To solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) and determine the number of solutions, we will follow a series of steps while justifying each step using the properties of equality.
Step 1: Distribute the \( \frac{3}{2} \) on the left side
Starting with the equation: \[ \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \]
Using the distributive property: \[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x \] Calculating the multiplication: \[ 6 - 12x + 5x = 1 - 7x \]
Step 2: Combine like terms
Now we simplify the left side: \[ 6 - 12x + 5x = 6 - 7x \] So the equation now looks like: \[ 6 - 7x = 1 - 7x \]
Step 3: Isolate the variable
We notice that both sides contain \(-7x\). Let's add \(7x\) to both sides to eliminate it: \[ 6 - 7x + 7x = 1 - 7x + 7x \] This simplifies to: \[ 6 = 1 \]
Step 4: Analyze the resulting statement
The equation \(6 = 1\) is a false statement. This indicates that there are no values of \(x\) that can satisfy the original equation.
Conclusion
Hence, the number of solutions to the equation \( \frac{3}{2}(4-8x) + 5x = 1 - 7x \) is: \[ \text{No solutions} \]
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