To solve for the value that fills in the blank in the equation \(-7x - 12 = (fill in the blank)x + 12\), we can rearrange the equation:
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Rewrite it as: \[ -7x - 12 = bx + 12 \]
where \(b\) is the value that we need to find.
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Add \(12\) to both sides: \[ -7x - 12 + 12 = bx + 12 + 12 \] Simplifying it gives: \[ -7x = bx + 24 \]
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Now, let's isolate \(x\): \[ -7x - bx = 24 \] Factoring out \(x\), \[ (-7 - b)x = 24 \]
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For this equation to hold true for all \(x\), the coefficient of \(x\) on the left (i.e., \(-7 - b\)) must equal \(0\). Therefore, we can set up the equation: \[ -7 - b = 0 \]
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Solving for \(b\): \[ b = -7 \]
Thus, the filled-in blank would be \(-7\). Therefore, the equation can be completed as: \[ -7x - 12 = -7x + 12 \]
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