To solve the equation \(-7x - 12 = __ x + 12\), we can rewrite it in the following way to isolate the blank:
\(-7x - 12 = bx + 12\)
Here, \(b\) represents the blank we need to fill in.
Now, let's rearrange the equation for clarity:
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Move \(bx\) to the left side: \(-7x - bx = 12 + 12\) \(-7x - bx = 24\)
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Combine like terms: \(- (7 + b)x = 24\)
For this equation to be balanced, both sides must equal zero or maintain the integrity of the equation.
Since we want to find the relationship for \(b\), we can express the equation in terms of \(b\):
If we want to equate coefficients of \(x\) on both sides, we can simply note that the left side must equal the right side's coefficient of \(x\), which is \(0\) (since there’s no \(x\) term explicitly given).
Thus: \[-7 - b = 0\]
From this, we solve for \(b\): \[b = -7\]
So, the blank can be filled with \( -7\).
Thus, the final answer is: \(-7x - 12 = -7x + 12\).