Justify your steps using the properties of equality and determine the number of solutions to the equation(4-8x)+5x1-7x. (4 points) B I

Answer 1

To solve the equation \( (4 - 8x) + 5x = 1 - 7x \), we will apply the properties of equality step-by-step and justify each move.

1. Combine like terms on the left side: \[ 4 - 8x + 5x = 1 - 7x \] This can be simplified as: \[ 4 - 3x = 1 - 7x \] (justifying this by using the property of combining like terms, i.e., \( -8x + 5x = -3x \)).

2. Add \( 7x \) to both sides: \[ 4 - 3x + 7x = 1 - 7x + 7x \] This simplifies to: \[ 4 + 4x = 1 \] (justifying this by the Addition Property of Equality, which states that if two expressions are equal, adding the same value to both sides keeps them equal).

3. Subtract 4 from both sides: \[ 4 + 4x - 4 = 1 - 4 \] This simplifies to: \[ 4x = -3 \] (using the Subtraction Property of Equality).

4. Divide both sides by 4: \[ x = -\frac{3}{4} \] (applying the Division Property of Equality, which states that if two expressions are equal, dividing both sides by the same non-zero value keeps them equal).

Thus, we have found that the equation \( (4 - 8x) + 5x = 1 - 7x \) has one solution: \[ x = -\frac{3}{4} \]

Conclusion:

The equation has one solution, \( x = -\frac{3}{4} \).