To solve the system of equations using the elimination method, we start with the two equations:
- \( 4x + 8y = 20 \) (Equation 1)
- \( -4x + 2y = -30 \) (Equation 2)
The goal of the elimination method is to eliminate one of the variables by adding or subtracting the equations.
Let's add Equation 1 and Equation 2:
\[ (4x + 8y) + (-4x + 2y) = 20 + (-30) \]
Simplifying both sides:
\[ 4x - 4x + 8y + 2y = 20 - 30 \] \[ 0 + 10y = -10 \]
Now, we can solve for \(y\):
\[ 10y = -10 \] \[ y = -1 \]
Next, we'll substitute \(y = -1\) back into one of the original equations to find the value of \(x\). We'll use Equation 1:
\[ 4x + 8(-1) = 20 \] \[ 4x - 8 = 20 \] \[ 4x = 20 + 8 \] \[ 4x = 28 \] \[ x = 7 \]
Now we have the solution:
\[ x = 7, \quad y = -1 \]
Thus, the solution to the system of equations is \( (x, y) = (7, -1) \).
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