To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we'll follow these steps:
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Distribute the 32: \[ 32(4 - 8x) = 32 \cdot 4 - 32 \cdot 8x = 128 - 256x \] So the equation becomes: \[ 128 - 256x + 5x = 1 - 7x \]
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Combine like terms on the left side: \[ 128 - 256x + 5x = 128 - 251x \] Now the equation is: \[ 128 - 251x = 1 - 7x \]
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Add \(251x\) to both sides to isolate the variable terms on one side: \[ 128 = 1 - 7x + 251x \] Simplifying the right side gives: \[ 128 = 1 + 244x \]
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Subtract 1 from both sides: \[ 128 - 1 = 244x \] This simplifies to: \[ 127 = 244x \]
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Divide both sides by 244 to solve for \(x\): \[ x = \frac{127}{244} \]
Now, we have found a single solution for the equation: \[ x = \frac{127}{244} \]
Conclusion on the number of solutions: Since we found \(x = \frac{127}{244}\) and it is a valid solution in the real number system, the equation \(32(4-8x)+5x=1-7x\) has exactly one solution.