Solving Linear–Quadratic Systems - Algebra 2
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What are the intersection points of $y=2x$ and $y=x^2$?
What are the intersection points of $y=2x$ and $y=x^2$?
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$(0,0)$ and $(2,4)$. Substitute $y = 2x$ into $y = x^2$.
$(0,0)$ and $(2,4)$. Substitute $y = 2x$ into $y = x^2$.
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What method solves a linear–quadratic system by combining equations to eliminate a variable?
What method solves a linear–quadratic system by combining equations to eliminate a variable?
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Elimination (after rewriting equations as needed). Combine equations to cancel out one variable.
Elimination (after rewriting equations as needed). Combine equations to cancel out one variable.
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What is the standard form of a horizontal parabola opening left or right?
What is the standard form of a horizontal parabola opening left or right?
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$x=ay^2+by+c$. Standard parabola form with horizontal axis of symmetry.
$x=ay^2+by+c$. Standard parabola form with horizontal axis of symmetry.
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What are the intersection points of $y=x-2$ and $y=x^2-4$?
What are the intersection points of $y=x-2$ and $y=x^2-4$?
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$(-1,-3)$ and $(2,0)$. Set $x - 2 = x^2 - 4$ and solve.
$(-1,-3)$ and $(2,0)$. Set $x - 2 = x^2 - 4$ and solve.
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What is the standard form of a vertical parabola opening up or down?
What is the standard form of a vertical parabola opening up or down?
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$y=ax^2+bx+c$. Standard parabola form with vertical axis of symmetry.
$y=ax^2+bx+c$. Standard parabola form with vertical axis of symmetry.
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What does $b^2-4ac=0$ mean for the number of intersection points?
What does $b^2-4ac=0$ mean for the number of intersection points?
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There is $1$ real intersection point (tangent). Zero discriminant means one repeated intersection.
There is $1$ real intersection point (tangent). Zero discriminant means one repeated intersection.
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What does $b^2-4ac>0$ mean for the number of intersection points?
What does $b^2-4ac>0$ mean for the number of intersection points?
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There are $2$ real intersection points. Positive discriminant means two distinct intersections.
There are $2$ real intersection points. Positive discriminant means two distinct intersections.
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What does the discriminant tell you about real intersections after substitution?
What does the discriminant tell you about real intersections after substitution?
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Sign of $b^2-4ac$ gives $0$, $1$, or $2$ real solutions. Discriminant determines intersection count possibilities.
Sign of $b^2-4ac$ gives $0$, $1$, or $2$ real solutions. Discriminant determines intersection count possibilities.
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What must every solution to a linear–quadratic system be written as?
What must every solution to a linear–quadratic system be written as?
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An ordered pair $(x,y)$. Solutions are coordinate points $(x,y)$.
An ordered pair $(x,y)$. Solutions are coordinate points $(x,y)$.
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What must you do after finding $x$-values from a substituted quadratic equation?
What must you do after finding $x$-values from a substituted quadratic equation?
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Substitute into the line to find corresponding $y$-values. Back-substitute $x$-values to complete solution pairs.
Substitute into the line to find corresponding $y$-values. Back-substitute $x$-values to complete solution pairs.
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What is the key algebraic step after substituting a linear equation into a quadratic equation?
What is the key algebraic step after substituting a linear equation into a quadratic equation?
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Solve the resulting quadratic in one variable. Use quadratic formula or factoring methods.
Solve the resulting quadratic in one variable. Use quadratic formula or factoring methods.
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What is the standard form of a circle centered at $(h,k)$ with radius $r$?
What is the standard form of a circle centered at $(h,k)$ with radius $r$?
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$(x-h)^2+(y-k)^2=r^2$. Circle equation with center and radius parameters.
$(x-h)^2+(y-k)^2=r^2$. Circle equation with center and radius parameters.
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What is the standard form of a horizontal parabola opening left or right?
What is the standard form of a horizontal parabola opening left or right?
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$x=ay^2+by+c$. Standard parabola form with horizontal axis of symmetry.
$x=ay^2+by+c$. Standard parabola form with horizontal axis of symmetry.
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What is the substituted equation after using $y=7-2x$ in $y=x^2$?
What is the substituted equation after using $y=7-2x$ in $y=x^2$?
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$x^2=7-2x$. Set the two expressions for $y$ equal.
$x^2=7-2x$. Set the two expressions for $y$ equal.
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What should you do first to solve $2x+y=7$ with $y=x^2$ by substitution?
What should you do first to solve $2x+y=7$ with $y=x^2$ by substitution?
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Rewrite the line as $y=7-2x$. Solve for $y$ to enable substitution method.
Rewrite the line as $y=7-2x$. Solve for $y$ to enable substitution method.
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What method solves a linear–quadratic system by replacing a variable using the linear equation?
What method solves a linear–quadratic system by replacing a variable using the linear equation?
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Substitution. Replace one variable using the linear equation.
Substitution. Replace one variable using the linear equation.
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What method solves a linear–quadratic system by combining equations to eliminate a variable?
What method solves a linear–quadratic system by combining equations to eliminate a variable?
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Elimination (after rewriting equations as needed). Combine equations to cancel out one variable.
Elimination (after rewriting equations as needed). Combine equations to cancel out one variable.
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What does it mean if a line and a parabola intersect at two points?
What does it mean if a line and a parabola intersect at two points?
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The system has $2$ real solutions. Two intersection points means two solution pairs.
The system has $2$ real solutions. Two intersection points means two solution pairs.
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What does it mean if a line is tangent to a circle in a linear–quadratic system?
What does it mean if a line is tangent to a circle in a linear–quadratic system?
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The system has exactly $1$ real solution. Tangency creates one point of contact.
The system has exactly $1$ real solution. Tangency creates one point of contact.
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What does it mean if a line does not intersect a circle on the coordinate plane?
What does it mean if a line does not intersect a circle on the coordinate plane?
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The system has $0$ real solutions. No intersection means no real solution pairs.
The system has $0$ real solutions. No intersection means no real solution pairs.
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What is the general form of a linear equation in two variables?
What is the general form of a linear equation in two variables?
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$Ax+By=C$. Standard form for any line equation.
$Ax+By=C$. Standard form for any line equation.
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What is the slope-intercept form of a line used for substitution?
What is the slope-intercept form of a line used for substitution?
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$y=mx+b$. Isolates $y$ for easy substitution into quadratics.
$y=mx+b$. Isolates $y$ for easy substitution into quadratics.
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Identify the correct solution set type if a line intersects a parabola at exactly one point.
Identify the correct solution set type if a line intersects a parabola at exactly one point.
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One real solution (the line is tangent to the parabola). Discriminant equals zero for tangent intersections.
One real solution (the line is tangent to the parabola). Discriminant equals zero for tangent intersections.
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What are the intersection points of $y=-2$ and $x^2+y^2=5$?
What are the intersection points of $y=-2$ and $x^2+y^2=5$?
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$(1,-2)$ and $(-1,-2)$. Set $y = -2$ in $x^2 + y^2 = 5$.
$(1,-2)$ and $(-1,-2)$. Set $y = -2$ in $x^2 + y^2 = 5$.
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What are the intersection points of $y=2$ and $x^2+y^2=5$?
What are the intersection points of $y=2$ and $x^2+y^2=5$?
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$(1,2)$ and $(-1,2)$. Set $y = 2$ in $x^2 + y^2 = 5$.
$(1,2)$ and $(-1,2)$. Set $y = 2$ in $x^2 + y^2 = 5$.
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What are the intersection points of $y=-1$ and $x^2+y^2=5$?
What are the intersection points of $y=-1$ and $x^2+y^2=5$?
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$(2,-1)$ and $(-2,-1)$. Set $y = -1$ in $x^2 + y^2 = 5$.
$(2,-1)$ and $(-2,-1)$. Set $y = -1$ in $x^2 + y^2 = 5$.
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What are the intersection points of $y=1$ and $x^2+y^2=5$?
What are the intersection points of $y=1$ and $x^2+y^2=5$?
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$(2,1)$ and $(-2,1)$. Set $y = 1$ in $x^2 + y^2 = 5$.
$(2,1)$ and $(-2,1)$. Set $y = 1$ in $x^2 + y^2 = 5$.
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What is the standard form of a vertical parabola opening up or down?
What is the standard form of a vertical parabola opening up or down?
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$y=ax^2+bx+c$. Standard parabola form with vertical axis of symmetry.
$y=ax^2+bx+c$. Standard parabola form with vertical axis of symmetry.
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What are the intersection points of $y=-x$ and $x^2+y^2=8$?
What are the intersection points of $y=-x$ and $x^2+y^2=8$?
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$(2,-2)$ and $(-2,2)$. Substitute $y = -x$ into circle equation.
$(2,-2)$ and $(-2,2)$. Substitute $y = -x$ into circle equation.
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What are the intersection points of $y=x$ and $x^2+y^2=8$?
What are the intersection points of $y=x$ and $x^2+y^2=8$?
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$(2,2)$ and $(-2,-2)$. Substitute $y = x$ into circle equation.
$(2,2)$ and $(-2,-2)$. Substitute $y = x$ into circle equation.
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