Let us start by analyzing System A:
[tex]\[
\begin{array}{c}
5x - y = -11 \quad \text{(1)} \\
3x - 2y = -8 \quad \text{(2)}
\end{array}
\][/tex]
We were instructed to obtain system B by replacing the second equation by adding it to the first equation multiplied by -2. Let's follow these steps:
1. Multiply equation (1) by -2:
[tex]\[
-2 \times (5x - y) = -2 \times (-11)
\][/tex]
[tex]\[
-10x + 2y = 22 \quad \text{(3)}
\][/tex]
2. Add equation (3) to equation (2):
[tex]\[
(3x - 2y) + (-10x + 2y) = -8 + 22
\][/tex]
[tex]\[
3x - 10x = -8 + 22
\][/tex]
[tex]\[
-7x = 14 \quad \text{(4)}
\][/tex]
Thus, System B is:
[tex]\[
\begin{array}{c}
5x - y = -11 \\
-7x = 14
\end{array}
\][/tex]
Now, let's verify if the solution [tex]\((-2, 1)\)[/tex] is valid for both systems.
For the first equation in System A and System B:
[tex]\[
5(-2) - 1 = -10 - 1 = -11
\][/tex]
The left-hand side equals the right-hand side, so this solution satisfies the first equation.
For the second equation in System A:
[tex]\[
3(-2) - 2(1) = -6 - 2 = -8
\][/tex]
The left-hand side equals the right-hand side, so this solution satisfies the second equation of System A.
For the second equation in System B:
[tex]\[
-7(-2) = 14
\][/tex]
The left-hand side equals the right-hand side, so this solution satisfies the second equation of System B as well.
Hence, the solution [tex]\((-2, 1)\)[/tex] is valid for both Systems A and B. Therefore, the correct answer is:
D. The second equation in system B is [tex]\(-7x = 14\)[/tex]. The solution to system B will be the same as the solution to system A.
