To solve the equation \(\frac{3}{2}(4 - 8x) + 5x = 1 - 7x\) and determine the number of solutions, we will follow these steps and justify each step using properties of equality.
Step 1: Distribute the \(\frac{3}{2}\) on the left side.
\[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x \] \[ 6 - 12x + 5x = 1 - 7x \]
Justification: This step uses the distributive property of multiplication over addition.
Step 2: Combine like terms on the left side.
\[ 6 - 12x + 5x = 6 - 7x \] \[ 6 - 7x = 1 - 7x \]
Justification: Combine \( -12x + 5x\) to get \( -7x\).
Step 3: Add \(7x\) to both sides to eliminate the \(x\) term on the right side.
\[ 6 - 7x + 7x = 1 - 7x + 7x \] \[ 6 = 1 \]
Justification: This step uses the addition property of equality. Adding the same value (in this case, \(7x\)) to both sides keeps the equation balanced.
Step 4: Analyze the resulting equation.
The equation simplifies to \(6 = 1\), which is a false statement and does not hold true under any circumstance.
Conclusion:
Since we reached a contradiction (a false statement), it implies that there is no solution to the original equation.
Final Answer: The equation \(\frac{3}{2}(4 - 8x) + 5x = 1 - 7x\) has no solutions.
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