To find the solution set of the equations [tex]\( y = x^2 + 2x + 7 \)[/tex] and [tex]\( y = x + 7 \)[/tex], we need to determine the points where these two equations intersect. This can be done by setting the equations equal to each other and solving for [tex]\( x \)[/tex].
Step 1: Set the equations equal to each other.
[tex]\[
x^2 + 2x + 7 = x + 7
\][/tex]
Step 2: Simplify the equation by moving all terms to one side.
[tex]\[
x^2 + 2x + 7 - (x + 7) = 0
\][/tex]
[tex]\[
x^2 + 2x - x + 7 - 7 = 0
\][/tex]
[tex]\[
x^2 + x = 0
\][/tex]
Step 3: Factor the equation.
[tex]\[
x(x + 1) = 0
\][/tex]
Step 4: Solve for [tex]\( x \)[/tex].
[tex]\[
x = 0 \quad \text{or} \quad x = -1
\][/tex]
Step 5: Substitute these values of [tex]\( x \)[/tex] back into either of the original equations to find the corresponding values of [tex]\( y \)[/tex].
For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 0 + 7 = 7
\][/tex]
For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = -1 + 7 = 6
\][/tex]
Thus, the points of intersection are [tex]\((0, 7)\)[/tex] and [tex]\((-1, 6)\)[/tex].
So, the solution set is:
[tex]\[
\{(0, 7), (-1, 6)\}
\][/tex]
Therefore, the correct answer is:
A. [tex]\(\{(0, 7), (-1, 6)\}\)[/tex]
