To solve the system of equations using the elimination method, we have the following equations:
- \(-7x - 4y = -11\) (Equation 1)
- \(2x + 8y = 10\) (Equation 2)
We want to eliminate one of the variables. To do this, let's first manipulate the equations so that the coefficients of \(y\) or \(x\) line up.
Let's eliminate \(y\). We can do this by manipulating both equations to have the same coefficient for \(y\).
- For Equation 1, we can multiply the entire equation by \(2\):
\[ 2(-7x - 4y) = 2(-11) \implies -14x - 8y = -22 \quad \text{(Equation 3)} \]
- For Equation 2, we can leave it as is:
\[ 2x + 8y = 10 \quad \text{(Equation 2)} \]
Now we can add Equation 3 and Equation 2 together:
\[ (-14x - 8y) + (2x + 8y) = -22 + 10 \]
This simplifies to:
\[ -14x + 2x - 8y + 8y = -22 + 10 \]
\[ -12x = -12 \]
Now, divide both sides by -12:
\[ x = 1 \]
Now that we have \(x\), we can substitute it back into one of the original equations to find \(y\). Let's use Equation 2:
\[ 2(1) + 8y = 10 \]
\[ 2 + 8y = 10 \]
Subtract 2 from both sides:
\[ 8y = 8 \]
Now, divide by 8:
\[ y = 1 \]
So, the solution to the system of equations is:
\[ \boxed{(1, 1)} \]
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