To solve the system of equations:
- \(-6x + 2y = -32\)
- \(4x + 3y = -9\)
We can use the substitution or elimination method. Let's use the elimination method.
First, let's rearrange equation 1 to solve for \(y\).
\[-6x + 2y = -32\]
Add \(6x\) to both sides:
\[2y = 6x - 32\]
Now, divide everything by 2:
\[y = 3x - 16\] (Equation 3)
Now, we can substitute equation 3 into equation 2:
\[4x + 3(3x - 16) = -9\]
Distributing \(3\):
\[4x + 9x - 48 = -9\]
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 equation 3 to find \(y\):
\[y = 3(3) - 16\]
\[y = 9 - 16\]
\[y = -7\]
So, the solution to the system of equations is:
\((x, y) = (3, -7)\)
To verify, we can plug these values back into the original equations:
- For \(-6(3) + 2(-7) = -32\):
\(-18 - 14 = -32\) → True
- For \(4(3) + 3(-7) = -9\):
\(12 - 21 = -9\) → True
Both equations are satisfied, so the solution is correct.
Final Answer: \( (x, y) = (3, -7) \)