To solve the system of equations
[tex]\[
\begin{cases}
-8x - 8y = 0 \\
-8x + 2y = -20
\end{cases}
\][/tex]
we can use the method of substitution or elimination. Here, I'll demonstrate both the methods for clarity, and we'll check which answer is correct among the given options.
### Method 1: Substitution Method
1. Let's take the first equation:
[tex]\[
-8x - 8y = 0
\][/tex]
2. From this equation, we can solve for [tex]\( y \)[/tex]:
[tex]\[
-8x = 8y \\
y = -x
\][/tex]
3. Substitute [tex]\( y = -x \)[/tex] into the second equation:
[tex]\[
-8x + 2(-x) = -20 \\
-8x - 2x = -20 \\
-10x = -20 \\
x = 2
\][/tex]
4. Now, substitute [tex]\( x = 2 \)[/tex] back into [tex]\( y = -x \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[
y = -2
\][/tex]
Thus, the solution to the system is [tex]\( (2, -2) \)[/tex].
### Method 2: Elimination Method
1. Write down the system of equations:
[tex]\[
\begin{cases}
-8x - 8y = 0 \\
-8x + 2y = -20
\end{cases}
\][/tex]
2. To eliminate [tex]\( x \)[/tex], we can subtract the first equation from the second equation:
[tex]\[
\big( -8x + 2y \big) - \big( -8x - 8y \big) = -20 - 0 \\
-8x + 2y + 8x + 8y = -20 \\
10y = -20 \\
y = -2
\][/tex]
3. Substitute [tex]\( y = -2 \)[/tex] back into the first equation to find [tex]\( x \)[/tex]:
[tex]\[
-8x - 8(-2) = 0 \\
-8x + 16 = 0 \\
-8x = -16 \\
x = 2
\][/tex]
Thus, the solution to the system is [tex]\( (2, -2) \)[/tex].
### Conclusion
The solution to the system of equations
[tex]\[
\begin{cases}
-8x - 8y = 0 \\
-8x + 2y = -20
\end{cases}
\][/tex]
is [tex]\((2, -2)\)[/tex].
Therefore, the correct answer is:
A. [tex]\((2, -2)\)[/tex]
