Which expression is equivalent to 9x + 6x + 2y + 3y?

1. INFER what needs to be done

• Given expression: \(\mathrm{9x + 6x + 2y + 3y}\)
• Strategy: Combine like terms (terms with the same variable)
• Like terms in this expression:
  • x terms: \(\mathrm{9x}\) and \(\mathrm{6x}\)
  • y terms: \(\mathrm{2y}\) and \(\mathrm{3y}\)

2. SIMPLIFY by combining the x terms

• Add the coefficients of the x terms: \(\mathrm{9 + 6 = 15}\)
• Result: \(\mathrm{15x}\)

3. SIMPLIFY by combining the y terms

• Add the coefficients of the y terms: \(\mathrm{2 + 3 = 5}\)
• Result: \(\mathrm{5y}\)

4. Write the final simplified expression

• \(\mathrm{15x + 5y}\)

Answer: D. \(\mathrm{15x + 5y}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when adding coefficients.

For example, they might calculate \(\mathrm{9 + 6 = 12}\) instead of 15, or \(\mathrm{2 + 3 = 8}\) instead of 5. These computational mistakes lead to expressions like \(\mathrm{12x + 8y}\), causing them to select Choice C (\(\mathrm{12x + 8y}\)).

Second Most Common Error:

Missing conceptual knowledge about like terms: Students don't understand that only terms with the same variable can be combined.

They might try to add all the coefficients together (\(\mathrm{9 + 6 + 2 + 3 = 20}\)) or combine unlike terms, leading to confusion and random guessing between the available choices.

The Bottom Line:

This problem tests both conceptual understanding of like terms and careful arithmetic. Students who rush through the addition or don't clearly identify which terms can be combined will select incorrect answers.