To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we will use the properties of equality and justify each step. Let's do this step by step.
Step 1: Distribute on the left side
Start with the original equation: \[ 32(4 - 8x) + 5x = 1 - 7x \] Using the Distributive Property: \[ 32 \cdot 4 - 32 \cdot 8x + 5x = 1 - 7x \] Calculating the multiplication: \[ 128 - 256x + 5x = 1 - 7x \] Combine like terms on the left side: \[ 128 - 251x = 1 - 7x \]
Step 2: Move all terms involving \( x \) to one side
Add \( 251x \) to both sides: \[ 128 = 1 - 7x + 251x \] This simplifies to: \[ 128 = 1 + 244x \]
Step 3: Move constant terms to the other side
Subtract \( 1 \) from both sides: \[ 128 - 1 = 244x \] Thus: \[ 127 = 244x \]
Step 4: Solve for \( x \)
Divide both sides by \( 244 \) using the Division Property of Equality: \[ x = \frac{127}{244} \]
Conclusion
The equation \( 32(4 - 8x) + 5x = 1 - 7x \) has one solution, which is: \[ x = \frac{127}{244} \]