To solve the equation \( \frac{3}{2} (4 - 8x) + 5x = 1 - 7x \) and justify each step using the properties of equality, we will follow these steps:
Step 1: Distribute on the left-hand side
We start with the equation: \[ \frac{3}{2} (4 - 8x) + 5x = 1 - 7x \] Distributing \( \frac{3}{2} \) gives: \[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x \] Calculating the distribution: \[ 6 - 12x + 5x = 1 - 7x \] Thus, we simplify the left-hand side: \[ 6 - 7x = 1 - 7x \]
Step 2: Add \( 7x \) to both sides
To isolate the constant term on the left, we add \( 7x \) to both sides: \[ 6 - 7x + 7x = 1 - 7x + 7x \] This simplifies to: \[ 6 = 1 \]
Step 3: Analyze the result
The equation \( 6 = 1 \) is a contradiction. Since this statement is not true, it indicates that there is no value of \( x \) that can satisfy the original equation.
Conclusion
Thus, there are no solutions to the equation \( \frac{3}{2} (4 - 8x) + 5x = 1 - 7x \).
Using properties of equality:
- We used the distributive property to expand \( \frac{3}{2}(4 - 8x) \).
- We used the addition property of equality to add \( 7x \) to both sides of the equation.
- The contradiction \( 6 = 1 \) indicates there are no solutions.
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