To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \) and justify the steps using the properties of equality, let’s break it down step by step.
1. **Distribute on the left side:**
\[
32(4) - 32(8x) + 5x = 1 - 7x
\]
\[
128 - 256x + 5x = 1 - 7x
\]
2. **Combine like terms on the left side:**
\[
128 - 251x = 1 - 7x
\]
3. **Isolate the variable \(x\):**
We want to get all \(x\) terms on one side and constant terms on 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:
\[
-244x = 1 - 128
\]
\[
-244x = -127
\]
4. **Solve for \(x\):**
Now divide both sides by \(-244\):
\[
x = \frac{-127}{-244} = \frac{127}{244}
\]
5. **Conclusion:**
We found one unique solution for \(x\), which is \(x = \frac{127}{244}\).
Therefore, the equation \( 32(4 - 8x) + 5x = 1 - 7x \) has exactly **one solution**.