PSAT Math Flashcards: Systems Of Polynomial Equations

What this deck covers

This deck focuses on Systems Of Polynomial Equations, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 20: What is the solution to the system $x^2+y^2=25$ and $y=0$?

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

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

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

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

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

Flashcard 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 $y$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: Infinitely many solutions. Identical equations represent the same curve.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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