To solve the system of equations by addition and to eliminate one variable, follow these steps:
1. Consider the given system of equations:
[tex]\[
\begin{array}{l}
3x - 6y = -15 \\
-2x + 5y = 14
\end{array}
\][/tex]
2. We need to eliminate one of the variables (either [tex]\(x\)[/tex] or [tex]\(y\)[/tex]) by making the coefficients of that variable equal and opposite.
3. Let's decide to eliminate [tex]\(x\)[/tex]. To do this, we need to manipulate the equations so that the coefficients of [tex]\(x\)[/tex] become equal and opposite.
4. To eliminate [tex]\(x\)[/tex], we can multiply the top equation by 2 and the bottom equation by 3. Doing this, the coefficients of [tex]\(x\)[/tex] will become 6 and -6, respectively.
5. Multiply the top equation by 2:
[tex]\[
2 \cdot (3x - 6y) = 2 \cdot (-15)
\][/tex]
This results in:
[tex]\[
6x - 12y = -30
\][/tex]
6. Multiply the bottom equation by 3:
[tex]\[
3 \cdot (-2x + 5y) = 3 \cdot 14
\][/tex]
This results in:
[tex]\[
-6x + 15y = 42
\][/tex]
7. Now, we have the modified system of equations:
[tex]\[
\begin{array}{l}
6x - 12y = -30 \\
-6x + 15y = 42
\end{array}
\][/tex]
8. Adding these two equations will eliminate the [tex]\(x\)[/tex] variable:
[tex]\[
(6x - 12y) + (-6x + 15y) = -30 + 42
\][/tex]
This simplifies to:
[tex]\[
3y = 12 \quad \Rightarrow \quad y = 4
\][/tex]
Therefore, to eliminate one variable by addition, you should multiply the top equation by 2 and the bottom equation by 3. Thus, the answer is:
[tex]\[
\boxed{\text{C. Multiply the top equation by 2 and the bottom equation by 3.}}
\][/tex]
