To find a true equivalent system for System B that includes the equation \(7x + 4y = -4\) from System A, we can look for a new equation that is a scalar multiple of a linear combination of the original equations of System A.
Let's analyze the options provided:
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Option 1: \(-12x - 12y = 24\)
We can simplify this equation by dividing everything by -12: \[ x + y = -2 \] This does not appear to relate to any known equivalent action on \(7x + 4y = -4\).
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Option 2: \(12x + 12y = 24\)
Similar to option 1, we can simplify this by dividing everything by 12: \[ x + y = 2 \] This equation also does not correlate with any scalar multiple of \(7x + 4y = -4\).
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Option 3: \(-14x - 8y = 8\)
If we divide this whole equation by -2, we get: \[ 7x + 4y = -4 \] which is indeed the same as one of the equations in System A.
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Option 4: \(-14x - 8y = -8\)
Simplifying this equation by dividing everything by -2 would result in: \[ 7x + 4y = 4 \] which is different from our original equation from System A.
From this analysis, Option 3 (\(-14x - 8y = 8\)) yields an equivalent equation to \(7x + 4y = -4\).
Thus, the equation that creates a true equivalent system is: Option 3: \(-14x - 8y = 8\).
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