Justify your steps using the properties of equality and determine the number of solutions to the equation 32(4−8x)+5x=1−7x in 1-2 sentences (4 points)

To solve the equation \(32(4 - 8x) + 5x = 1 - 7x\), first use the distributive property to expand the left side:

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

This simplifies to:

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

Combine like terms:

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

Next, isolate \(x\) by adding \(251x\) and subtracting \(1\) from both sides:

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

This results in:

\[
127 = 244x
\]

Dividing both sides by \(244\) gives:

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

Since the equation leads to a single specific solution for \(x\), the number of solutions is 1.