which represents the solution(s) of the system of equations, $y = -x^{2}+6x + 16$ and $y=-4x + 37$…

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

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

Answer

Explanation:

Step1: Set the two equations equal

Since $y$ is equal in both equations, we set $-x^{2}+6x + 16=-4x + 37$.

Step2: Rearrange to standard quadratic form

Move all terms to one - side: $-x^{2}+6x + 4x+16 - 37 = 0$, which simplifies to $-x^{2}+10x - 21 = 0$. Multiply through by - 1 to get $x^{2}-10x + 21 = 0$.

Step3: Factor the quadratic equation

We factor $x^{2}-10x + 21$ as $(x - 3)(x - 7)=0$.

Step4: Solve for x

Using the zero - product property, if $(x - 3)(x - 7)=0$, then $x-3 = 0$ or $x - 7=0$. So $x = 3$ or $x = 7$.

Step5: Find the corresponding y - values

When $x = 3$, substitute into $y=-4x + 37$, we get $y=-4\times3 + 37=-12 + 37 = 25$. When $x = 7$, substitute into $y=-4x + 37$, we get $y=-4\times7+37=-28 + 37 = 9$.

Answer:

C. $(3,25)$ and $(7,9)$