Question:Which expression is equivalent to \((7\mathrm{a} + 4\mathrm{b}) - (3\mathrm{a} - 2\mathrm{b})\)?

1. TRANSLATE the problem information

• Given expression: \((7\mathrm{a} + 4\mathrm{b}) - (3\mathrm{a} - 2\mathrm{b})\)
• Need to find: An equivalent simplified expression

2. INFER the solution strategy

• When subtracting expressions in parentheses, we must distribute the negative sign to every term in the second expression
• After distribution, we'll combine like terms to simplify

3. SIMPLIFY by distributing the negative sign

• \((7\mathrm{a} + 4\mathrm{b}) - (3\mathrm{a} - 2\mathrm{b})\)
• The negative sign in front of \((3\mathrm{a} - 2\mathrm{b})\) affects both terms inside:
• \(= 7\mathrm{a} + 4\mathrm{b} - 3\mathrm{a} - (-2\mathrm{b})\)
• \(= 7\mathrm{a} + 4\mathrm{b} - 3\mathrm{a} + 2\mathrm{b}\)

4. SIMPLIFY by combining like terms

• Group the 'a' terms: \(7\mathrm{a} - 3\mathrm{a} = 4\mathrm{a}\)
• Group the 'b' terms: \(4\mathrm{b} + 2\mathrm{b} = 6\mathrm{b}\)
• Final result: \(4\mathrm{a} + 6\mathrm{b}\)

Answer: B. \(4\mathrm{a} + 6\mathrm{b}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Not properly distributing the negative sign to the second term in the parentheses.

Students often treat \(-(3\mathrm{a} - 2\mathrm{b})\) as \(-3\mathrm{a} - 2\mathrm{b}\) instead of \(-3\mathrm{a} + 2\mathrm{b}\). They forget that the negative sign changes \(-2\mathrm{b}\) to \(+2\mathrm{b}\). This gives them \(7\mathrm{a} + 4\mathrm{b} - 3\mathrm{a} - 2\mathrm{b} = 4\mathrm{a} + 2\mathrm{b}\).

This may lead them to select Choice A (\(4\mathrm{a} + 2\mathrm{b}\)).

Second Most Common Error:

Conceptual confusion about like terms: Adding coefficients instead of following proper algebraic operations.

Some students might incorrectly add \(7\mathrm{a} + 3\mathrm{a} = 10\mathrm{a}\) instead of computing \(7\mathrm{a} - 3\mathrm{a} = 4\mathrm{a}\), perhaps because they see both 'a' terms and think "combine" means "add." This leads to \(10\mathrm{a} + 6\mathrm{b}\).

This may lead them to select Choice D (\(10\mathrm{a} + 6\mathrm{b}\)).

The Bottom Line:

The key challenge is correctly handling negative signs when they appear in front of parentheses - this requires careful attention to how the distributive property works with subtraction.