To solve the system of equations:
[tex]\[
\begin{cases}
x + 4y = -2 \\
x - 4y = -8
\end{cases}
\][/tex]
We can use the method of addition (or elimination) to eliminate one of the variables.
1. Add the two equations together:
[tex]\[
(x + 4y) + (x - 4y) = -2 + (-8)
\][/tex]
This simplifies to:
[tex]\[
x + x + 4y - 4y = -2 - 8
\][/tex]
Which further simplifies to:
[tex]\[
2x = -10
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[
2x = -10 \implies x = \frac{-10}{2} \implies x = -5
\][/tex]
3. Substitute [tex]\(x = -5\)[/tex] back into one of the original equations to find [tex]\(y\)[/tex]:
Using the first equation, [tex]\(x + 4y = -2\)[/tex]:
[tex]\[
-5 + 4y = -2
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
Add 5 to both sides to isolate the term with [tex]\(y\)[/tex]:
[tex]\[
4y = -2 + 5
\][/tex]
[tex]\[
4y = 3
\][/tex]
Divide both sides by 4:
[tex]\[
y = \frac{3}{4}
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
x = -5, \; y = \frac{3}{4}
\][/tex]
The solution does not match any of the given options directly. However, verifying our solution by plugging [tex]\(x = -5\)[/tex] and [tex]\(y = \frac{3}{4}\)[/tex] back into the original equations confirms that the solved values are correct. Thus, the provided answer choices might involve a mistake.
Nevertheless, the correct solution is:
[tex]\[
x = -5, \; y = \frac{3}{4}
\][/tex]
Since none of the given multiple-choice options are correct as provided, it would be best to check the problem statement or the answer choices again.
