To solve the equation \(32(4 - 8x) + 5x = 1 - 7x\) and justify the steps using properties of equality, follow these steps:

Question

### Step 1: Distribute
Distribute the \(32\) across \((4 - 8x)\):

\[
32 \cdot 4 - 32 \cdot 8x + 5x = 1 - 7x
\]

Calculating the terms:

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

### Step 2: Combine like terms
Combine the \(x\) terms on the left side:

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

### Step 3: Get all \(x\) terms on one side
Add \(7x\) to both sides:

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

This simplifies to:

\[
128 - 244x = 1
\]

### Step 4: Isolate the constant
Subtract \(128\) from both sides:

\[
-244x = 1 - 128
\]

Calculating the right side:

\[
-244x = -127
\]

### Step 5: Solve for \(x\)
Divide both sides by \(-244\):

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

This simplifies to:

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

### Conclusion
The solution to the equation is \(x = \frac{127}{244}\). Since we found a single unique solution, the number of solutions to the equation is **1**.

### Justification with Properties of Equality
1. **Distributive Property**: Used to distribute \(32\) across the expression \( (4 - 8x) \).
2. **Combining Like Terms**: Followed to simplify both sides of the equation.
3. **Addition Property of Equality**: Applied to add \(7x\) to both sides and \(128\) to isolate the variable.
4. **Division Property of Equality**: Used to divide both sides by \(-244\) to solve for \(x\).

Thus, we have a logical progression of steps that demonstrate the equality and justify the solution methodically.

make this smalller

Answer 1

To solve the equation \(32(4 - 8x) + 5x = 1 - 7x\), we follow these steps:

Step 1: Distribute

Distributing \(32\):

\[ 32 \cdot 4 - 32 \cdot 8x + 5x = 1 - 7x \]

Calculating the terms gives us:

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

Step 2: Combine like terms

Combining \(x\) terms on the left side yields:

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

Step 3: Get all \(x\) terms on one side

Adding \(7x\) to both sides:

\[ 128 - 244x = 1 \]

Step 4: Isolate the constant

Subtracting \(128\) from both sides:

\[ -244x = -127 \]

Step 5: Solve for \(x\)

Dividing both sides by \(-244\):

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

Conclusion

The solution is \(x = \frac{127}{244}\), indicating there is 1 unique solution.

Justification with Properties of Equality

1. Distributive Property: Used to expand \(32(4 - 8x)\).
2. Combining Like Terms: Simplified the equation.
3. Addition Property of Equality: Added \(7x\) to both sides.
4. Subtraction Property of Equality: Isolated terms by subtracting \(128\).
5. Division Property of Equality: Divided to solve for \(x\).

This provides a clear, concise process to derive the solution.