Which expression is equivalent to 4p + 3q - p + 5q - 2p?7p + 8qp + 8q3p + 8qp...

1. INFER which terms can be combined

• Given expression: \(\mathrm{4p + 3q - p + 5q - 2p}\)
• Like terms are those with the same variable:
  • p terms: \(\mathrm{4p, -p, -2p}\)
  • q terms: \(\mathrm{3q, 5q}\)

2. SIMPLIFY by combining like terms

• Combine p terms: \(\mathrm{4p - p - 2p}\)
  \(\mathrm{= (4 - 1 - 2)p}\)
  \(\mathrm{= 1p}\)
  \(\mathrm{= p}\)
• Combine q terms: \(\mathrm{3q + 5q}\)
  \(\mathrm{= (3 + 5)q}\)
  \(\mathrm{= 8q}\)

3. Write the final simplified expression

• \(\mathrm{4p + 3q - p + 5q - 2p}\)
  \(\mathrm{= p + 8q}\)

Answer: B (\(\mathrm{p + 8q}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill with sign handling: Students often make sign errors when combining the p terms, particularly confusing themselves with \(\mathrm{-p}\) and \(\mathrm{-2p}\). They might calculate \(\mathrm{4p - p - 2p}\) as \(\mathrm{(4 - 1 + 2)p = 5p}\) instead of \(\mathrm{(4 - 1 - 2)p = p}\), not properly tracking that both \(\mathrm{-p}\) and \(\mathrm{-2p}\) are subtractions.

This may lead them to select Choice A (\(\mathrm{7p + 8q}\)) if they also make an error with the q terms, or create confusion that leads to guessing.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students might successfully combine one type of term but miss combining all instances. For example, they might see \(\mathrm{4p - p = 3p}\) but forget about the \(\mathrm{-2p}\) at the end, or they might combine some but not all of the q terms.

This leads to incorrect intermediate results and may cause them to select Choice C (\(\mathrm{3p + 8q}\)) or get confused and guess.

The Bottom Line:

This problem tests careful tracking of signs and systematic organization of like terms - students who rush or don't methodically group terms before combining often make avoidable errors.