Let y be a nonzero real number. The expression 15x/y - 15xy can be written in the form \(\frac{\mathrm{kx}(1 -...

1. TRANSLATE the problem information

• Given: The expression \(\frac{15\mathrm{x}}{\mathrm{y}} - 15\mathrm{xy}\) needs to be written as \(\frac{\mathrm{kx}(1 - \mathrm{y}²)}{\mathrm{y}}\)
• Find: The value of constant k

2. INFER the approach

• The target form \(\frac{\mathrm{kx}(1 - \mathrm{y}²)}{\mathrm{y}}\) suggests I should factor out a common term involving x
• Since both terms contain 15x, I should factor that out first
• Then I'll need to manipulate what remains to match \(\frac{(1 - \mathrm{y}²)}{\mathrm{y}}\)

3. SIMPLIFY by factoring out the common term

• Factor 15x from both terms: \(\frac{15\mathrm{x}}{\mathrm{y}} - 15\mathrm{xy} = 15\mathrm{x}(\frac{1}{\mathrm{y}} - \mathrm{y})\)
• Now I have the x coefficient (15x) separated, and need to work on \((\frac{1}{\mathrm{y}} - \mathrm{y})\)

4. SIMPLIFY the expression in parentheses

• I need to combine \(\frac{1}{\mathrm{y}} - \mathrm{y}\) using a common denominator
• Rewrite y as \(\frac{\mathrm{y}²}{\mathrm{y}}\): \(\frac{1}{\mathrm{y}} - \mathrm{y} = \frac{1}{\mathrm{y}} - \frac{\mathrm{y}²}{\mathrm{y}}\)
• Combine over common denominator y: \(\frac{1 - \mathrm{y}²}{\mathrm{y}}\)

5. INFER the final result

• Substituting back: \(15\mathrm{x}(\frac{1}{\mathrm{y}} - \mathrm{y}) = \frac{15\mathrm{x}(1 - \mathrm{y}²)}{\mathrm{y}}\)
• Comparing with \(\frac{\mathrm{kx}(1 - \mathrm{y}²)}{\mathrm{y}}\), I can see that \(\mathrm{k} = 15\)

Answer: D. 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students struggle with combining the fractions \(\frac{1}{\mathrm{y}} - \mathrm{y}\) correctly. They might try to subtract directly without finding a common denominator, leading to incorrect expressions like \(\frac{1-\mathrm{y}}{\mathrm{y}}\) instead of \(\frac{1-\mathrm{y}²}{\mathrm{y}}\). This algebraic error prevents them from reaching the correct form and may lead them to select Choice A (5) or Choice B (10) if they attempt to work backwards from incorrect intermediate steps.

Second Most Common Error:

Poor INFER reasoning: Students may not recognize that factoring out 15x is the key first step. Instead, they might try to manipulate the original expression \(\frac{15\mathrm{x}}{\mathrm{y}} - 15\mathrm{xy}\) directly into the target form, leading to confusion about how to handle the different terms. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires recognizing the factoring strategy and executing fraction operations carefully. The key insight is that the target form gives you a roadmap - factor out the x-coefficient first, then work on matching the remaining expression.