To determine the values that would make the given equation \(3(5x + 3) = 4x + ____ x + 9\) have zero solutions, we first need to simplify and analyze the equation.
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Expand the left side: \[ 3(5x + 3) = 15x + 9 \]
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Rewrite the right side: Let the missing value be \(k\). Thus, the right side becomes: \[ 4x + kx + 9 = (4 + k)x + 9 \]
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Now we equate both sides: \[ 15x + 9 = (4 + k)x + 9 \]
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Subtract 9 from both sides: \[ 15x = (4 + k)x \]
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Rearranging gives: \[ 15x - (4 + k)x = 0 \] \[ (15 - (4 + k))x = 0 \] \[ (15 - 4 - k)x = 0 \] \[ (11 - k)x = 0 \]
To have zero solutions, we need the coefficient of \(x\) to be zero. Therefore, we set: \[ 11 - k = 0 \]
- Solving for \(k\): \[ k = 11 \]
Thus, the missing value \(k\) that would make the equation have zero solutions is:
\[ \boxed{11} \]