To solve the system of equations using the elimination method, we have the following equations:
- \( 2x + 8y = 8 \)
- \( 2x - 4y = -16 \)
First, let's align the equations for clarity:
\[ \begin{align*} (1) & \quad 2x + 8y = 8 \ (2) & \quad 2x - 4y = -16 \ \end{align*} \]
Next, we can eliminate \(x\) by subtracting one equation from the other. To do this, we can first multiply equation (2) by -1 so that we can eliminate \(2x\):
\[ -1 \cdot (2x - 4y) = -1 \cdot (-16) \implies -2x + 4y = 16 \]
Now we can rewrite our system:
- \( 2x + 8y = 8 \)
- \( -2x + 4y = 16 \)
Next, we can add the two equations together:
\[ (2x + 8y) + (-2x + 4y) = 8 + 16 \]
This simplifies to:
\[ 0x + 12y = 24 \]
Thus, we can solve for \(y\):
\[ 12y = 24 \quad \implies \quad y = \frac{24}{12} = 2 \]
Now that we have the value for \(y\), let's substitute \(y = 2\) back into one of the original equations to find \(x\). We'll use equation (1):
\[ 2x + 8(2) = 8 \]
This simplifies to:
\[ 2x + 16 = 8 \]
Now, we isolate \(2x\):
\[ 2x = 8 - 16 \quad \implies \quad 2x = -8 \quad \implies \quad x = -4 \]
Thus, the solution to the system of equations is:
\[ (x, y) = (-4, 2) \]
The final answer is:
\[ \boxed{(-4, 2)} \]