Systems of Polynomial Equations - PSAT Math
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What are the solutions to the system $x^2+y^2=25$ and $x=0$?
What are the solutions to the system $x^2+y^2=25$ and $x=0$?
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$(0,-5)$ and $(0,5)$. Circle intersects $y$-axis where $y^2=25$.
$(0,-5)$ and $(0,5)$. Circle intersects $y$-axis where $y^2=25$.
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What is the solution set of the system $y=x^2+1$ and $y-x^2=1$?
What is the solution set of the system $y=x^2+1$ and $y-x^2=1$?
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Infinitely many solutions. Both equations represent the same parabola.
Infinitely many solutions. Both equations represent the same parabola.
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What is the solution to the system $y=x^2+1$ and $y=x^2+3$?
What is the solution to the system $y=x^2+1$ and $y=x^2+3$?
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No solution. Parallel parabolas never intersect.
No solution. Parallel parabolas never intersect.
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Find the solutions to the system $x-y=1$ and $x^2-y^2=5$.
Find the solutions to the system $x-y=1$ and $x^2-y^2=5$.
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$(3,2)$. Factor as $(x+y)(x-y)=5$ with $x-y=1$.
$(3,2)$. Factor as $(x+y)(x-y)=5$ with $x-y=1$.
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Find the solutions to the system $x+y=5$ and $xy=6$.
Find the solutions to the system $x+y=5$ and $xy=6$.
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$(2,3)$ and $(3,2)$. Solve quadratic $x^2-5x+6=0$ from substitution.
$(2,3)$ and $(3,2)$. Solve quadratic $x^2-5x+6=0$ from substitution.
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Find the solutions to the system $y=x+1$ and $y^2=4$.
Find the solutions to the system $y=x+1$ and $y^2=4$.
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$(-3,-2)$ and $(1,2)$. From $y^2=4$, $y=pm^2$, then find $x$ values.
$(-3,-2)$ and $(1,2)$. From $y^2=4$, $y=pm^2$, then find $x$ values.
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What are the solutions to the system $x^2+y^2=25$ and $y=0$?
What are the solutions to the system $x^2+y^2=25$ and $y=0$?
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$(-5,0)$ and $(5,0)$. Circle intersects $x$-axis where $x^2=25$.
$(-5,0)$ and $(5,0)$. Circle intersects $x$-axis where $x^2=25$.
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What are the solutions to the system $y=x^2-1$ and $y=0$?
What are the solutions to the system $y=x^2-1$ and $y=0$?
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$(-1,0)$ and $(1,0)$. The parabola crosses the $x$-axis at $x=pm^1$.
$(-1,0)$ and $(1,0)$. The parabola crosses the $x$-axis at $x=pm^1$.
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What are the solutions to the system $y=x^2$ and $x=3$?
What are the solutions to the system $y=x^2$ and $x=3$?
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$(3,9)$. Substitute $x=3$ into $y=x^2$.
$(3,9)$. Substitute $x=3$ into $y=x^2$.
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What does it mean if substitution produces a true identity such as $0=0$?
What does it mean if substitution produces a true identity such as $0=0$?
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Infinitely many solutions (same curve). The equations represent the same curve.
Infinitely many solutions (same curve). The equations represent the same curve.
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What is the maximum possible number of real solutions to a system of a cubic and a line?
What is the maximum possible number of real solutions to a system of a cubic and a line?
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$3$. A cubic and line can intersect at most 3 times.
$3$. A cubic and line can intersect at most 3 times.
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What is the maximum possible number of real solutions to a system of two quadratic equations?
What is the maximum possible number of real solutions to a system of two quadratic equations?
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$4$. Two parabolas can intersect at most 4 times.
$4$. Two parabolas can intersect at most 4 times.
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What is the maximum possible number of real intersection points of $y=ax^2+bx+c$ and $y=mx+b$?
What is the maximum possible number of real intersection points of $y=ax^2+bx+c$ and $y=mx+b$?
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$2$. A parabola and line can intersect at most twice.
$2$. A parabola and line can intersect at most twice.
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What does each solution $(x,y)$ of a polynomial system represent on a graph?
What does each solution $(x,y)$ of a polynomial system represent on a graph?
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An intersection point of the graphs. Solutions occur where the curves meet.
An intersection point of the graphs. Solutions occur where the curves meet.
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What method solves a system by setting two expressions for $y$ equal: $y=f(x)$ and $y=g(x)$?
What method solves a system by setting two expressions for $y$ equal: $y=f(x)$ and $y=g(x)$?
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Substitution: solve $f(x)=g(x)$, then find $y$. Setting equal eliminates $y$ to solve for $x$ first.
Substitution: solve $f(x)=g(x)$, then find $y$. Setting equal eliminates $y$ to solve for $x$ first.
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What does it mean if substitution produces a contradiction such as $0=5$?
What does it mean if substitution produces a contradiction such as $0=5$?
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No solution (no intersection). The curves never meet.
No solution (no intersection). The curves never meet.
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Identify the best next step to solve $y=x^2$ and $y=2x+3$ by substitution.
Identify the best next step to solve $y=x^2$ and $y=2x+3$ by substitution.
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Set $x^2=2x+3$. Substitute the second equation into the first.
Set $x^2=2x+3$. Substitute the second equation into the first.
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What are the $x$-values solving the system $y=x^2$ and $y=2x$?
What are the $x$-values solving the system $y=x^2$ and $y=2x$?
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$x=0$ or $x=2$. Solve $x^2=2x$: $x(x-2)=0$.
$x=0$ or $x=2$. Solve $x^2=2x$: $x(x-2)=0$.
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What are the solutions to the system $y=x^2$ and $y=2x$?
What are the solutions to the system $y=x^2$ and $y=2x$?
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$(0,0)$ and $(2,4)$. Substitute $x$-values back to find $y$-values.
$(0,0)$ and $(2,4)$. Substitute $x$-values back to find $y$-values.
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What is the maximum possible number of real intersections of $y=x^3$ and $y=mx+b$?
What is the maximum possible number of real intersections of $y=x^3$ and $y=mx+b$?
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$3$. A cubic and line can intersect at most three times.
$3$. A cubic and line can intersect at most three times.
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Identify the system’s solutions in the coordinate plane when $y=f(x)$ and $y=g(x)$ are given.
Identify the system’s solutions in the coordinate plane when $y=f(x)$ and $y=g(x)$ are given.
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All intersection points $(x,y)$ of the two graphs. Solutions occur where the curves meet in the plane.
All intersection points $(x,y)$ of the two graphs. Solutions occur where the curves meet in the plane.
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What equation represents the $x$-coordinates of intersection points of $y=f(x)$ and $y=g(x)$?
What equation represents the $x$-coordinates of intersection points of $y=f(x)$ and $y=g(x)$?
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Solve $f(x)-g(x)=0$ for $x$. Rearranging gives the equation whose roots are x-coordinates.
Solve $f(x)-g(x)=0$ for $x$. Rearranging gives the equation whose roots are x-coordinates.
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What method solves a system by setting two expressions for the same variable equal, such as $y=f(x)$ and $y=g(x)$?
What method solves a system by setting two expressions for the same variable equal, such as $y=f(x)$ and $y=g(x)$?
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Substitution: set $f(x)=g(x)$ and solve. This eliminates one variable by equating the expressions.
Substitution: set $f(x)=g(x)$ and solve. This eliminates one variable by equating the expressions.
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Solve the system $y=x^2$ and $y=4$ for all solutions $(x,y)$.
Solve the system $y=x^2$ and $y=4$ for all solutions $(x,y)$.
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$(-2,4)$ and $(2,4)$. Set $x^2=4$ to get $x=\pm 2$, both with $y=4$.
$(-2,4)$ and $(2,4)$. Set $x^2=4$ to get $x=\pm 2$, both with $y=4$.
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What are the $x$-values of solutions to $y=x^2$ and $y=x$?
What are the $x$-values of solutions to $y=x^2$ and $y=x$?
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$x=0$ or $x=1$. Solve $x^2=x$: $x(x-1)=0$.
$x=0$ or $x=1$. Solve $x^2=x$: $x(x-1)=0$.
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Solve the system $y=x^2-1$ and $y=0$; what are the solutions?
Solve the system $y=x^2-1$ and $y=0$; what are the solutions?
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$(1,0)$ and $(-1,0)$. $x^2-1=0$ gives $x=\pm 1$, then $y=0$ for both.
$(1,0)$ and $(-1,0)$. $x^2-1=0$ gives $x=\pm 1$, then $y=0$ for both.
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Solve the system $y=x^2$ and $y=x$; what are the solutions?
Solve the system $y=x^2$ and $y=x$; what are the solutions?
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$(0,0)$ and $(1,1)$. $x^2=x$ gives $x(x-1)=0$, so $x=0$ or $x=1$.
$(0,0)$ and $(1,1)$. $x^2=x$ gives $x(x-1)=0$, so $x=0$ or $x=1$.
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Solve the system $y=x^2-2$ and $y=x$; what are the solutions?
Solve the system $y=x^2-2$ and $y=x$; what are the solutions?
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$(-1,-1)$ and $(2,2)$. $x^2-2=x$ gives $x^2-x-2=0$, factoring to $(x+1)(x-2)=0$.
$(-1,-1)$ and $(2,2)$. $x^2-2=x$ gives $x^2-x-2=0$, factoring to $(x+1)(x-2)=0$.
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Solve the system $y=x^2+1$ and $y=2x$; what are the solutions?
Solve the system $y=x^2+1$ and $y=2x$; what are the solutions?
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$(1,2)$. $x^2+1=2x$ gives $x^2-2x+1=0$, so $(x-1)^2=0$.
$(1,2)$. $x^2+1=2x$ gives $x^2-2x+1=0$, so $(x-1)^2=0$.
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Solve the system $y=x^2-4x+3$ and $y=0$; what are the solutions?
Solve the system $y=x^2-4x+3$ and $y=0$; what are the solutions?
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$(1,0)$ and $(3,0)$. Factor: $(x-1)(x-3)=0$ gives $x=1,3$ with $y=0$.
$(1,0)$ and $(3,0)$. Factor: $(x-1)(x-3)=0$ gives $x=1,3$ with $y=0$.
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