To solve the system of equations using the elimination method, we have the following two equations:
- \( 3x - 4y = -4 \)
- \( x + 3y = -10 \)
Let's eliminate \( x \) by manipulating these equations.
First, we can express \( x \) from the second equation:
\[ x = -10 - 3y \]
Next, we will substitute this expression for \( x \) into the first equation:
\[ 3(-10 - 3y) - 4y = -4 \]
Now, simplify this equation:
\[ -30 - 9y - 4y = -4 \]
Combine like terms:
\[ -30 - 13y = -4 \]
Now, isolate \( y \):
\[ -13y = -4 + 30 \] \[ -13y = 26 \] \[ y = \frac{26}{-13} = -2 \]
Now that we have \( y \), we'll find \( x \) using the expression we had for \( x \):
\[ x = -10 - 3(-2) = -10 + 6 = -4 \]
So the solution to the system of equations is:
\[ (x, y) = (-4, -2) \]
Thus, the solution is:
\[ \boxed{(-4, -2)} \]