To solve the system of equations using the elimination method, we start with the given equations:
- \( 2x + 8y = 8 \)
- \( 2x - 4y = -16 \)
To eliminate one of the variables, we can subtract one equation from the other. First, let's simplify each equation:
From Equation 1, we can divide the entire equation by 2 to make it easier to work with:
\[ x + 4y = 4 \quad \text{(Equation 1 simplified)} \]
From Equation 2, we can also divide the entire equation by 2:
\[ x - 2y = -8 \quad \text{(Equation 2 simplified)} \]
Now we have the simplified system:
- \( x + 4y = 4 \)
- \( x - 2y = -8 \)
Next, we can eliminate \( x \) by subtracting Equation 2 from Equation 1:
\[ (x + 4y) - (x - 2y) = 4 - (-8) \]
This simplifies to:
\[ x + 4y - x + 2y = 4 + 8 \] \[ 6y = 12 \]
Now, solve for \( y \):
\[ y = 2 \]
Now that we have \( y \), we can substitute it back into one of the simplified equations to find \( x \). We will use Equation 1:
\[ x + 4(2) = 4 \] \[ x + 8 = 4 \] \[ x = 4 - 8 \] \[ x = -4 \]
So the solution to the system of equations is:
\[ x = -4, \quad y = 2 \]
Thus, the final answer is \((-4, 2)\).
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