To determine whether the equation \(-6y + 4(2y + 7) = 14(8y + 16) + 24\) has one solution, no solution, or infinitely many solutions, we'll simplify both sides of the equation step by step.
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Expand both sides of the equation:
Left side: \[ -6y + 4(2y + 7) = -6y + 8y + 28 = 2y + 28 \]
Right side: \[ 14(8y + 16) + 24 = 112y + 224 + 24 = 112y + 248 \]
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Set the expanded expressions equal to each other: \[ 2y + 28 = 112y + 248 \]
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Rearranging to isolate \(y\): \[ 2y - 112y = 248 - 28 \] \[ -110y = 220 \]
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Solve for \(y\): \[ y = \frac{220}{-110} = -2 \]
Since we found a specific value for \(y\), the equation \(-6y + 4(2y + 7) = 14(8y + 16) + 24\) has exactly one solution: \(y = -2\).
Therefore, the answer is that the equation has one solution.