To solve the system of equations using the elimination method, we start with the two given equations:
- \(-3x + 4y = 4\)
- \(-6x + 6y = -6\)
First, we can manipulate the equations to facilitate elimination. We can multiply the first equation by 2 to align the coefficients of \(x\):
\[ 2(-3x + 4y) = 2(4) \]
This gives us:
\[ -6x + 8y = 8 \quad (3) \]
Now we rewrite our system of equations:
- \(-6x + 8y = 8\)
- \(-6x + 6y = -6\)
Next, we can subtract equation (4) from equation (3):
\[ (-6x + 8y) - (-6x + 6y) = 8 - (-6) \]
This simplifies to:
\[ 8y - 6y = 8 + 6 \] \[ 2y = 14 \] \[ y = 7 \]
Now that we have the value of \(y\), we can substitute it back into one of the original equations to find \(x\). Let's use equation (1):
\[ -3x + 4(7) = 4 \] \[ -3x + 28 = 4 \] \[ -3x = 4 - 28 \] \[ -3x = -24 \] \[ x = 8 \]
Thus, the solution to the system of equations is:
\[ \boxed{(8, 7)} \]