To solve the system of equations:
- \(-9x + y = 12\) \
- \(2x + 4y = -28\)
We can use either substitution or elimination. Let's use substitution.
From equation (1), we can express \(y\) in terms of \(x\):
\[ y = 12 + 9x \]
Now, we can substitute this expression for \(y\) into equation (2):
\[ 2x + 4(12 + 9x) = -28 \]
Now, distribute the \(4\):
\[ 2x + 48 + 36x = -28 \]
Combine like terms:
\[ 38x + 48 = -28 \]
Now, subtract \(48\) from both sides:
\[ 38x = -28 - 48 \] \[ 38x = -76 \]
Now, divide by \(38\):
\[ x = \frac{-76}{38} = \frac{-38}{19} = -2 \]
Now that we have \(x\), we can substitute it back into the expression we found for \(y\):
\[ y = 12 + 9(-2) \] \[ y = 12 - 18 \] \[ y = -6 \]
Thus, the solution to the system of equations is:
\[ (x, y) = (-2, -6) \]
To verify, plug \(x = -2\) and \(y = -6\) back into the original equations.
For equation (1):
\[ -9(-2) + (-6) = 18 - 6 = 12 \quad \text{(True)} \]
For equation (2):
\[ 2(-2) + 4(-6) = -4 - 24 = -28 \quad \text{(True)} \]
Both equations are satisfied, so the solution is confirmed:
\[ \boxed{(-2, -6)} \]