To solve the system of equations
- \(-9x + y = 12\) (Equation 1)
- \(2x + 4y = -28\) (Equation 2)
let's first manipulate both equations to express them in a more manageable form.
Starting with Equation 1, we can express \(y\) in terms of \(x\):
\[ y = 9x + 12 \]
Now, we can substitute this expression for \(y\) into Equation 2:
\[ 2x + 4(9x + 12) = -28 \]
Expanding that gives:
\[ 2x + 36x + 48 = -28 \]
Combine like terms:
\[ 38x + 48 = -28 \]
Now, isolate \(x\) by subtracting 48 from both sides:
\[ 38x = -28 - 48 \]
\[ 38x = -76 \]
Now, divide by 38:
\[ x = \frac{-76}{38} = -2 \]
Now that we have the value of \(x\), let's substitute it back into the equation for \(y\):
\[ y = 9(-2) + 12 \]
\[ y = -18 + 12 \]
\[ y = -6 \]
So, the solution to the system of equations is
\[ \boxed{(-2, -6)}. \]
To verify, we can substitute \(x = -2\) 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 hold, confirming that our solution \( (-2, -6) \) is correct.