Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = -3x^2 + 6x + 122$
$4 = 3x - y$
answer
two solutions
and
submit answer

Explanation:

Step1: Solve the linear equation for y

From \( 4 = 3x - y \), we can rearrange it to get \( y = 3x - 4 \).

Step2: Substitute y into the quadratic equation

Substitute \( y = 3x - 4 \) into \( y = -3x^2 + 6x + 122 \). So we have:
\( 3x - 4 = -3x^2 + 6x + 122 \)

Step3: Rearrange into standard quadratic form

Bring all terms to one side:
\( 3x^2 - 3x - 126 = 0 \)
Divide through by 3:
\( x^2 - x - 42 = 0 \)

Step4: Factor the quadratic equation

Factor \( x^2 - x - 42 \). We need two numbers that multiply to -42 and add to -1. Those numbers are -7 and 6. So:
\( (x - 7)(x + 6) = 0 \)

Step5: Solve for x

Set each factor equal to zero:
\( x - 7 = 0 \) gives \( x = 7 \)
\( x + 6 = 0 \) gives \( x = -6 \)

Step6: Find the corresponding y values

For \( x = 7 \):
\( y = 3(7) - 4 = 21 - 4 = 17 \)
For \( x = -6 \):
\( y = 3(-6) - 4 = -18 - 4 = -22 \)

Answer:

\((7, 17)\) and \((-6, -22)\)