Question

find all solutions of the system of equations algebraically. write your solutions as coordinate points. y = -3x² - 9x + 199 3x + y = 10 answer two solutions and submit answer

Explanation:

Step1: Solve the linear equation for y

From \( 3x + y = 10 \), we get \( y = 10 - 3x \).

Step2: Substitute y into the quadratic equation

Substitute \( y = 10 - 3x \) into \( y = -3x^2 - 9x + 199 \):
\[
10 - 3x = -3x^2 - 9x + 199
\]

Step3: Rearrange into standard quadratic form

Bring all terms to one side:
\[
3x^2 + 6x - 189 = 0
\]
Divide by 3:
\[
x^2 + 2x - 63 = 0
\]

Step4: Factor the quadratic equation

Factor \( x^2 + 2x - 63 \):
\[
(x + 9)(x - 7) = 0
\]

Step5: Solve for x

Set each factor to zero:
\( x + 9 = 0 \) gives \( x = -9 \); \( x - 7 = 0 \) gives \( x = 7 \).

Step6: Find corresponding y values

For \( x = -9 \), \( y = 10 - 3(-9) = 10 + 27 = 37 \).
For \( x = 7 \), \( y = 10 - 3(7) = 10 - 21 = -11 \).

Answer:

\((-9, 37)\) and \((7, -11)\)