To solve the system of equations using substitution, we'll start with the two given equations:
- \(-6x - 9y = 3\)
- \(2x + 2y = -6\)
First, we can simplify the second equation by dividing everything by 2:
\[ x + y = -3 \]
Now we can solve for \(y\) in terms of \(x\):
\[ y = -3 - x \]
Next, we'll substitute this expression for \(y\) into the first equation:
\(-6x - 9y = 3\)
Substituting \(y\):
\[ -6x - 9(-3 - x) = 3 \]
Now we simplify the left side:
\[ -6x + 27 + 9x = 3 \]
Combining like terms gives:
\[ 3x + 27 = 3 \]
Next, we isolate \(x\) by subtracting 27 from both sides:
\[ 3x = 3 - 27 \] \[ 3x = -24 \]
Now, divide by 3:
\[ x = -8 \]
Now that we have \(x\), we can substitute it back to find \(y\):
\[ y = -3 - (-8) \] \[ y = -3 + 8 \] \[ y = 5 \]
Thus, the solution to the system of equations is:
\[ (x, y) = (-8, 5) \]
To verify, we can substitute these values back into the original equations:
- For \(-6(-8) - 9(5) = 3\):
\[ 48 - 45 = 3 \quad \text{(True)} \]
- For \(2(-8) + 2(5) = -6\):
\[ -16 + 10 = -6 \quad \text{(True)} \]
Both equations hold true, confirming our solution is correct:
\(\boxed{(-8, 5)}\)
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