Algebra Flashcards: Solving Linear Quadratic Systems

What this deck covers

This deck focuses on Solving Linear Quadratic Systems, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

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 form should solutions to a system in two variables be written in?

Answer: As ordered pairs $(x,y)$. Solutions are coordinate pairs showing intersection points.

Flashcard 2: Find the intersection points of y=2x+3 and y=x^2+3.

Answer: $(0,3)$ and $(2,7)$. Set $2x+3=x^2+3$ and solve the quadratic.

Flashcard 3: Find the intersection points of y=x and x^2+y^2=2.

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

Flashcard 4: Find the intersection points of y=-1 and y=x^2.

Answer: No real solution. Parabola $y=x^2$ never reaches negative values.

Flashcard 5: Find the intersection points of y=x+2 and y= -x^2+2.

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

Flashcard 6: Identify the number of real solutions if substitution gives $(x-2)^2=0$.

Answer: $1$ real solution. The equation $(x-2)^2=0$ has one repeated solution.

Flashcard 7: Find the intersection points of $y=3$ and $y=x^2$.

Answer: $(-\sqrt{3},3)$ and $(\sqrt{3},3)$. Set $3=x^2$ and solve for x-values.

Flashcard 8: Find the intersection points of y=1 and x^2+y^2=2.

Answer: $( -1,1)$ and $(1,1)$. Set $y=1$ in the circle equation and solve.

Flashcard 9: Find the intersection points of y=x-2 and y=x^2-2.

Answer: $(0,-2)$ and $(1,-1)$. Set $x-2=x^2-2$ and solve for intersections.

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

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

Flashcard 11: Find the intersection points of y=4-x^2 and y=0.

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

Flashcard 12: Find the intersection points of y=1 and x^2+y^2=2.

Answer: $( -1,1)$ and $(1,1)$. Set $y=1$ in the circle equation and solve.

Flashcard 13: What is the key algebraic step to solve $y=2x+1$ and $y=x^2$ together?

Answer: Set $2x+1=x^2$. Set the expressions for $y$ equal to each other.

Flashcard 14: Find the intersection points of y=x and y=x^2.

Answer: $(0,0)$ and $(1,1)$. Set $x=x^2$ and solve for the intersection points.

Flashcard 15: What does $b^2-4ac>0$ tell you about a linear-quadratic system after substitution?

Answer: There are $2$ distinct real solutions. Positive discriminant means two intersection points exist.

Flashcard 16: Find the intersection points of y=x+1 and y=x^2.

Answer: $( -1,0)$ and $(2,3)$. Set $x+1=x^2$ and solve the quadratic.

Flashcard 17: Find the intersection points of y=1 and y= -x^2+2.

Answer: $( -1,1)$ and $(1,1)$. Set $1=-x^2+2$ and solve for $x$.

Flashcard 18: What does $b^2-4ac<0$ tell you about a linear-quadratic system after substitution?

Answer: There are $0$ real solutions (no real intersections). Negative discriminant means no real intersection points exist.

Flashcard 19: What does it mean if a line is tangent to a parabola in a linear-quadratic system?

Answer: The system has $1$ real solution (one intersection point). The line touches the parabola at exactly one point.

Flashcard 20: Find the intersection points of y=2x and y=x^2.

Answer: $(0,0)$ and $(2,4)$. Set $2x=x^2$ and solve for intersection points.

Flashcard 21: Identify the correct substitution for the system y=-3x and x^2+y^2=3.

Answer: Replace $y$ with $-3x$ in $x^2+y^2=3$. Substitute the linear expression into the circle equation.

Flashcard 22: Identify the number of real solutions if substitution gives $x^2-9=0$.

Answer: $2$ real solutions. The equation $x^2=9$ has two distinct solutions.

Flashcard 23: Find the intersection points of y=2x and y=x^2.

Answer: $(0,0)$ and $(2,4)$. Set $2x=x^2$ and solve for intersection points.

Flashcard 24: Find the intersection points of y=-2x-3 and y=x^2.

Answer: No real solution. The discriminant is negative, so no real solutions exist.

Flashcard 25: Find the intersection points of y= -2x+1 and x^2+y^2=5.

Answer: $(0,1)$ and $(2,-3)$. Substitute $y=-2x+1$ into $x^2+y^2=5$ and solve.

Flashcard 26: What does it mean if a line and a parabola intersect at exactly two points?

Answer: The system has $2$ real solutions (two intersection points). Two distinct points where the curves cross.

Flashcard 27: Find the intersection points of y=-x+2 and y= -x^2+2.

Answer: $(0,2)$ and $(1,1)$. Set $-x+2=-x^2+2$ and solve for intersections.

Flashcard 28: Find the intersection points of y=4-x^2 and y=3.

Answer: $( -1,3)$ and $(1,3)$. Set $4-x^2=3$ and solve for $x$.

Flashcard 29: What method replaces one variable using a linear equation to solve a linear-quadratic system?

Answer: Substitution. Replace one variable with an expression from the linear equation.

Flashcard 30: What is the key algebraic step to solve y=2x+1 and y=x^2 together?

Answer: Set $2x+1=x^2$. Set the expressions for $y$ equal to each other.

Flashcard 31: What is the usual maximum number of real intersection points between a line and a parabola?

Answer: At most $2$ real intersection points. A line can intersect a parabola in at most two places.

Flashcard 32: What does $b^2-4ac<0$ tell you about a linear-quadratic system after substitution?

Answer: There are $0$ real solutions (no real intersections). Negative discriminant means no real intersection points exist.

Flashcard 33: Find the intersection points of y=-x and x^2+y^2=2.

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

Flashcard 34: Find the intersection points of y=0 and x^2+y^2=9.

Answer: $( -3,0)$ and $(3,0)$. Set $y=0$ in the circle equation and solve.

Flashcard 35: What method replaces one variable using a linear equation to solve a linear-quadratic system?

Answer: Substitution. Replace one variable with an expression from the linear equation.

Flashcard 36: Find the intersection points of y=-2x+1 and y=x^2.

Answer: ($( -1,1)$ and $(2,4)$). Set $-2x+1=x^2$ and solve the resulting equation.

Flashcard 37: What is the standard form of a parabola that opens up or down with vertex at the origin?

Answer: $y=ax^2$. Basic parabola form opening vertically.

Flashcard 38: Find the intersection points of y=x-2 and y=x^2-2.

Answer: $(0,-2)$ and $(1,-1)$. Set $x-2=x^2-2$ and solve for intersections.

Flashcard 39: Find the intersection points of x=0 and x^2+y^2=9.

Answer: $(0,-3)$ and $(0,3)$. Set $x=0$ in the circle equation and solve.

Flashcard 40: Find the intersection points of y=-x and y=x^2.

Answer: $(0,0)$ and $(-1,1)$. Set $-x=x^2$ and solve the resulting quadratic.

Flashcard 41: What is the usual maximum number of real intersection points between a line and a circle?

Answer: At most $2$ real intersection points. A line can intersect a circle in at most two places.

Flashcard 42: What is the usual maximum number of real intersection points between a line and a parabola?

Answer: At most $2$ real intersection points. A line can intersect a parabola in at most two places.

Flashcard 43: Find the intersection points of y=2 and x^2+y^2=1.

Answer: No real solution. The line $y=2$ is outside the unit circle.

Flashcard 44: Find the intersection points of y=0 and x^2+y^2=9.

Answer: $( -3,0)$ and $(3,0)$. Set $y=0$ in the circle equation and solve.

Flashcard 45: Find the $y$-value if $y= -2x+5$ and $x=3$ is a solution from substitution.

Answer: $y=-1$. Substitute $x=3$ into $y=-2x+5$.

Flashcard 46: Find the intersection points of y=-2x-3 and y=x^2.

Answer: No real solution. The discriminant is negative, so no real solutions exist.

Flashcard 47: What is the discriminant used for a quadratic equation $ax^2+bx+c=0$?

Answer: $b^2-4ac$. Formula that determines the nature of quadratic solutions.

Flashcard 48: Find the intersection points of y= -1 and x^2+y^2=5.

Answer: $( -2,-1)$ and $(2,-1)$. Set $y=-1$ in $x^2+y^2=5$ and solve for $x$.

Flashcard 49: Find the intersection points of y=x and y=x^2.

Answer: $(0,0)$ and $(1,1)$. Set $x=x^2$ and solve for the intersection points.

Flashcard 50: Find the intersection points of y=2x+1 and x^2+y^2=5.

Answer: $(0,1)$ and $( -2,-3)$. Substitute $y=2x+1$ into $x^2+y^2=5$ and solve.

Flashcard 51: Identify the number of real solutions if substitution gives $5x^2+1=0$.

Answer: $0$ real solutions. The equation $5x^2=-1$ has no real solutions.

Flashcard 52: What is the discriminant used for a quadratic equation $ax^2+bx+c=0$?

Answer: $b^2-4ac$. Formula that determines the nature of quadratic solutions.

Flashcard 53: Find the intersection points of y=4-x^2 and y=0.

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

Flashcard 54: Find the $y$-value if $y=3x-4$ and $x=-2$ is a solution from substitution.

Answer: $y=-10$. Substitute $x=-2$ into $y=3x-4$.

Flashcard 55: What does $b^2-4ac=0$ tell you about a linear-quadratic system after substitution?

Answer: There is exactly $1$ real solution (a tangent intersection). Zero discriminant means the line is tangent to the curve.

Flashcard 56: Find the intersection points of y=4-x^2 and y=5.

Answer: No real solution. The parabola's maximum value is $4$, less than $5$.

Flashcard 57: Find the intersection points of y=-x and y=x^2.

Answer: $(0,0)$ and $(-1,1)$. Set $-x=x^2$ and solve the resulting quadratic.

Flashcard 58: What is the standard form of a line used for substitution if it is already solved for $y$?

Answer: $y=mx+b$. Slope-intercept form ready for direct substitution.

Flashcard 59: What is the usual maximum number of real intersection points between a line and a circle?

Answer: At most $2$ real intersection points. A line can intersect a circle in at most two places.

Flashcard 60: Identify the number of real solutions if substitution gives $(x-2)^2=0$.

Answer: $1$ real solution. The equation $(x-2)^2=0$ has one repeated solution.

Flashcard 61: Identify the number of real solutions if substitution gives $x^2-9=0$.

Answer: $2$ real solutions. The equation $x^2=9$ has two distinct solutions.

Flashcard 62: Find the intersection points of y=4-x^2 and y=3.

Answer: $( -1,3)$ and $(1,3)$. Set $4-x^2=3$ and solve for $x$.

Flashcard 63: What must you always do after finding $x$-values from substitution in a system?

Answer: Substitute back to find the matching $y$-value(s). Complete the solution by finding corresponding coordinates.

Flashcard 64: Find the intersection points of y=-2x+1 and y=x^2.

Answer: ($( -1,1)$ and $(2,4)$). Set $-2x+1=x^2$ and solve the resulting equation.

Flashcard 65: Find the intersection points of y=-1 and y=x^2.

Answer: No real solution. Parabola $y=x^2$ never reaches negative values.

Flashcard 66: What does it mean if a line is tangent to a parabola in a linear-quadratic system?

Answer: The system has $1$ real solution (one intersection point). The line touches the parabola at exactly one point.

Flashcard 67: What is the standard form of a circle centered at the origin with radius $r$?

Answer: $x^2+y^2=r^2$. Circle equation with center at origin.

Flashcard 68: What must you always do after finding $x$-values from substitution in a system?

Answer: Substitute back to find the matching $y$-value(s). Complete the solution by finding corresponding coordinates.

Flashcard 69: Find the intersection points of y=3 and x^2+y^2=10.

Answer: $( -1,3)$ and $(1,3)$. Set $y=3$ in $x^2+y^2=10$ and solve for $x$.

Flashcard 70: Find the intersection points of y=0 and y=x^2-4.

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

Flashcard 71: Find the intersection points of y= -1 and x^2+y^2=5.

Answer: $( -2,-1)$ and $(2,-1)$. Set $y=-1$ in $x^2+y^2=5$ and solve for $x$.

Flashcard 72: Find the intersection points of y=-x+2 and y= -x^2+2.

Answer: $(0,2)$ and $(1,1)$. Set $-x+2=-x^2+2$ and solve for intersections.

Flashcard 73: What does it mean if a line does not intersect a parabola on a graph?

Answer: The system has $0$ real solutions (no intersection points). The curves never meet on the coordinate plane.

Flashcard 74: Find the intersection points of y=x+2 and y= -x^2+2.

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

Flashcard 75: Find the intersection points of y=3 and y=x^2.

Answer: ( -\sqrt{3},3) and $(\sqrt{3},3)$. Set $3=x^2$ and solve for $x$-values.

Flashcard 76: Find the intersection points of y=4-x^2 and y=5.

Answer: No real solution. The parabola's maximum value is $4$, less than $5$.

Flashcard 77: Find the intersection points of y=-x and x^2+y^2=2.

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

Flashcard 78: What is the standard form of a parabola that opens up or down with vertex at the origin?

Answer: $y=ax^2$. Basic parabola form opening vertically.

Flashcard 79: Find the intersection points of y=2 and x^2+y^2=1.

Answer: No real solution. The line $y=2$ is outside the unit circle.

Flashcard 80: What does it mean if a line and a parabola intersect at exactly two points?

Answer: The system has $2$ real solutions (two intersection points). Two distinct points where the curves cross.

Flashcard 81: Find the intersection points of x=0 and x^2+y^2=9.

Answer: $(0,-3)$ and $(0,3)$. Set $x=0$ in the circle equation and solve.

Flashcard 82: Find the intersection points of y=3 and x^2+y^2=10.

Answer: $( -1,3)$ and $(1,3)$. Set $y=3$ in $x^2+y^2=10$ and solve for $x$.

Flashcard 83: What form should solutions to a system in two variables be written in?

Answer: As ordered pairs $(x,y)$. Solutions are coordinate pairs showing intersection points.

Flashcard 84: What does $b^2-4ac=0$ tell you about a linear-quadratic system after substitution?

Answer: There is exactly $1$ real solution (a tangent intersection). Zero discriminant means the line is tangent to the curve.

Flashcard 85: Find the $y$-value if $y=3x-4$ and $x=-2$ is a solution from substitution.

Answer: $y=-10$. Substitute $x=-2$ into $y=3x-4$.

Flashcard 86: Find the intersection points of y= -2x+1 and x^2+y^2=5.

Answer: $(0,1)$ and $(2,-3)$. Substitute $y=-2x+1$ into $x^2+y^2=5$ and solve.

Flashcard 87: Identify the number of real solutions if substitution gives $5x^2+1=0$.

Answer: $0$ real solutions. The equation $5x^2=-1$ has no real solutions.

Flashcard 88: Find the intersection points of y=2x+1 and x^2+y^2=5.

Answer: $(0,1)$ and $( -2,-3)$. Substitute $y=2x+1$ into $x^2+y^2=5$ and solve.

Flashcard 89: What does $b^2-4ac>0$ tell you about a linear-quadratic system after substitution?

Answer: There are $2$ distinct real solutions. Positive discriminant means two intersection points exist.

Flashcard 90: Find the intersection points of y=1 and y= -x^2+2.

Answer: $( -1,1)$ and $(1,1)$. Set $1=-x^2+2$ and solve for $x$.

Flashcard 91: What is the standard form of a line used for substitution if it is already solved for $y$?

Answer: $y=mx+b$. Slope-intercept form ready for direct substitution.

Flashcard 92: Find the intersection points of y=x+1 and y=x^2.

Answer: $( -1,0)$ and $(2,3)$. Set $x+1=x^2$ and solve the quadratic.

Flashcard 93: Find the $y$-value if $y= -2x+5$ and $x=3$ is a solution from substitution.

Answer: $y=-1$. Substitute $x=3$ into $y=-2x+5$.

Flashcard 94: What does it mean if a line does not intersect a parabola on a graph?

Answer: The system has $0$ real solutions (no intersection points). The curves never meet on the coordinate plane.

Flashcard 95: Identify the correct substitution for the system y=-3x and x^2+y^2=3.

Answer: Replace $y$ with $-3x$ in $x^2+y^2=3$. Substitute the linear expression into the circle equation.

Flashcard 96: Find the intersection points of y=2x+3 and y=x^2+3.

Answer: $(0,3)$ and $(2,7)$. Set $2x+3=x^2+3$ and solve the quadratic.

Flashcard 97: Find the intersection points of y=0 and y=x^2-4.

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