To solve the given system of equations:
[tex]\[
\begin{cases}
-2x - 2y = 0 \\
-5x - 7y = 12
\end{cases}
\][/tex]
we will use the method of elimination. First, let's work to eliminate one of the variables.
Step 1: Eliminate [tex]\( y \)[/tex]
Since we already have [tex]\(-2x - 2y = 0\)[/tex], we can multiply this equation by 3.5 to make the coefficients of [tex]\( y \)[/tex] the same in both equations:
[tex]\[
3.5(-2x - 2y) = 3.5(0) \\
-7x - 7y = 0
\][/tex]
So the system of equations now looks like this:
[tex]\[
\begin{cases}
-7x - 7y = 0 \\
-5x - 7y = 12
\end{cases}
\][/tex]
Step 2: Subtract the first equation from the second
So, we subtract:
[tex]\[
(-5x - 7y) - (-7x - 7y) = 12 - 0
\][/tex]
This simplifies to:
[tex]\[
-5x + 7x - 7y + 7y = 12 \\
2x = 12 \\
x = 6
\][/tex]
Step 3: Substitute [tex]\( x \)[/tex] back into one of the original equations
Now that we have [tex]\( x = 6 \)[/tex], we substitute [tex]\( x \)[/tex] back into the first equation [tex]\( -2x - 2y = 0 \)[/tex]:
[tex]\[
-2(6) - 2y = 0 \\
-12 - 2y = 0 \\
-2y = 12 \\
y = -6
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
\boxed{x = 6, y = -6}
\][/tex]
