To solve the system of equations using the elimination method, we have the following equations:
- \( 2x + y = 9 \) (Equation 1)
- \( 8x - 2y = 6 \) (Equation 2)
First, we will manipulate the equations to eliminate one of the variables. Let's eliminate \(y\). To do that, we can multiply Equation 1 by 2, so that the coefficients of \(y\) in both equations will match:
\[ 2(2x + y) = 2(9) \]
This results in:
\[ 4x + 2y = 18 \quad \text{(Equation 3)} \]
Now we can rewrite our system:
- \( 4x + 2y = 18 \) (Equation 3)
- \( 8x - 2y = 6 \) (Equation 2)
Next, we will add Equation 3 and Equation 2 to eliminate \(y\):
\[ (4x + 2y) + (8x - 2y) = 18 + 6 \]
Simplifying, we get:
\[ 4x + 8x + 2y - 2y = 24 \]
This simplifies to:
\[ 12x = 24 \]
Now divide both sides by 12:
\[ x = 2 \]
Now that we have \(x\), we can substitute this value back into one of the original equations to find \(y\). Let's use Equation 1:
\[ 2(2) + y = 9 \]
This simplifies to:
\[ 4 + y = 9 \]
So,
\[ y = 9 - 4 \] \[ y = 5 \]
Thus, the solution to the system of equations is:
\[ (x, y) = (2, 5) \]
So, the correct answer is:
(2, 5)