Question

yy
equals=
x squared plus 2 x plus 9x2+2x+9
yy
equals=
negative x plus 7−x+7
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Part 1
Select the correct choice and fill in any answer boxes in your choice below.
A.
The​ solution(s) is/are enter your response here.
​(Type an ordered pair. Use a comma to separate answers as​ needed.)

Answer 1

To solve the equation \(y = x^2 + 2x + 9\) and \(y = -x + 7\), we need to set them equal to each other:

\[
x^2 + 2x + 9 = -x + 7
\]

Now, we will move all the terms to one side:

\[
x^2 + 2x + 9 + x - 7 = 0
\]

Combine like terms:

\[
x^2 + 3x + 2 = 0
\]

Next, we can factor the quadratic equation:

\[
(x + 1)(x + 2) = 0
\]

Setting each factor to zero gives us:

1. \(x + 1 = 0 \Rightarrow x = -1\)
2. \(x + 2 = 0 \Rightarrow x = -2\)

Now, we need to find the corresponding \(y\) values for each \(x\). We can use either equation. Let’s use \(y = -x + 7\):

For \(x = -1\):

\[
y = -(-1) + 7 = 1 + 7 = 8
\]

For \(x = -2\):

\[
y = -(-2) + 7 = 2 + 7 = 9
\]

Thus, the solutions are:

- When \(x = -1\), \(y = 8\) → point \((-1, 8)\)
- When \(x = -2\), \(y = 9\) → point \((-2, 9)\)

So, the solutions are:

A. The solution(s) is/are \((-1, 8), (-2, 9)\)