To eliminate one variable when adding the two equations, we need to make the coefficients of either \(y\) or \(x\) equal in magnitude but opposite in sign.
The given equations are:
- \(5x + 13y = 232\)
- \(12x + 7y = 218\)
To eliminate \(y\):
- The coefficient of \(y\) in the first equation is \(13\).
- The coefficient of \(y\) in the second equation is \(7\).
To eliminate \(y\), we want to make the coefficients equal. We can achieve this by multiplying the first equation by \(-7\) and the second equation by \(13\) (so that the coefficients are \(-91\) and \(91\), which will cancel out when we add the equations):
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Multiply the first equation by \(-7\): \[ -7(5x + 13y) = -7(232) \implies -35x - 91y = -1624 \]
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Multiply the second equation by \(13\): \[ 13(12x + 7y) = 13(218) \implies 156x + 91y = 2834 \]
Adding these two equations together: \[ (-35x - 91y) + (156x + 91y) = -1624 + 2834 \] This results in: \[ 121x = 1210 \quad \text{(y is eliminated)} \]
To eliminate \(x\):
To eliminate \(x\), we compare the coefficients of \(x\):
- The coefficient of \(x\) in the first equation is \(5\).
- The coefficient of \(x\) in the second equation is \(12\).
To eliminate \(x\), we can multiply the first equation by \(12\) and the second equation by \(-5\):
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Multiply the first equation by \(12\): \[ 12(5x + 13y) = 12(232) \implies 60x + 156y = 2784 \]
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Multiply the second equation by \(-5\): \[ -5(12x + 7y) = -5(218) \implies -60x - 35y = -1090 \]
Adding these two equations together: \[ (60x + 156y) + (-60x - 35y) = 2784 - 1090 \] This yields: \[ 121y = 1694 \quad \text{(x is eliminated)} \]
Summary:
To eliminate \(y\):
- First equation by \(-7\) and second equation by \(13\).
To eliminate \(x\):
- First equation by \(12\) and second equation by \(-5\).
So, the correct answers can be summarized as:
- To eliminate \(y\): The first equation can be multiplied by \(-7\) and the second equation by \(13\).
- To eliminate \(x\): The first equation can be multiplied by \(12\) and the second equation by \(-5\).
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