Question

which represents the solution(s) of the system of equations, $y = -x^2 + 6x + 16$ and $y = -4x + 37$? determine the solution set algebraically.
\bigcirc \\ (3, 25)
\bigcirc \\ (-3, 49)
\bigcirc \\ (3, 25) and (7, 9)
\bigcirc \\ (-3, 49) and (-7, 65)

Explanation:

Step1: Set equations equal

Since both equal \( y \), set \( -x^2 + 6x + 16 = -4x + 37 \).

Step2: Rearrange to quadratic

Move all terms to left: \( -x^2 + 6x + 16 + 4x - 37 = 0 \) → \( -x^2 + 10x - 21 = 0 \). Multiply by -1: \( x^2 - 10x + 21 = 0 \).

Step3: Factor quadratic

Factor: \( (x - 3)(x - 7) = 0 \). So \( x = 3 \) or \( x = 7 \).

Step4: Find corresponding y

For \( x = 3 \): \( y = -4(3) + 37 = 25 \). For \( x = 7 \): \( y = -4(7) + 37 = 9 \).

Answer:

(3, 25) and (7, 9)