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Justify your steps using the properties of equality and determine the number of solutions to the equation 32(4−8x)+5x=1−7x

.(4 points)make it easy to copy and paste

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To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \) and determine the number of solutions, we will follow these steps, justifying each step using the properties of equality:

  1. Distribute on the left side: \[ 32(4 - 8x) + 5x = 128 - 256x + 5x \]

    • Justification: Distributive Property

  2. Combine like terms on the left side: \[ 128 - 251x = 1 - 7x \]

    • Justification: Combine like terms (i.e., \(-256x + 5x = -251x\))

  3. Add \( 251x \) to both sides to isolate \( x \): \[ 128 = 1 + 251x - 7x \]

    • Justification: Addition Property of Equality

  4. Combine like terms on the right side: \[ 128 = 1 + 244x \]

    • Justification: Combine like terms (i.e., \( 251x - 7x = 244x \))

  5. Subtract 1 from both sides: \[ 127 = 244x \]

    • Justification: Subtraction Property of Equality

  6. Divide both sides by 244: \[ x = \frac{127}{244} \]

    • Justification: Division Property of Equality

  7. Conclusion: The solution to the equation is \( x = \frac{127}{244} \), which means there is exactly one solution to the equation.

Thus, the number of solutions to the equation \( 32(4 - 8x) + 5x = 1 - 7x \) is 1.