To solve the system of equations given by:
- \( 4x - 8y = -36 \)
- \( -6x + y = 32 \)
we can first simplify and rewrite these equations.
Starting with the first equation:
Equation 1:
\[ 4x - 8y = -36 \]
We can simplify it by dividing the entire equation by 4:
\[ x - 2y = -9 \]
Now we have:
\[ x = 2y - 9 \] (Equation 3)
Now for the second equation:
Equation 2:
\[ -6x + y = 32 \]
We can express \( y \) in terms of \( x \):
\[ y = 6x + 32 \] (Equation 4)
Now we can substitute Equation 3 into Equation 4 for \( x \):
Substituting \( x \) from Equation 3 into Equation 4:
\[ y = 6(2y - 9) + 32 \]
Now, distribute the 6:
\[ y = 12y - 54 + 32 \]
Combining like terms gives:
\[ y = 12y - 22 \]
Now, isolate \( y \):
\[ -11y = -22 \]
Dividing both sides by \(-11\):
\[ y = 2 \]
Now that we have \( y \), substitute \( y = 2 \) back into Equation 3 to find \( x \):
\[ x = 2(2) - 9 \]
\[ x = 4 - 9 \]
\[ x = -5 \]
So the solution to the system of equations is:
\[ (x, y) = (-5, 2) \]
Thus, the final answer is:
\[ \boxed{(-5, 2)} \]