To solve the equation \( \frac{3}{2} (4 - 8x) + 5x = 1 - 7x \), we will follow the steps systematically, justifying each step using properties of equality.
Step 1: Distribute \(\frac{3}{2}\) on the left side.
We start by distributing \( \frac{3}{2} \) to both terms inside the parentheses: \[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x \] Calculating each term: \[ 6 - 12x + 5x = 1 - 7x \] (Justification: Distributive Property)
Step 2: Combine like terms on the left side.
Now, we combine the \(x\) terms: \[ 6 - 12x + 5x = 6 - 7x \] This simplifies to: \[ 6 - 7x = 1 - 7x \] (Justification: Combining Like Terms)
Step 3: Add \(7x\) to both sides.
To isolate the constant on the left side, we will add \(7x\) to both sides of the equation: \[ 6 - 7x + 7x = 1 - 7x + 7x \] This gives: \[ 6 = 1 \] (Justification: Addition Property of Equality)
Step 4: Analyze the result.
The equation \(6 = 1\) is a contradiction; it is not true for any value of \(x\).
Conclusion
Since we reached a contradiction, there are no solutions to the equation \( \frac{3}{2} (4 - 8x) + 5x = 1 - 7x\).
In summary, we determined that the number of solutions to the equation is zero, justified by applying the properties of equality, and identifying that a false statement results from our manipulation of the equation.