Which expression is equivalent to \(2(4\mathrm{p} + 5\mathrm{q}) - (\mathrm{p} - 3\mathrm{q})\)?

1. INFER the solution strategy

Looking at \(2(4\mathrm{p} + 5\mathrm{q}) - (\mathrm{p} - 3\mathrm{q})\), I need to:

• Apply distributive property first
• Handle the subtraction of the entire expression \((\mathrm{p} - 3\mathrm{q})\)
• Combine like terms at the end

2. SIMPLIFY by distributing

Distribute the 2 to both terms inside the first parentheses:

• \(2(4\mathrm{p} + 5\mathrm{q}) = 2 \times 4\mathrm{p} + 2 \times 5\mathrm{q} = 8\mathrm{p} + 10\mathrm{q}\)

3. SIMPLIFY the subtraction

Handle \(-(\mathrm{p} - 3\mathrm{q})\) by distributing the negative sign:

• \(-(\mathrm{p} - 3\mathrm{q}) = -\mathrm{p} - (-3\mathrm{q}) = -\mathrm{p} + 3\mathrm{q}\)

4. SIMPLIFY by combining terms

Now I have: \((8\mathrm{p} + 10\mathrm{q}) + (-\mathrm{p} + 3\mathrm{q})\)

• Combine the p terms: \(8\mathrm{p} - \mathrm{p} = 7\mathrm{p}\)
• Combine the q terms: \(10\mathrm{q} + 3\mathrm{q} = 13\mathrm{q}\)
• Result: \(7\mathrm{p} + 13\mathrm{q}\)

Answer: C) \(7\mathrm{p} + 13\mathrm{q}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with sign distribution: Students incorrectly handle \(-(\mathrm{p} - 3\mathrm{q})\) by writing \(-(\mathrm{p} - 3\mathrm{q}) = -\mathrm{p} - 3\mathrm{q}\) instead of \(-\mathrm{p} + 3\mathrm{q}\). They forget that subtracting a negative term creates a positive term.

Following this error: \(8\mathrm{p} + 10\mathrm{q} - \mathrm{p} - 3\mathrm{q} = 7\mathrm{p} + 7\mathrm{q}\)

This leads them to select Choice B (\(7\mathrm{p} + 7\mathrm{q}\))

Second Most Common Error:

Poor INFER reasoning about order of operations: Students might try to combine terms before fully distributing, or handle the subtraction before distributing the 2, leading to calculation confusion and mixed-up coefficients.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

The key challenge is correctly handling the subtraction of an entire expression. Students must remember that the negative sign affects every term inside the parentheses, changing signs appropriately.