To solve the system of equations using the elimination method, we have the following equations:
- \( 9x - 7y = 22 \) (Equation 1)
- \( x + 3y = -24 \) (Equation 2)
To eliminate one variable, we can manipulate Equation 2 to have the same coefficient for \(x\) as in Equation 1. First, we can multiply Equation 2 by \(9\):
\[ 9(x + 3y) = 9(-24) \]
This results in:
\[ 9x + 27y = -216 \quad \text{(Equation 3)} \]
Now, we have the following system of equations:
- \( 9x - 7y = 22 \) (Equation 1)
- \( 9x + 27y = -216 \) (Equation 3)
Next, we will subtract Equation 1 from Equation 3 to eliminate \(x\):
\[ (9x + 27y) - (9x - 7y) = -216 - 22 \]
This simplifies to:
\[ 9x + 27y - 9x + 7y = -238 \]
Thus, we get:
\[ 34y = -238 \]
Now we can solve for \(y\):
\[ y = \frac{-238}{34} = -7 \]
Now that we have \(y\), we can substitute this value back into Equation 2 to solve for \(x\):
\[ x + 3(-7) = -24 \]
This simplifies to:
\[ x - 21 = -24 \]
Adding \(21\) to both sides gives:
\[ x = -3 \]
Therefore, the solution to the system of equations is:
\[ x = -3, \quad y = -7 \]
We can write the final answer as:
\[ (x, y) = (-3, -7) \]