Question:Let p and q be real numbers. If (t - 4)/3 = 3p + q, which equation correctly expresses t...

1. TRANSLATE the problem setup

• Given equation: \(\frac{\mathrm{t - 4}}{3} = 3\mathrm{p} + \mathrm{q}\)
• Goal: Express \(\mathrm{t}\) in terms of \(\mathrm{p}\) and \(\mathrm{q}\)

2. SIMPLIFY by eliminating the fraction

• Multiply both sides by 3 to clear the denominator:
  \(3 \times \frac{\mathrm{t - 4}}{3} = 3 \times (3\mathrm{p} + \mathrm{q})\)
• Left side: \(\mathrm{t - 4}\)
• Right side: \(9\mathrm{p} + 3\mathrm{q}\) (using distributive property)
• Result: \(\mathrm{t - 4} = 9\mathrm{p} + 3\mathrm{q}\)

3. SIMPLIFY to isolate t completely

• Add 4 to both sides:
  \(\mathrm{t - 4 + 4} = 9\mathrm{p} + 3\mathrm{q} + 4\)
• Final result: \(\mathrm{t} = 9\mathrm{p} + 3\mathrm{q} + 4\)

Answer: B (\(\mathrm{t} = 9\mathrm{p} + 3\mathrm{q} + 4\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make distribution errors when multiplying \(3(3\mathrm{p} + \mathrm{q})\), getting \(3\mathrm{p} + 3\mathrm{q}\) instead of \(9\mathrm{p} + 3\mathrm{q}\).

This incorrect calculation gives \(\mathrm{t} = 3\mathrm{p} + 3\mathrm{q} + 4\), but since \(3\mathrm{p} + 3\mathrm{q} \neq 3\mathrm{p} + \mathrm{q}\), students may try to "fix" this by removing the +4, leading them to select Choice D (\(\mathrm{t} = 3\mathrm{p} + \mathrm{q} + 4\)).

Second Most Common Error:

Incomplete SIMPLIFY execution: Students correctly find \(\mathrm{t - 4} = 9\mathrm{p} + 3\mathrm{q}\) but forget the final step of adding 4 to both sides.

They incorrectly think this means \(\mathrm{t} = 9\mathrm{p} + 3\mathrm{q} - 4\), leading them to select Choice A (\(\mathrm{t} = 9\mathrm{p} + 3\mathrm{q} - 4\)).

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires careful attention to distributing coefficients and completing all steps to fully isolate the target variable.