Question

Justify your steps using the properties of equality and determine the number of solutions to the equation 32(4−8x)+5x=1−7x .(4 points)

Answer 1

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we will follow a systematic approach using properties of equality. Let's go step by step:

1. **Distribute on the left side**:

\[
32(4 - 8x) = 32 \cdot 4 - 32 \cdot 8x
\]

Calculating each term:

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

Therefore, the equation becomes:

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

We can combine the \(x\) terms:

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

2. **Move all terms involving \(x\) to one side and constant terms to the other side**:

Let's add \(251x\) to both sides:

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

Combine like terms on the right side:

\[
128 = 1 + 244x
\]

3. **Isolate the variable \(x\)**:

Subtract \(1\) from both sides:

\[
128 - 1 = 244x
\]

\[
127 = 244x
\]

Now, divide both sides by \(244\):

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

4. **Check for uniqueness**:

Since we found one specific value for \(x\) (i.e., \(x = \frac{127}{244}\)), we conclude that this equation has exactly **one solution**.

### Summary:
- We used the **distributive property** to expand \(32(4 - 8x)\).
- We combined like terms using the **properties of equality**.
- We isolated \(x\) using **addition and multiplication properties of equality**.
- The conclusion is that the equation \(32(4 - 8x) + 5x = 1 - 7x\) has exactly **one solution**: \(x = \frac{127}{244}\).