To solve the system of equations given by:
- \(-6x + 2y = -32\)
- \(4x + 3y = -9\)
we can use either substitution or elimination. Let's solve it using the elimination method.
First, we can manipulate the first equation to eliminate \(y\):
- Divide the entire first equation by 2 to simplify it:
\[ -3x + y = -16 \quad \text{(equation 1)} \]
So now we have:
- \(-3x + y = -16\)
- \(4x + 3y = -9\)
Now, let's express \(y\) in terms of \(x\) from equation 1:
\[ y = 3x - 16 \]
Next, substitute this expression for \(y\) into equation 2:
\[ 4x + 3(3x - 16) = -9 \]
Distributing the 3:
\[ 4x + 9x - 48 = -9 \]
Combine like terms:
\[ 13x - 48 = -9 \]
Add 48 to both sides:
\[ 13x = 39 \]
Now, divide by 13:
\[ x = 3 \]
Now substitute \(x = 3\) back into the expression for \(y\):
\[ y = 3(3) - 16 \] \[ y = 9 - 16 \] \[ y = -7 \]
So the solution to the system of equations is:
\[ (x, y) = (3, -7) \]
Thus, the final answer is:
\[ (3, -7) \]