Let's check each equation by substituting \( x = -7 \) into them.
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For the equation \( 4x - 23 = 5 \): \[ 4(-7) - 23 = -28 - 23 = -51 \quad \text{(not equal to 5)} \]
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For the equation \( 5x + 11 = -24 \): \[ 5(-7) + 11 = -35 + 11 = -24 \quad \text{(equal to -24)} \]
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For the equation \( -3x - 8 = -29 \): \[ -3(-7) - 8 = 21 - 8 = 13 \quad \text{(not equal to -29)} \]
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For the equation \( -x + 3 = -4 \): \[ -(-7) + 3 = 7 + 3 = 10 \quad \text{(not equal to -4)} \]
The only equation that has the solution of \( x = -7 \) is:
\( 5x + 11 = -24 \).