For what value of w does \(\mathrm{w - 10 = 2(w + 5)}\) ?

1. TRANSLATE the problem information

• Given: \(\mathrm{w - 10 = 2(w + 5)}\)
• Find: The value of w that makes this equation true

2. INFER the solution approach

• I need to isolate w by eliminating the parentheses first, then moving all w terms to one side
• The distributive property will help me expand the right side

3. SIMPLIFY by applying the distributive property

• Distribute 2 to both terms inside parentheses: \(\mathrm{2(w + 5) = 2w + 10}\)
• Equation becomes: \(\mathrm{w - 10 = 2w + 10}\)

4. SIMPLIFY by collecting variable terms

• Subtract w from both sides: \(\mathrm{-10 = w + 10}\)
• This moves all w terms to one side

5. SIMPLIFY to isolate the variable

• Subtract 10 from both sides: \(\mathrm{-20 = w}\)
• Therefore: \(\mathrm{w = -20}\)

Answer: D. -20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Incomplete distribution of the 2

Students might only distribute the 2 to the first term: \(\mathrm{w - 10 = 2w + 5}\) (instead of \(\mathrm{2w + 10}\)). Then solving:

\(\mathrm{w - 2w = 5 + 10}\)

\(\mathrm{-w = 15}\)

giving \(\mathrm{w = -15}\).

This may lead them to select Choice C (-15).

Second Most Common Error:

Poor SIMPLIFY execution: Sign errors during algebraic manipulation

Students might incorrectly handle signs when moving terms between sides, such as treating subtraction as addition or making errors with negative coefficients. These calculation mistakes typically result from rushing through the algebra.

This may lead them to select Choice A (5) or Choice B (0).

The Bottom Line:

This problem requires careful attention to both the distributive property and sign management. Students who work systematically through each algebraic step while double-checking their signs will succeed, while those who rush or skip steps in the algebraic manipulation will likely make errors that lead directly to the wrong answer choices.