To solve the system of equations using the elimination method, we have the following:
- \( x + y = -8 \) (Equation 1)
- \( x - y = -6 \) (Equation 2)
We can eliminate \(y\) by adding both equations together.
Adding (1) and (2):
\[ (x + y) + (x - y) = -8 + (-6) \]
This simplifies to:
\[ 2x = -14 \]
Now, divide both sides by 2:
\[ x = -7 \]
Now that we have \(x\), we can substitute \(x\) back into one of the original equations to find \(y\). We'll use Equation 1:
\[ -7 + y = -8 \]
To solve for \(y\), add 7 to both sides:
\[ y = -8 + 7 \]
\[ y = -1 \]
So the solution to the system of equations is:
\[ (x, y) = (-7, -1) \]
Thus, the correct answer is:
\[ \boxed{(-7, -1)} \]