To solve the equation \( \frac{3}{4}(4 - 8x) + 5x = 1 - 7x \), we will justify our steps using properties of equality and determine the number of solutions.
Step 1: Distribute the \( \frac{3}{4} \)
Using the distributive property of equality, we first distribute \( \frac{3}{4} \) across the expression \( (4 - 8x) \).
\[ \frac{3}{4} \cdot 4 - \frac{3}{4} \cdot 8x + 5x = 1 - 7x \]
Calculating \( \frac{3}{4} \cdot 4 \) gives us 3, and calculating \( \frac{3}{4} \cdot 8x \) gives us \( 6x \).
\[ 3 - 6x + 5x = 1 - 7x \]
Step 2: Combine like terms on the left side
Now combine the like terms \( -6x + 5x \):
\[ 3 - x = 1 - 7x \]
Step 3: Isolate the variable \( x \)
Next, we want to isolate \( x \). To do this, we can add \( 7x \) to both sides (using the addition property of equality):
\[ 3 - x + 7x = 1 - 7x + 7x \]
This simplifies to:
\[ 3 + 6x = 1 \]
Step 4: Isolate the constant term
Next, subtract 3 from both sides (again applying the addition property of equality):
\[ 6x = 1 - 3 \]
This simplifies to:
\[ 6x = -2 \]
Step 5: Solve for \( x \)
Now, divide both sides by 6 (using the division property of equality):
\[ x = \frac{-2}{6} \]
This simplifies to:
\[ x = \frac{-1}{3} \]
Conclusion: Determine the number of solutions
The equation \( 3/4(4 - 8x) + 5x = 1 - 7x \) has been solved and we found one solution:
\[ x = -\frac{1}{3} \]
Since we arrived at one specific value for \( x \) through valid algebraic manipulation, we conclude that there is exactly one solution to the equation.