To solve this problem, we need to go through the following steps:
1. Identify System A:
[tex]\[
\begin{cases}
5x - y = -11 \\
3x - 2y = -8 \\
\end{cases}
\][/tex]
The given solution for this system is [tex]\( (-2, 1) \)[/tex].
2. Transform the second equation for System B:
The process described involves replacing the second equation of System A with a new equation. This new equation is obtained by adding the second equation of System A to the first equation of System A multiplied by -2.
Multiply the first equation [tex]\(5x - y = -11\)[/tex] by -2:
[tex]\[
-2(5x - y) = -2(-11)
\][/tex]
Simplifying, we get:
[tex]\[
-10x + 2y = 22
\][/tex]
Now, add this result to the second equation of System A [tex]\(3x - 2y = -8\)[/tex]:
[tex]\[
(3x - 2y) + (-10x + 2y) = -8 + 22
\][/tex]
Simplifying the left-hand side and the right-hand side:
[tex]\[
3x - 10x + 2y - 2y = 14
\][/tex]
[tex]\[
-7x = 14
\][/tex]
3. Construct System B:
Based on the above transformation, the second equation of System B is:
[tex]\[
-7x = 14
\][/tex]
So, System B is:
[tex]\[
\begin{cases}
5x - y = -11 \\
-7x = 14 \\
\end{cases}
\][/tex]
4. Check the Correctness of the Answers:
- Option A: 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.
- Option B: The second equation in System B is [tex]\( 7x = 30 \)[/tex]. The solution to System B will be the same as the solution to System A.
- Option C: The second equation in System B is [tex]\( -7x = 14 \)[/tex]. The solution to System B will not be the same as the solution to System A.
- Option D: The second equation in System B is [tex]\( 7x = 30 \)[/tex]. The solution to System B will not be the same as the solution to System A.
Since the second equation we derived for System B is [tex]\( -7x = 14 \)[/tex] and given the solution to System A is [tex]\((-2, 1)\)[/tex], we solve for [tex]\(x\)[/tex] from [tex]\(-7x = 14\)[/tex]:
[tex]\[
x = 14 / -7 \implies x = -2
\][/tex]
Substituting [tex]\(x = -2\)[/tex] back in, we retain consistency with System A's solution.
Hence, the correct answer is:
A. The second equation in system [tex]\(B\)[/tex] is [tex]\(-7x = 14\)[/tex]. The solution to system [tex]\(B\)[/tex] will be the same as the solution to system [tex]\(A\)[/tex].
