To eliminate the \(x\)-terms in the given system of equations, we want to make the coefficients of \(x\) in both equations equal (but opposite in sign) so that when we add both equations, the \(x\)-terms will cancel out.
The original equations are:
- \(3x + 5y = 5\)
- \(-2x - 8y = 6\)
To eliminate the \(x\)-terms, we can multiply both equations by appropriate values.
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For the first equation \(3x + 5y = 5\), we can multiply it by \(2\) to make the coefficient of \(x\) equal to \(6\):
\[ 2(3x + 5y) = 2(5) \implies 6x + 10y = 10 \]
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For the second equation \(-2x - 8y = 6\), we can multiply it by \(3\) to make the coefficient of \(x\) equal to \(-6\):
\[ 3(-2x - 8y) = 3(6) \implies -6x - 24y = 18 \]
Now we have the two modified equations:
- \(6x + 10y = 10\)
- \(-6x - 24y = 18\)
Now, when we add these two equations together, the \(x\)-terms will cancel out:
\[ (6x - 6x) + (10y - 24y) = 10 + 18 \]
This results in:
\[ -14y = 28 \]
Now you can solve for \(y\).
In summary, to eliminate the \(x\)-terms, you would multiply the first equation by \(2\) and the second equation by \(3\):
- \(3x + 5y = 5\) multiply by 2 on both sides.
- \(-2x - 8y = 6\) multiply by 3 on both sides.
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