To solve the equation \(32(4 - 8x) + 5x = 1 - 7x\), I first distribute \(32\) on the left side. This gives us \(128 - 256x + 5x = 1 - 7x\). Next, I can combine the like terms on the left side, resulting in \(128 - 251x = 1 - 7x\).
Using the properties of equality, I will add \(7x\) to both sides to isolate \(x\): \(128 - 251x + 7x = 1\). This simplifies to \(128 - 244x = 1\). Then, I subtract \(128\) from both sides, yielding \(-244x = 1 - 128\), which simplifies to \(-244x = -127\).
Now, I divide both sides by \(-244\), giving me \(x = \frac{-127}{-244}\) or \(x = \frac{127}{244}\). Since I obtained a single solution, the equation has one solution. Therefore, the number of solutions to the equation is 1.