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 \( 32(4 - 8x) + 5x = 1 - 7x \), we will use properties of equality to simplify and isolate \( x \).

### Step 1: Distribute the 32 on the left side.

\[
32(4 - 8x) = 32 \cdot 4 - 32 \cdot 8x = 128 - 256x
\]

So, the equation becomes:

\[
128 - 256x + 5x = 1 - 7x
\]

### Step 2: Combine like terms on the left side.

Combining the \( x \) terms:

\[
128 - 256x + 5x = 128 - 251x
\]

Now, the equation is:

\[
128 - 251x = 1 - 7x
\]

### Step 3: Move all \( x \) terms to one side and constant terms to the other side.

To do this, add \( 251x \) to both sides:

\[
128 = 1 - 7x + 251x
\]

Which simplifies to:

\[
128 = 1 + 244x
\]

Now, subtract \( 1 \) from both sides:

\[
128 - 1 = 244x
\]

This simplifies to:

\[
127 = 244x
\]

### Step 4: Divide both sides by \( 244 \) to solve for \( x \).

\[
x = \frac{127}{244}
\]

### Step 5: Determine the number of solutions.

Since we have found one specific solution, the number of solutions to the equation \( 32(4 - 8x) + 5x = 1 - 7x \) is **1**.

Thus, the justification of the steps used the properties of equality, specifically the distributive property, the property of combining like terms, and the additive and multiplicative properties of equality.

**Final Answer:** There is 1 solution: \( x = \frac{127}{244} \).