Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = 2x^2 - 5x - 87$
$x + y = 9$
answer
two solutions
and
submit answer

Explanation:

Step1: Solve for y from the linear equation

From \( x + y = 9 \), we get \( y = 9 - x \).

Step2: Substitute y into the quadratic equation

Substitute \( y = 9 - x \) into \( y = 2x^2 - 5x - 87 \):
\( 9 - x = 2x^2 - 5x - 87 \)

Step3: Rearrange into standard quadratic form

\( 2x^2 - 5x - 87 - 9 + x = 0 \)
\( 2x^2 - 4x - 96 = 0 \)
Divide by 2: \( x^2 - 2x - 48 = 0 \)

Step4: Factor the quadratic equation

Factor \( x^2 - 2x - 48 \): \( (x - 8)(x + 6) = 0 \)

Step5: Solve for x

Set each factor to zero:
\( x - 8 = 0 \) gives \( x = 8 \)
\( x + 6 = 0 \) gives \( x = -6 \)

Step6: Find corresponding y values

For \( x = 8 \), \( y = 9 - 8 = 1 \)
For \( x = -6 \), \( y = 9 - (-6) = 15 \)

Answer:

\((8, 1)\) and \((-6, 15)\)