To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we first distribute the \( 32 \) on the left side. Using the distributive property, we have:
\[ 32 \cdot 4 - 32 \cdot 8x = 128 - 256x. \]
Now the equation becomes:
\[ 128 - 256x + 5x = 1 - 7x. \]
Next, we combine like terms on the left side. Combining \(-256x\) and \(5x\) gives:
\[ 128 - 251x = 1 - 7x. \]
Then, we can add \(7x\) to both sides to eliminate the \(7x\) from the right side:
\[ 128 - 251x + 7x = 1. \]
This simplifies to:
\[ 128 - 244x = 1. \]
To isolate \(x\), we first subtract \(128\) from both sides:
\[ -244x = 1 - 128, \]
which simplifies to:
\[ -244x = -127. \]
Now, divide both sides by \(-244\):
\[ x = \frac{-127}{-244} = \frac{127}{244}. \]
Since we obtained a unique value for \(x\), there is exactly one solution to the equation \( 32(4 - 8x) + 5x = 1 - 7x \).