Solving Linear–Quadratic Systems - Algebra 1

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

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

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

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

3. Find the intersection points of $y=x+2$ and $y= -x^2+2$.

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

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

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

5. Find the intersection points of $y=3$ and $y=x^2$.

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

6. Find the intersection points of $y=4-x^2$ and $y=0$.

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

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

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

8. What is the key algebraic step to solve $y=2x+1$ and $y=x^2$ together?

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

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

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

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

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

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

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

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

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

13. Find the intersection points of $y=2x$ and $y=x^2$.

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

14. Find the intersection points of $y=-2x-3$ and $y=x^2$.

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

15. Find the intersection points of $y=-x+2$ and $y= -x^2+2$.

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

16. What is the key algebraic step to solve $y=2x+1$ and $y=x^2$ together?

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

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

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

18. What method replaces one variable using a linear equation to solve a linear-quadratic system?

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

19. Find the intersection points of $y=x-2$ and $y=x^2-2$.

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

20. Find the intersection points of $x=0$ and $x^2+y^2=9$.

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

21. Find the intersection points of $y=-x$ and $y=x^2$.

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

22. What is the usual maximum number of real intersection points between a line and a circle?

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

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

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

24. Find the intersection points of $y=0$ and $x^2+y^2=9$.

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

25. What is the discriminant used for a quadratic equation $ax^2+bx+c=0$?

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

26. Find the intersection points of $y=x$ and $y=x^2$.

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

27. Find the intersection points of $y=2x+1$ and $x^2+y^2=5$.

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

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

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

29. Find the intersection points of $y=4-x^2$ and $y=5$.

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

30. Find the intersection points of $y=4-x^2$ and $y=3$.

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

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

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

32. Find the intersection points of $y=-1$ and $y=x^2$.

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

33. What does it mean if a line is tangent to a parabola in a linear-quadratic system?

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

34. What is the standard form of a circle centered at the origin with radius $r$?

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

35. What must you always do after finding $x$-values from substitution in a system?

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

36. Find the intersection points of $y=3$ and $x^2+y^2=10$.

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

37. Find the intersection points of $y= -1$ and $x^2+y^2=5$.

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

38. Find the intersection points of $y=3$ and $y=x^2$.

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

39. Find the intersection points of $y=-x$ and $x^2+y^2=2$.

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

40. What is the standard form of a parabola that opens up or down with vertex at the origin?

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

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

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

42. What does it mean if a line and a parabola intersect at exactly two points?

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

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

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

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

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

45. Find the intersection points of $y= -2x+1$ and $x^2+y^2=5$.

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

46. What is the standard form of a line used for substitution if it is already solved for $y$?

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

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

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

48. What does it mean if a line does not intersect a parabola on a graph?

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

49. Identify the correct substitution for the system $y=-3x$ and $x^2+y^2=3$.

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

50. Find the intersection points of $y=2x+3$ and $y=x^2+3$.

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

51. Find the intersection points of $y=0$ and $y=x^2-4$.

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

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