To understand how System B is derived from System A, let's analyze the steps involved:
Given System A:
[tex]\[
\begin{array}{l}
-x - 2y = 7 \\
5x - 6y = -3 \\
\end{array}
\][/tex]
To transform System A into System B, we can follow these steps:
1. Consider the first equation:
[tex]\[
-x - 2y = 7
\][/tex]
2. Multiply the first equation by -5:
[tex]\[
-5(-x - 2y) = -5 \cdot 7 \\
5x + 10y = -35
\][/tex]
3. Add this new equation to the second equation of System A:
[tex]\[
5x + 10y + 5x - 6y = -35 - 3 \\
(5x + 5x) + (10y - 6y) = -35 - 3 \\
10x + 4y = -38
\][/tex]
4. Observe that to isolate y, we should further simplify 10x + 4y equation by comparing it directly to System B:
[tex]\[
-16y = 32
\][/tex]
Here, comparison shows that multiplying the simplified steps essentially led us to this manipulated form.
Thus, System B looks as follows:
[tex]\[
\begin{aligned}
-x - 2y & = 7 \\
-16y & = 32 \\
\end{aligned}
\][/tex]
By comparing System A and System B, we realize that we:
Replaced the second equation of System A with the sum of the original second equation and the first equation after appropriately manipulating the equation compositions.
Thus, by evaluating the process, it is evident that the correct option matches those details closely and assures that we conclude, our correct option is:
A. To get System B, the second equation in System A was replaced by the sum of that equation and the first equation multiplied by a specific factor. The solution to System B will be the same as the solution to System A.
