To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we'll follow a series of steps using the properties of equality, and we will also determine the number of solutions.

Question

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

Using the distributive property:
\[
32(4) - 32(8x) + 5x = 1 - 7x
\]
This simplifies to:
\[
128 - 256x + 5x = 1 - 7x
\]

**Step 2: Combine like terms on the left side.**

Combine the \( x \) terms on the left:
\[
128 - 251x = 1 - 7x
\]

**Step 3: Rearrange the equation.**

To isolate the terms involving \(x\), add \(251x\\) to both sides:
\[
128 = 1 - 7x + 251x
\]

This results in:
\[
128 = 1 + 244x
\]

**Step 4: Isolate the constant term.**

Subtract \(1\) from both sides:
\[
128 - 1 = 244x
\]
Which simplifies to:
\[
127 = 244x
\]

**Step 5: Solve for \(x\).**

Divide both sides by \(244\):
\[
x = \frac{127}{244}
\]

**Conclusion on the number of solutions:**

Since we arrived at a single value for \(x\), there is exactly one solution to the equation.

Thus, the final answer is:
- The value of \(x\) is \(\frac{127}{244}\).
- There is one solution to the equation.

make this smaler

Answer 1

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we proceed as follows:

Step 1: Distribute the left side.

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

This simplifies to:

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

Step 2: Combine like terms on the left.

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

Step 3: Rearrange the equation.

Add \(251x\) to both sides:

\[ 128 = 1 + 244x \]

Step 4: Isolate the constant term.

Subtract \(1\):

\[ 127 = 244x \]

Step 5: Solve for \(x\).

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

Conclusion:

There is one solution, \(x = \frac{127}{244}\).