Which of the following expressions is equivalent to \((4\mathrm{x} + 5\mathrm{y}) - (2\mathrm{x} - 3\mathrm{y})\)?

1. INFER the meaning of subtracting parentheses

• The expression \((4\mathrm{x} + 5\mathrm{y}) - (2\mathrm{x} - 3\mathrm{y})\) means we're subtracting the entire second binomial
• Key insight: Subtracting a binomial is equivalent to adding its opposite
• This means we distribute the negative sign to every term inside the second parentheses

2. SIMPLIFY by distributing the negative sign

• \((4\mathrm{x} + 5\mathrm{y}) - (2\mathrm{x} - 3\mathrm{y})\) becomes:
• \(4\mathrm{x} + 5\mathrm{y} - 2\mathrm{x} + 3\mathrm{y}\)
• Notice: \(-2\mathrm{x}\) comes from \(-(+2\mathrm{x})\) and \(+3\mathrm{y}\) comes from \(-(-3\mathrm{y})\)

3. SIMPLIFY by combining like terms

• Group the x terms: \(4\mathrm{x} - 2\mathrm{x} = 2\mathrm{x}\)
• Group the y terms: \(5\mathrm{y} + 3\mathrm{y} = 8\mathrm{y}\)
• Final result: \(2\mathrm{x} + 8\mathrm{y}\)

Answer: B (\(2\mathrm{x} + 8\mathrm{y}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that the negative sign must be distributed to both terms in the second parentheses

Students might incorrectly think \((4\mathrm{x} + 5\mathrm{y}) - (2\mathrm{x} - 3\mathrm{y}) = 4\mathrm{x} + 5\mathrm{y} - 2\mathrm{x} - 3\mathrm{y}\), forgetting that subtracting \(-3\mathrm{y}\) actually gives \(+3\mathrm{y}\). This leads to combining terms as: \(4\mathrm{x} - 2\mathrm{x} + 5\mathrm{y} - 3\mathrm{y} = 2\mathrm{x} + 2\mathrm{y}\).

This may lead them to select Choice A (\(2\mathrm{x} + 2\mathrm{y}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when combining like terms

Students correctly distribute the negative sign to get \(4\mathrm{x} + 5\mathrm{y} - 2\mathrm{x} + 3\mathrm{y}\), but then make calculation errors. They might combine incorrectly: \(4\mathrm{x} - 2\mathrm{x} = 6\mathrm{x}\) (adding instead of subtracting) while correctly getting \(5\mathrm{y} + 3\mathrm{y} = 8\mathrm{y}\).

This may lead them to select Choice D (\(6\mathrm{x} + 8\mathrm{y}\)).

The Bottom Line:

The key challenge is understanding that subtracting a binomial means distributing the negative to every term inside the parentheses, which changes the sign of each term. Students who miss this fundamental concept will consistently get subtraction of binomials wrong.