[tex]$
\begin{array}{l}
y=-x-11 \\
y=x^2-5x-7
\end{array}
$[/tex]

Which of the following is a solution to the system of equations?

A. [tex]$(-2, -9)$[/tex]
B. [tex]$(0, -7)$[/tex]
C. [tex]$(2, -13)$[/tex]
D. [tex]$(3, -13)$[/tex]



Answer :

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]