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

.(4 points)

1 answer

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \) and justify each step using properties of equality, we'll proceed as follows:

  1. Distribute the \( 32 \) on the left side: \[ 32 \cdot 4 - 32 \cdot 8x + 5x = 1 - 7x \] \[ 128 - 256x + 5x = 1 - 7x \]

    • Justification: Distributive Property.

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

    • Justification: Combining Like Terms.

  3. Add \( 7x \) to both sides to move \( x \)-terms to one side: \[ 128 - 251x + 7x = 1 \] \[ 128 - 244x = 1 \]

    • Justification: Addition Property of Equality.

  4. Subtract \( 128 \) from both sides: \[ -244x = 1 - 128 \] \[ -244x = -127 \]

    • Justification: Subtraction Property of Equality.

  5. Divide both sides by \( -244 \): \[ x = \frac{-127}{-244} \] \[ x = \frac{127}{244} \]

    • Justification: Division Property of Equality.

Since we obtained a single value for \( x \) (which is \( \frac{127}{244} \)), we conclude that the equation has exactly one solution.

Final Answer: The number of solutions to the equation \( 32(4 - 8x) + 5x = 1 - 7x \) is one solution \(\left( x = \frac{127}{244} \right)\).