To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \) and justify each step using properties of equality, we will follow the algebraic steps carefully.
Step 1: Distribute the \( 32 \) on the left side
Using the distributive property, we can expand \( 32(4 - 8x) \):
\[ 32 \cdot 4 - 32 \cdot 8x = 128 - 256x \]
Now our equation looks like this:
\[ 128 - 256x + 5x = 1 - 7x \]
Step 2: Combine like terms on the left side
Combine the \( x \) terms on the left side:
\[ 128 - 251x = 1 - 7x \]
Step 3: Isolate the \( x \) terms
Next, we can add \( 7x \) to both sides to move all \( x \) terms to the left:
\[ 128 - 251x + 7x = 1 \] \[ 128 - 244x = 1 \]
Step 4: Isolate the constant
Now, subtract \( 128 \) from both sides:
\[ -244x = 1 - 128 \] \[ -244x = -127 \]
Step 5: Solve for \( x \)
Now, divide both sides by \( -244 \) to solve for \( x \):
\[ x = \frac{-127}{-244} = \frac{127}{244} \]
Conclusion:
The original equation \( 32(4 - 8x) + 5x = 1 - 7x \) has a unique solution, which is
\[ x = \frac{127}{244}. \]
Number of Solutions
Since we arrived at one specific value for \( x \), we conclude that there is one solution to the equation.