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 go through it step by step and justify each step using the properties of equality.

### Step 1: Distribute on the left side

Starting with:
\[
32(4 - 8x) + 5x = 1 - 7x
\]

We can distribute \(32\) on the left side:
\[
32 \cdot 4 - 32 \cdot 8x + 5x = 1 - 7x
\]

Calculating the products:
\[
128 - 256x + 5x = 1 - 7x
\]

### Step 2: Combine like terms

Now, combine the like terms on the left hand side:
\[
128 - 251x = 1 - 7x
\]

### Step 3: Isolate the variable

Next, we want to isolate \(x\). We can start by moving all \(x\) terms to one side and constant terms to the other side. Let's add \(7x\) to both sides:
\[
128 - 251x + 7x = 1
\]
This simplifies to:
\[
128 - 244x = 1
\]

Now, subtract \(128\) from both sides:
\[
-244x = 1 - 128
\]
This simplifies to:
\[
-244x = -127
\]

### Step 4: Solve for \(x\)

Now, divide both sides by \(-244\) to solve for \(x\):
\[
x = \frac{-127}{-244} = \frac{127}{244}
\]

### Conclusion: Determine the number of solutions

The equation \(32(4 - 8x) + 5x = 1 - 7x\) simplifies to a single linear equation with one solution:
\[
x = \frac{127}{244}
\]

Thus, the number of solutions to the equation is **one**.