Two parabolas can intersect at most twice.

202. What are the solutions to the system $x^2+y^2=25$ and $x=0$?

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

203. What is the standard substitution step for solving $y=f(x)$ and $y=g(x)$?

     Set $f(x)=g(x)$ and solve for $x$. Since both equal $y$, set the expressions equal to each other.

204. What is the maximum possible number of real solutions to a system of a line and a cubic?

     $3$. A line can cross a cubic curve up to 3 times.

205. Identify the number of solutions if the system is $y=x^2$ and $y=x^2$.

     Infinitely many solutions. Identical equations have all points in common.

206. What is the key check to avoid extraneous solutions after substitution in a system?

     Substitute each solution back into both original equations. Verify solutions satisfy both equations to catch errors.

207. What are the intersection points of $y=x^2$ and $y=-x+2$?

     $(1,1)$ and $(-2,4)$. Solve $x^2=-x+2$ to get $x^2+x-2=0$, factor as $(x+2)(x-1)=0$.

208. Identify the meaning of a repeated solution when solving $f(x)=g(x)$ from a system.

     A tangency (intersection with multiplicity $2$ or more). The curves touch without crossing at that point.

209. Solve the system $y=x^2$ and $y=-x$ for $(x,y)$.

     $(0,0)$ and $(-1,1)$. Set $x^2=-x$; rearrange to $x^2+x=0$ and factor.

210. Find $k$ so the system $y=x^2$ and $y=k$ has exactly one real solution.

     $k=0$. Horizontal line $y=0$ touches vertex of $y=x^2$ at origin.

211. What is the solution set of the system $y=x^2$ and $y=-x$?

     $(0,0)$ and $( -1,1)$. Solve $x^2=-x$ to get $x^2+x=x(x+1)=0$, so $x=0$ or $x=-1$.

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

     $(x,y)$ is a solution if it satisfies both equations simultaneously. The point must make both equations true when substituted.

213. Identify the intersection points of $y=x^2-1$ and $y=0$.

     $(1,0)$ and $(-1,0)$. Set $x^2-1=0$; factor $(x-1)(x+1)=0$.

214. What is the maximum possible number of intersection points of $y=$ (cubic) and $y=$ (quadratic)?

     Up to $3$ intersection points. A cubic and quadratic can meet at most 3 times.

215. What are the intersection points of $y=x^2$ and $y=-x$?

     $(0,0)$ and $(-1,1)$. Solve $x^2=-x$ to get $x=0$ or $x=-1$.

216. What are all solution points to the system $y=x^2$ and $y=x$?

     $(0,0)$ and $(1,1)$. Substitute $x$-values back: $y=0^2=0$ and $y=1^2=1$.

217. What does it mean if a polynomial system has no real solutions?

     The graphs do not intersect in the real plane. The curves never meet in real coordinates.

218. What does it mean if substitution produces an identity such as $0=0$?

     Infinitely many solutions (the equations are equivalent). The equations represent the same curve.

219. Identify the solution set of $\begin{cases}y=x^2\\y=-x^2\end{cases}$.

     $\{(0,0)\}$. Set $x^2=-x^2$: $2x^2=0$, so $x=0$ only.

220. What is the solution set of the system $y=x^2-1$ and $y=0$?

     $( -1,0)$ and $(1,0)$. Set $x^2-1=0$ to get $x=pm 1$.

221. What does it mean if substitution produces a true identity such as $0=0$?

     Infinitely many solutions (same curve). The equations represent the same curve.

222. Solve the system $y=x^2+1$ and $y=2x+1$ for $(x,y)$.

     $(0,1)$ and $(2,5)$. Set $x^2+1=2x+1$; simplify to $x^2-2x=0$.

223. What are the solutions to the system $y=x^2-1$ and $y=0$?

     $(-1,0)$ and $(1,0)$. The parabola crosses the $x$-axis at $x=pm^1$.

224. What is the maximum possible number of real intersections of $y=x^2$ and $y=mx+b$?

     $2$. A parabola and line can intersect at most twice.

225. What should you do after solving for $x$ values from substitution in a system?

     Substitute each $x$ back to find the corresponding $y$. Each $x$-value pairs with a $y$-value to form complete solutions.

Study these flashcards