If u - 3 = 6/(t - 2), what is t in terms of u?

1. TRANSLATE the problem information

• Given equation: \(\mathrm{u - 3 = \frac{6}{t - 2}}\)
• Goal: Express t in terms of u

2. INFER the most effective approach

• The fraction on the right side makes this equation complex to work with
• Strategy: Eliminate the fraction by multiplying both sides by \(\mathrm{(t - 2)}\)
• This will give us a simpler equation to manipulate

3. SIMPLIFY by eliminating the fraction

• Multiply both sides by \(\mathrm{(t - 2)}\):
  \(\mathrm{(t - 2)(u - 3) = 6}\)

4. SIMPLIFY by expanding the left side

• Use distributive property: \(\mathrm{(t - 2)(u - 3) = tu - 3t - 2u + 6}\)
• So: \(\mathrm{tu - 3t - 2u + 6 = 6}\)

5. SIMPLIFY by collecting like terms

• Subtract 6 from both sides: \(\mathrm{tu - 3t - 2u = 0}\)
• Rearrange to group t terms: \(\mathrm{t(u - 3) - 2u = 0}\)

6. SIMPLIFY to isolate t

• Add 2u to both sides: \(\mathrm{t(u - 3) = 2u}\)
• Divide both sides by \(\mathrm{(u - 3)}\): \(\mathrm{t = \frac{2u}{u - 3}}\)

Answer: D. \(\mathrm{t = \frac{2u}{u - 3}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when expanding \(\mathrm{(t - 2)(u - 3)}\) or when rearranging terms. For example, they might get the distribution wrong and write \(\mathrm{tu - 3t + 2u - 6}\) instead of \(\mathrm{tu - 3t - 2u + 6}\), leading to incorrect intermediate steps and ultimately a wrong final answer.

This may lead them to select Choice B \(\mathrm{\left(\frac{2u + 9}{u}\right)}\) or cause confusion and guessing.

Second Most Common Error:

Incomplete INFER reasoning: Students recognize they need to eliminate the fraction but don't complete the full solution process. They might correctly get to \(\mathrm{t - 2 = \frac{6}{u - 3}}\) but then stop or make errors in the final steps of combining fractions.

This may lead them to select Choice C \(\mathrm{\left(\frac{1}{u - 3}\right)}\) after getting confused about how to finish.

The Bottom Line:

This problem requires sustained algebraic manipulation through multiple steps. Success depends on methodical execution of distributive property, careful sign tracking, and persistence through the complete solution process.