To solve the system of equations using the elimination method, we have:
- \( 9x - 7y = 22 \) (Equation 1)
- \( x + 3y = -24 \) (Equation 2)
First, we will manipulate Equation 2 so that it can be easily eliminated with Equation 1. We can multiply Equation 2 by 9 to match the coefficient of \( x \) in Equation 1:
\[ 9(x + 3y) = 9(-24) \]
This gives us:
\[ 9x + 27y = -216 \quad (Equation 3) \]
Now we have a new equation (Equation 3) and will write the system again:
- \( 9x - 7y = 22 \) (Equation 1)
- \( 9x + 27y = -216 \) (Equation 3)
Next, we'll subtract Equation 1 from Equation 3 to eliminate \( x \):
\[ (9x + 27y) - (9x - 7y) = -216 - 22 \]
This simplifies to:
\[ 27y + 7y = -238 \]
\[ 34y = -238 \]
Now, solve for \( y \):
\[ y = \frac{-238}{34} = -7 \]
Next, we substitute \( y = -7 \) back into Equation 2 to solve for \( x \):
\[ x + 3(-7) = -24 \]
\[ x - 21 = -24 \]
Adding 21 to both sides gives us:
\[ x = -3 \]
So, we have \( x = -3 \) and \( y = -7 \). The solution is:
\[ (x, y) = (-3, -7) \]
Looking at the answer choices:
A) (3, -9)
B) (0, -8)
C) (-3, -7)
D) (4, 2)
The correct answer is C) (-3, -7).