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

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**, \( x = -\frac{3}{4} \).