To solve the system of equations:
- \(-6x + 22y = -32\) (Equation 1)
- \(4x + 3y = -9\) (Equation 2)
We can use either the substitution method or the elimination method. Here, I will use the elimination method.
First, let's work with both equations. We can manipulate them to eliminate \(x\) or \(y\).
Step 1: Simplify Equation 1. Divide the first equation by 2 to make calculations easier:
\[ -3x + 11y = -16 \quad \text{(Equation 1 simplified)} \]
Step 2: Multiply Equation 2 to facilitate elimination. To eliminate \(x\), we can multiply Equation 2 by \(3\) so that the coefficients of \(x\) match:
\[ 3(4x + 3y) = 3(-9) \]
This gives us:
\[ 12x + 9y = -27 \quad \text{(Equation 2 multiplied by 3)} \]
Step 3: Eliminate \(x\). Now, let's multiply Equation 1 by \(4\) to align it with the new Equation 2 format:
\[ 4(-3x + 11y) = 4(-16) \]
This simplifies to:
\[ -12x + 44y = -64 \quad \text{(New Equation 1)} \]
Now we can set both expressions equal to eliminate \(y\):
- \(12x + 9y = -27\) (from transformed Equation 2)
- \(-12x + 44y = -64\) (from modified Equation 1)
Step 4: Add the equations to eliminate \(x\).
Adding this system:
\[ (12x + 9y) + (-12x + 44y) = -27 + (-64) \]
This simplifies to:
\[ 53y = -91 \]
Step 5: Solve for \(y\).
Divide both sides by \(53\):
\[ y = -\frac{91}{53} \]
Step 6: Substitute back to find \(x\). Now plug this \(y\) value back into one of the original equations. We’ll use Equation 2:
\[ 4x + 3\left(-\frac{91}{53}\right) = -9 \] \[ 4x - \frac{273}{53} = -9 \]
Convert \(-9\) into a fraction with the same denominator (53):
\[ -9 = -\frac{477}{53} \]
So now the equation is:
\[ 4x - \frac{273}{53} = -\frac{477}{53} \]
Step 7: Solve for \(x\).
Add \(\frac{273}{53}\) to both sides:
\[ 4x = -\frac{477}{53} + \frac{273}{53} \] \[ 4x = -\frac{204}{53} \]
Now, divide by 4:
\[ x = -\frac{204}{53} \cdot \frac{1}{4} = -\frac{51}{53} \]
Final result:
The solution to the system of equations is:
\[ x = -\frac{51}{53}, \quad y = -\frac{91}{53} \]
Thus, the solution is:
\[ \left( -\frac{51}{53}, -\frac{91}{53} \right) \]