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

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we'll follow these steps:

1. **Distribute the 32:**
\[
32(4 - 8x) = 32 \cdot 4 - 32 \cdot 8x = 128 - 256x
\]
So the equation becomes:
\[
128 - 256x + 5x = 1 - 7x
\]

2. **Combine like terms on the left side:**
\[
128 - 256x + 5x = 128 - 251x
\]
Now the equation is:
\[
128 - 251x = 1 - 7x
\]

3. **Add \(251x\) to both sides to isolate the variable terms on one side:**
\[
128 = 1 - 7x + 251x
\]
Simplifying the right side gives:
\[
128 = 1 + 244x
\]

4. **Subtract 1 from both sides:**
\[
128 - 1 = 244x
\]
This simplifies to:
\[
127 = 244x
\]

5. **Divide both sides by 244 to solve for \(x\):**
\[
x = \frac{127}{244}
\]

Now, we have found a single solution for the equation:
\[
x = \frac{127}{244}
\]

**Conclusion on the number of solutions:**
Since we found \(x = \frac{127}{244}\) and it is a valid solution in the real number system, the equation \(32(4-8x)+5x=1-7x\) has exactly **one solution**.