Let's solve the equation step by step and justify each step using the properties of equality.
The given equation is:
\[ \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \]
Step 1: Distribute \(\frac{3}{2}\) on the left side.
Using the distributive property, we can calculate:
\[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x \] \[ = 6 - 12x \]
So, we can rewrite the equation as:
\[ 6 - 12x + 5x = 1 - 7x \]
Step 2: Combine like terms on the left side.
Combine \(-12x\) and \(5x\):
\[ 6 - 7x = 1 - 7x \]
Step 3: Add \(7x\) to both sides.
This step is justified by the addition property of equality which states that if you add the same value to both sides of an equation, the equality is maintained:
\[ 6 - 7x + 7x = 1 - 7x + 7x \] \[ 6 = 1 \]
Step 4: Analyze the resulting statement.
The equation \(6 = 1\) is a contradiction, which means that there are no values of \(x\) that will satisfy the equation.
Conclusion
Since we reached a contradiction, we conclude that the original equation has no solutions.
Thus, the number of solutions to the equation is zero.
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