To determine which of the given points is a solution to the system of equations:
[tex]\[
\begin{cases}
y = -x - 11 \\
y = x^2 - 5x - 7
\end{cases}
\][/tex]
we will substitute each point into both equations and check if both equations are satisfied simultaneously.
Given points to check:
A. [tex]\((-2, -9)\)[/tex]
B. [tex]\((0, -7)\)[/tex]
C. [tex]\((2, -13)\)[/tex]
D. [tex]\((3, -13)\)[/tex]
1. Checking Point [tex]\((-2, -9)\)[/tex]:
- For [tex]\(y = -x - 11\)[/tex]:
[tex]\[
y = -(-2) - 11 = 2 - 11 = -9
\][/tex]
Point (-2, -9) satisfies the first equation.
- For [tex]\(y = x^2 - 5x - 7\)[/tex]:
[tex]\[
y = (-2)^2 - 5(-2) - 7 = 4 + 10 - 7 = 7
\][/tex]
Point (-2, -9) does not satisfy the second equation.
So, [tex]\((-2, -9)\)[/tex] is not a solution.
2. Checking Point [tex]\((0, -7)\)[/tex]:
- For [tex]\(y = -x - 11\)[/tex]:
[tex]\[
y = -0 - 11 = -11
\][/tex]
Point (0, -7) does not satisfy the first equation.
Thus, [tex]\((0, -7)\)[/tex] is not a solution.
3. Checking Point [tex]\((2, -13)\)[/tex]:
- For [tex]\(y = -x - 11\)[/tex]:
[tex]\[
y = -2 - 11 = -13
\][/tex]
Point (2, -13) satisfies the first equation.
- For [tex]\(y = x^2 - 5x - 7\)[/tex]:
[tex]\[
y = 2^2 - 5(2) - 7 = 4 - 10 - 7 = -13
\][/tex]
Point (2, -13) satisfies the second equation.
So, [tex]\((2, -13)\)[/tex] is a solution.
4. Checking Point [tex]\((3, -13)\)[/tex]:
- For [tex]\(y = -x - 11\)[/tex]:
[tex]\[
y = -3 - 11 = -14
\][/tex]
Point (3, -13) does not satisfy the first equation.
Hence, [tex]\((3, -13)\)[/tex] is not a solution.
After checking all points, we find that the only point that satisfies both equations is:
(C) [tex]\((2, -13)\)[/tex]
