To solve the equation \( \frac{3}{2}(4-8x) + 5x = 1 - 7x \), we will follow a series of steps, justifying each step using the properties of equality.
Step 1: Distribute \( \frac{3}{2} \) on the left side.
\[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x \]
Calculating each term:
\[ 6 - 12x + 5x = 1 - 7x \]
Step 2: Combine like terms on the left side.
\[ 6 - 7x = 1 - 7x \]
Step 3: Add \( 7x \) to both sides to eliminate the \( -7x \) term on both sides.
\[ 6 = 1 \]
This simplifies to \( 6 = 1 \), which is a contradiction.
Conclusion: The equation \( 6 = 1 \) indicates that there is no solution to the original equation. Therefore, we would say:
- The equation has no solutions.
This process uses the properties of equality like the distributive property and the addition property of equality to manipulate and simplify the equation.