If 42/x = 7x, what is the value of 7x^2?

1. TRANSLATE the problem information

• Given equation: \(\frac{42}{\mathrm{x}} = 7\mathrm{x}\)
• Find: The value of \(7\mathrm{x}^2\)

2. INFER the most efficient approach

• Key insight: We don't need to find x individually - we can get \(7\mathrm{x}^2\) directly
• Strategy: Multiply both sides by x to eliminate the fraction and isolate \(7\mathrm{x}^2\)

3. SIMPLIFY by multiplying both sides by x

• Starting equation: \(\frac{42}{\mathrm{x}} = 7\mathrm{x}\)
• Multiply both sides by x: \(\mathrm{x} \cdot \frac{42}{\mathrm{x}} = \mathrm{x} \cdot (7\mathrm{x})\)
• Left side: \(\mathrm{x} \cdot \frac{42}{\mathrm{x}} = 42\)
• Right side: \(\mathrm{x} \cdot (7\mathrm{x}) = 7\mathrm{x}^2\)
• Result: \(42 = 7\mathrm{x}^2\)

4. Identify the final answer

• Since \(42 = 7\mathrm{x}^2\), the value of \(7\mathrm{x}^2\) is 42

Answer: C. 42

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Taking the longer route of solving for x first, then computing \(7\mathrm{x}^2\)

Students might think: "I need to find x, then calculate \(7\mathrm{x}^2\)." They solve \(\frac{42}{\mathrm{x}} = 7\mathrm{x}\) by cross-multiplying to get \(42 = 7\mathrm{x}^2\), then \(\mathrm{x}^2 = 6\), so \(\mathrm{x} = \pm\sqrt{6}\). Then they calculate \(7\mathrm{x}^2 = 7(6) = 42\). While this gives the correct answer, it's unnecessarily complex and creates more opportunities for calculation errors.

Second Most Common Error:

Poor SIMPLIFY execution: Making algebraic manipulation errors

Students correctly identify the strategy to multiply by x, but make errors like:

• Forgetting that \(\mathrm{x} \cdot \frac{42}{\mathrm{x}} = 42\)
• Incorrectly computing \(\mathrm{x} \cdot (7\mathrm{x}) = 7\mathrm{x}\) instead of \(7\mathrm{x}^2\)
• Getting confused with signs or operations

This may lead them to select Choice B (7) or cause confusion leading to guessing.

The Bottom Line:

This problem rewards recognizing that the most direct path is often the best path. The beauty lies in seeing that multiplying both sides by x immediately gives you exactly what you're looking for: \(7\mathrm{x}^2\).