To determine how many solutions the equation \( 9z + 10 = 1 - 3(5 - 3z) \) has, we will first simplify the equation.
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Distribute \( -3 \) on the right-hand side: \[ 9z + 10 = 1 - 15 + 9z \] Simplifying the right-hand side: \[ 9z + 10 = -14 + 9z \]
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Next, we will subtract \( 9z \) from both sides: \[ 10 = -14 \]
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The statement \( 10 = -14 \) is a contradiction; it is never true.
Since we've arrived at a contradiction, this implies that there are no solutions to the original equation.
Thus, the answer is:
0 solutions.