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

Question

### 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.

make this like A 6TH GRADER WROTE IT

Answer 1

To solve the equation \(32(4 - 8x) + 5x = 1 - 7x\), let's take it step by step!

Step 1: Distribute

First, we need to use the distributive property on \(32(4 - 8x)\). This means we multiply \(32\) by both \(4\) and \(-8x\):

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

Calculating that gives us:

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

Step 2: Combine like terms

Now, let's combine the \(x\) terms on the left side. We have \(-256x + 5x\):

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

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

To move all the \(x\) terms to one side, we add \(7x\) to both sides of the equation:

\[ 128 - 244x = 1 \]

Step 4: Isolate the constant

Next, we want to isolate \(x\). First, subtract \(128\) from both sides:

\[ -244x = -127 \]

Step 5: Solve for \(x\)

Finally, we divide both sides by \(-244\) to find \(x\):

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

Conclusion

The solution we found is \(x = \frac{127}{244}\). That means there is 1 unique solution!

Justification with Properties of Equality

1. Distributive Property: We used it to expand \(32(4 - 8x)\).
2. Combining Like Terms: We simplified the equation.
3. Addition Property of Equality: We added \(7x\) to both sides.
4. Subtraction Property of Equality: We isolated the numbers by subtracting \(128\).
5. Division Property of Equality: We divided to find \(x\).

So that's how I solved the equation step by step!