Justify your steps using the properties of equality and determine the number of solutions to the equation 3/2(4−8x)+5x=1−7x.

Answer 1

To solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \), we will follow systematic steps and justify each action using properties of equality.

1. Distribute \( \frac{3}{2} \) on the left side: \[ \frac{3}{2}(4) - \frac{3}{2}(8x) + 5x = 1 - 7x \] This gives: \[ 6 - 12x + 5x = 1 - 7x \] (Justified using the Distributive Property.)

2. Combine like terms on the left side: \[ 6 - 7x = 1 - 7x \] (Justified as \( -12x + 5x = -7x \).)

3. Add \( 7x \) to both sides to eliminate \( -7x \) from both sides: \[ 6 = 1 \] (Justified using the Addition Property of Equality.)

4. Observe the result: The equation \( 6 = 1 \) is a contradiction, meaning that there are no values of \( x \) that can satisfy the original equation.

Thus, the original equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) has no solutions.