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

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=-x^{2}+6x + 16) and (y=-4x + 37), we set (-x^{2}+6x + 16=-4x + 37).

Step2: Rearrange to standard - quadratic form.

Add (x^{2}) to both sides, subtract (6x) from both sides and subtract 16 from both sides: (0=x^{2}-10x + 21).

Step3: Factor the quadratic equation.

We find two numbers that multiply to 21 and add up to - 10. The numbers are -3 and -7. So, (x^{2}-10x + 21=(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), (y=-4\times3+37=-12 + 37=25). When (x = 7), (y=-4\times7+37=-28 + 37=9).

Answer:

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