To determine how System B was derived from System A, let's carefully review the necessary steps. We have:
System A:
[tex]\[
\begin{aligned}
-x - 2y &= 7 \quad \text{(Equation 1)} \\
5x - 6y &= -3 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
System B:
[tex]\[
\begin{aligned}
-x - 2y &= 7 \quad \text{(Equation 3)} \\
-16y &= 32 \quad \text{(Equation 4)}
\end{aligned}
\][/tex]
We need to ascertain what operations on System A will result in System B.
First, observe that Equation 3 is identical to Equation 1, so the transformation must involve Equation 2 being replaced by a new equation derived from Equation 1 and Equation 2.
To transform Equation 2 (5x - 6y = -3) into Equation 4 (-16y = 32), notice that the new equation needs to eliminate the [tex]\(x\)[/tex] term and result in an equation involving only [tex]\(y\)[/tex]. This can be achieved by combining Equation 2 with a modified version of Equation 1.
Let's multiply Equation 1 by 5:
[tex]\[ -5x - 10y = 35 \quad \text{(Equation 1 multiplied by 5)} \][/tex]
Now, add this new equation to Equation 2:
[tex]\[
\begin{aligned}
(-5x - 10y) + (5x - 6y) &= 35 + (-3) \\
-5x + 5x - 10y - 6y &= 35 - 3 \\
-16y &= 32
\end{aligned}
\][/tex]
Equation 4 (-16y = 32) is exactly the result we get. Hence, the correct transformation involves replacing Equation 2 with the sum of Equation 2 and Equation 1 multiplied by 5.
Thus, the correct option is:
D. To get system B, the second equation in system A was replaced by the sum of that equation and the first equation multiplied by 5. The solution to system B will be the same as the solution to system A.
