Question

question
find all solutions of the system of equations algebraically. write your solutions as coordinate points.
$y = 3x^2 - 16x - 100$
$x + y = 8$
answer attempt 1 out of 2
two solutions
and

Explanation:

Step1: Express y from the linear equation

From \( x + y = 8 \), we get \( y = 8 - x \).

Step2: Substitute y into the quadratic equation

Substitute \( y = 8 - x \) into \( y = 3x^2 - 16x - 100 \):
\( 8 - x = 3x^2 - 16x - 100 \)

Step3: Rearrange into standard quadratic form

\( 3x^2 - 16x - 100 - 8 + x = 0 \)
\( 3x^2 - 15x - 108 = 0 \)
Divide by 3: \( x^2 - 5x - 36 = 0 \)

Step4: Factor the quadratic equation

Factor \( x^2 - 5x - 36 \): \( (x - 9)(x + 4) = 0 \)

Step5: Solve for x

\( x - 9 = 0 \) gives \( x = 9 \); \( x + 4 = 0 \) gives \( x = -4 \)

Step6: Find corresponding y values

For \( x = 9 \), \( y = 8 - 9 = -1 \)
For \( x = -4 \), \( y = 8 - (-4) = 12 \)

Answer:

\((9, -1)\) and \((-4, 12)\)