Let's solve the system of equations using the elimination method.
Given system:
[tex]\[
\begin{cases}
-4x - 2y = -12 \\
2x + 4y = -12
\end{cases}
\][/tex]
Step 1: Simplify the equations if possible.
Looking at our system, each term can be divided by 2 to make calculations easier:
[tex]\[
\begin{cases}
-2x - y = -6 \quad \text{ (Dividing the first equation by 2)} \\
x + 2y = -6 \quad \text{(Dividing the second equation by 2)}
\end{cases}
\][/tex]
Our simplified system of equations is now:
[tex]\[
\begin{cases}
-2x - y = -6 \\
x + 2y = -6
\end{cases}
\][/tex]
Step 2: Align the equations to eliminate one variable.
To eliminate [tex]\(y\)[/tex], we can multiply the second equation by 2 so that the coefficients of [tex]\(y\)[/tex] in both equations will be the same:
[tex]\[
\begin{cases}
-2x - y = -6 \\
2(x + 2y) = 2(-6)
\end{cases}
\][/tex]
This results in:
[tex]\[
\begin{cases}
-2x - y = -6 \\
2x + 4y = -12
\end{cases}
\][/tex]
Step 3: Add the equations to eliminate [tex]\(x\)[/tex].
By adding these two equations together, the [tex]\(x\)[/tex] terms will cancel out:
[tex]\[
(-2x - y) + (2x + 4y) = -6 + (-12)
\][/tex]
Simplifying, we get:
[tex]\[
-2x + 2x - y + 4y = -6 - 12
\][/tex]
[tex]\[
3y = -18
\][/tex]
Step 4: Solve for [tex]\(y\)[/tex].
[tex]\[
y = \frac{-18}{3}
\][/tex]
[tex]\[
y = -6
\][/tex]
Step 5: Substitute [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex].
Using the simplified second equation [tex]\(x + 2y = -6\)[/tex]:
[tex]\[
x + 2(-6) = -6
\][/tex]
[tex]\[
x - 12 = -6
\][/tex]
Adding 12 to both sides:
[tex]\[
x = 6
\][/tex]
So, the solution to the system of equations is:
[tex]\[
x = 6, \quad y = -6
\][/tex]
Therefore, the solution of the system is:
[tex]\[
\boxed{6} \quad \text{and} \quad \boxed{-6}
\][/tex]
