15 - k = m + nThe given equation relates the variables k, m, and n. Which equation correctly expresses...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{15 - k = m + n}\)
• Goal: Express k in terms of m and n (isolate k)

2. INFER the solving strategy

• To isolate k, I need to get k by itself on one side of the equation
• I can add k to both sides, then subtract the other terms from both sides

3. SIMPLIFY by adding k to both sides

\(\mathrm{15 - k + k = m + n + k}\)

\(\mathrm{15 = m + n + k}\)

4. SIMPLIFY by subtracting m and n from both sides

\(\mathrm{15 - m - n = m + n + k - m - n}\)

\(\mathrm{15 - m - n = k}\)

Answer: \(\mathrm{k = 15 - m - n}\) (Choice B)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when moving terms across the equals sign. They might incorrectly think that when moving m + n to the left side, they should add them instead of subtract them.

This incorrect reasoning: "\(\mathrm{15 = m + n + k}\), so \(\mathrm{k = 15 + m + n}\)" leads them to select Choice A (\(\mathrm{k = 15 + m + n}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly get to \(\mathrm{15 = m + n + k}\) but then make errors with the order of operations or sign handling when isolating k, particularly getting confused about which terms are positive or negative.

This may lead them to select Choice C (\(\mathrm{k = -15 + m + n}\)) or Choice D (\(\mathrm{k = -15 - m - n}\)).

The Bottom Line:

This problem tests careful algebraic manipulation. The key is systematically moving terms while keeping track of signs - a skill that requires both methodical thinking and attention to detail.