Question

Solve the system by substitution. y=−x2+2x+10 y=x+4 (1 point) Responses (-1,3) (2,6) (-1,3) (2,6) (0,4) (4,8) (0,4) (4,8) (1,5) (5,9) (1,5) (5,9) (3,7) (-2,2) (3,7) (-2,2)

Answer 1

To solve the system of equations by substitution, we have:

1. \( y = -x^2 + 2x + 10 \) (Equation 1)
2. \( y = x + 4 \) (Equation 2)

Since both equations are set equal to \( y \), we can set the right-hand sides of these equations equal to each other:

\[
-x^2 + 2x + 10 = x + 4
\]

Now, let's rearrange this equation to bring all terms to one side:

\[
-x^2 + 2x + 10 - x - 4 = 0
\]

This simplifies to:

\[
-x^2 + x + 6 = 0
\]

To make coefficients easier to work with, let's multiply the entire equation by -1:

\[
x^2 - x - 6 = 0
\]

Next, we can factor this quadratic:

\[
(x - 3)(x + 2) = 0
\]

Setting each factor equal to zero gives us the solutions for \( x \):

\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]

Now we will substitute these \( x \) values back into Equation 2 to find the corresponding \( y \) values.

For \( x = 3 \):
\[
y = 3 + 4 = 7
\]
So one solution is \( (3, 7) \).

For \( x = -2 \):
\[
y = -2 + 4 = 2
\]
So the other solution is \( (-2, 2) \).

Thus, the solutions to the system of equations are \( (3, 7) \) and \( (-2, 2) \).

From your provided responses, the correct answers match with:

- (3, 7)
- (-2, 2)

So the final answers from the choices given are:
- (3, 7)
- (-2, 2)