Question:A number p satisfies 7 - 2p = -5.What is the value of 21 - 6p?Answer Format: Enter your answer...

1. TRANSLATE the problem information

• Given information:
  • \(7 - 2\mathrm{p} = -5\) (equation involving p)
  • Need to find: \(21 - 6\mathrm{p}\)
• What this tells us: We have one equation with p, and we need to evaluate a different expression involving p.

2. INFER the most efficient approach

• Key insight: Notice that \(21 - 6\mathrm{p}\) looks similar to \(7 - 2\mathrm{p}\), but scaled up
• We can factor: \(21 - 6\mathrm{p} = 3(7 - 2\mathrm{p})\)
• This means we can use the given equation directly without solving for p!

3. SIMPLIFY using the scaling relationship

• Since \(21 - 6\mathrm{p} = 3(7 - 2\mathrm{p})\) and we know \(7 - 2\mathrm{p} = -5\):
• \(21 - 6\mathrm{p} = 3(-5) = -15\)

Alternative approach: Solve for p first

• From \(7 - 2\mathrm{p} = -5\): SIMPLIFY to get \(-2\mathrm{p} = -12\), so \(\mathrm{p} = 6\)
• SIMPLIFY the target expression: \(21 - 6\mathrm{p} = 21 - 6(6) = 21 - 36 = -15\)

Answer: \(-15\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the scaling relationship between the expressions

Students often don't notice that \(21 - 6\mathrm{p} = 3(7 - 2\mathrm{p})\), so they immediately try to solve \(7 - 2\mathrm{p} = -5\) for p. While this works, they may make arithmetic errors during the process. Some students get confused about what to do with the original equation and abandon systematic solution.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Sign errors with negative numbers

Students correctly identify they need to solve for p, but make errors like:

• \(7 - 2\mathrm{p} = -5\) → \(-2\mathrm{p} = -5 + 7 = 2\) (wrong sign)
• Or \(\mathrm{p} = 6\) but then \(21 - 6\mathrm{p} = 21 - 6(6) = 21 + 36 = 57\) (sign error in substitution)

This may lead them to select incorrect positive values or abandon the problem.

The Bottom Line:

This problem rewards pattern recognition - seeing that one expression is a scaled version of another. Students who miss this insight face more arithmetic steps and more opportunities for computational errors.