Question:\(\mathrm{p = (q - r) / 3}\)The given equation relates the positive numbers p, q, and r. Which equation correctly...

1. INFER the solution strategy

• Goal: Isolate \(\mathrm{q}\) on one side of the equation
• Current obstacle: \(\mathrm{q}\) is part of an expression in the numerator of a fraction
• Strategy: Eliminate the fraction first, then isolate \(\mathrm{q}\)

2. SIMPLIFY to eliminate the fraction

• Multiply both sides by 3 to clear the denominator:
  • Left side: \(3 \times \mathrm{p} = 3\mathrm{p}\)
  • Right side: \(3 \times [(\mathrm{q} - \mathrm{r}) / 3] = \mathrm{q} - \mathrm{r}\)
• Result: \(3\mathrm{p} = \mathrm{q} - \mathrm{r}\)

3. SIMPLIFY to isolate the variable

• Add \(\mathrm{r}\) to both sides to get \(\mathrm{q}\) by itself:
  • Left side: \(3\mathrm{p} + \mathrm{r}\)
  • Right side: \(\mathrm{q} - \mathrm{r} + \mathrm{r} = \mathrm{q}\)
• Result: \(3\mathrm{p} + \mathrm{r} = \mathrm{q}\)

4. Rewrite in standard form

• \(\mathrm{q} = 3\mathrm{p} + \mathrm{r}\)

Answer: B. \(\mathrm{q} = 3\mathrm{p} + \mathrm{r}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign error when isolating \(\mathrm{q}\)

Students correctly get to \(3\mathrm{p} = \mathrm{q} - \mathrm{r}\), but then subtract \(\mathrm{r}\) from both sides instead of adding \(\mathrm{r}\). This gives them \(3\mathrm{p} - \mathrm{r} = \mathrm{q}\), leading them to select Choice A (\(\mathrm{q} = 3\mathrm{p} - \mathrm{r}\)).

Second Most Common Error:

Inadequate SIMPLIFY execution: Incomplete algebraic manipulation

Students multiply both sides by 3 correctly to get \(3\mathrm{p} = \mathrm{q} - \mathrm{r}\), but then get confused about the next step and think they need to put everything over 3 again. This leads them to write \(\mathrm{q} = (3\mathrm{p} + \mathrm{r})/3\), causing them to select Choice D.

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires careful attention to signs and completing each step fully before moving to the next.