Justify your steps using the properties of equality and determine the number of solutions to the equation 32(4−8x)+5x=1−7x32(4−8x)+5x=1−7x.(4 points)

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), let's apply the properties of equality and simplify the expression step by step:

1. **Distribute on the left side**:
\[
32 \times 4 - 32 \times 8x + 5x = 1 - 7x
\]
This simplifies to:
\[
128 - 256x + 5x = 1 - 7x
\]
Combine like terms on the left side:
\[
128 - 251x = 1 - 7x
\]

2. **Move all \( x \) terms to one side and constants to the other**:
Add \( 251x \) to both sides and subtract \( 1 \) from both sides:
\[
128 - 1 = -7x + 251x
\]
This simplifies to:
\[
127 = 244x
\]

3. **Isolate \( x \)**:
Divide both sides by \( 244 \):
\[
x = \frac{127}{244}
\]

The equation has one solution, \( x = \frac{127}{244} \).

### Summary of properties of equality used:

- **Distributive Property**: Used to expand \( 32(4 - 8x) \).
- **Combining Like Terms**: Simplified the equation by combining terms involving \( x \).
- **Addition Property of Equality**: Added and subtracted terms on both sides to isolate \( x \).
- **Division Property of Equality**: Divided both sides by \( 244 \) to solve for \( x \).

Thus, the equation has exactly one solution.