Question:If 72/(k-2) = -9, what is the value of k?

1. TRANSLATE the problem setup

• Given equation: \(\frac{72}{\mathrm{k}-2} = -9\)
• Goal: Find the value of k
• Strategy: Eliminate the fraction to create a linear equation

2. SIMPLIFY by eliminating the fraction

• Multiply both sides by \((\mathrm{k}-2)\) to clear the denominator:
  \(\frac{72}{\mathrm{k}-2} \cdot (\mathrm{k}-2) = -9 \cdot (\mathrm{k}-2)\)
• This gives us: \(72 = -9(\mathrm{k}-2)\)

3. SIMPLIFY using the distributive property

• Distribute the -9: \(72 = -9\mathrm{k} + 18\)
• Be careful with the signs: \(-9 \times \mathrm{k} = -9\mathrm{k}\) and \(-9 \times (-2) = +18\)

4. SIMPLIFY to isolate the variable term

• Subtract 18 from both sides: \(72 - 18 = -9\mathrm{k}\)
• This gives us: \(54 = -9\mathrm{k}\)

5. SIMPLIFY to find k

• Divide both sides by -9: \(\mathrm{k} = \frac{54}{-9} = -6\)

Answer: k = -6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with negative signs: Students correctly eliminate the fraction but make errors when distributing the negative number.

They might write: \(72 = -9\mathrm{k} - 18\) (forgetting that \(-9 \times (-2) = +18\))

This leads to: \(72 + 18 = -9\mathrm{k}\), so \(90 = -9\mathrm{k}\), giving \(\mathrm{k} = -10\)

This causes confusion as they get an answer that doesn't verify when substituted back.

Second Most Common Error:

Poor TRANSLATE reasoning about fraction elimination: Students attempt to cross-multiply incorrectly or try to subtract terms inappropriately.

They might try: \(72 + 9 = \mathrm{k} - 2\) or similar incorrect manipulations.

This leads to confusion and guessing on the final answer.

The Bottom Line:

Success on this problem requires careful attention to signs throughout the algebraic manipulation, especially when distributing negative coefficients and performing division with negative numbers.