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.