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 \( 32(4 - 8x) + 5x = 1 - 7x \) and justify each step using properties of equality, we will follow the algebraic steps carefully.

### Step 1: Distribute the \( 32 \) on the left side

Using the **distributive property**, we can expand \( 32(4 - 8x) \):

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

Now our equation looks like this:

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

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

Combine the \( x \) terms on the left side:

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

### Step 3: Isolate the \( x \) terms

Next, we can add \( 7x \) to both sides to move all \( x \) terms to the left:

\[
128 - 251x + 7x = 1
\]
\[
128 - 244x = 1
\]

### Step 4: Isolate the constant

Now, subtract \( 128 \) from both sides:

\[
-244x = 1 - 128
\]
\[
-244x = -127
\]

### Step 5: Solve for \( x \)

Now, divide both sides by \( -244 \) to solve for \( x \):

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

### Conclusion:

The original equation \( 32(4 - 8x) + 5x = 1 - 7x \) has a unique solution, which is

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

### Number of Solutions

Since we arrived at one specific value for \( x \), we conclude that there is **one solution** to the equation.