Question

question
find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = -2x^2 + 11x + 89$
$1 = 3x - y$
answer attempt 1 out of 2
two solutions
and

Explanation:

Step1: Solve the linear equation for \( y \)

From \( 1 = 3x - y \), we can rearrange it to get \( y = 3x - 1 \).

Step2: Substitute \( y = 3x - 1 \) into the quadratic equation

Substitute \( y \) in \( y = -2x^2 + 11x + 89 \) with \( 3x - 1 \):
\[
3x - 1 = -2x^2 + 11x + 89
\]

Step3: Rearrange the equation to standard quadratic form

Bring all terms to one side:
\[
2x^2 - 8x - 90 = 0
\]
Divide through by 2:
\[
x^2 - 4x - 45 = 0
\]

Step4: Factor the quadratic equation

Factor \( x^2 - 4x - 45 \):
\[
(x - 9)(x + 5) = 0
\]

Step5: Solve for \( x \)

Set each factor equal to zero:

• \( x - 9 = 0 \) gives \( x = 9 \)
• \( x + 5 = 0 \) gives \( x = -5 \)

Step6: Find the corresponding \( y \) values

For \( x = 9 \):
\[
y = 3(9) - 1 = 26
\]
For \( x = -5 \):
\[
y = 3(-5) - 1 = -16
\]

Answer:

\((9, 26)\) and \((-5, -16)\)