To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we begin by distributing the \( 32 \) on the left side. Using the distributive property, we have:
\[ 32 \cdot 4 - 32 \cdot 8x = 128 - 256x \]
Thus, the equation becomes:
\[ 128 - 256x + 5x = 1 - 7x \]
Next, we combine like terms on the left side. Adding \( -256x \) and \( 5x \):
\[ 128 - 251x = 1 - 7x \]
Now, we can isolate the variable by adding \( 251x \) to both sides. This gives us:
\[ 128 = 1 - 7x + 251x \]
Simplifying the right side, we have:
\[ 128 = 1 + 244x \]
Next, we subtract \( 1 \) from both sides:
\[ 127 = 244x \]
To solve for \( x \), we divide both sides by \( 244 \):
\[ x = \frac{127}{244} \]
Since we found a specific value for \( x \), which is \( \frac{127}{244} \), we conclude that the equation has one solution.