Question:\(16 - (\mathrm{k} + 5) = 3\)What value of k is the solution to the given equation?-{8}81418

1. TRANSLATE the problem information

• Given equation: \(16 - (\mathrm{k} + 5) = 3\)
• Find: the value of k

2. INFER the solution approach

• The parentheses with a negative sign in front requires distributing first
• Then we'll need to combine like terms and isolate k

3. SIMPLIFY by distributing the negative sign

• \(16 - (\mathrm{k} + 5)\) becomes \(16 - \mathrm{k} - 5\)
• The negative distributes to both k and 5: \(-(\mathrm{k} + 5) = -\mathrm{k} - 5\)

4. SIMPLIFY by combining like terms

• \(16 - \mathrm{k} - 5 = 3\)
• Combine the constants: \(16 - 5 = 11\)
• Result: \(11 - \mathrm{k} = 3\)

5. SIMPLIFY to isolate the variable

• Subtract 11 from both sides: \(-\mathrm{k} = 3 - 11\)
• Simplify the right side: \(-\mathrm{k} = -8\)
• Multiply both sides by -1: \(\mathrm{k} = 8\)

Answer: B. 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly get to \(-\mathrm{k} = -8\) but forget the final step of multiplying by -1 to solve for k.

They stop at \(-\mathrm{k} = -8\) and think this means \(\mathrm{k} = -8\), not recognizing that they need one more step to isolate k completely.

This leads them to select Choice A (-8).

Second Most Common Error:

Poor INFER reasoning about sign distribution: Students don't properly distribute the negative sign and instead treat \(16 - (\mathrm{k} + 5)\) as \(16 - \mathrm{k} + 5\).

This gives them \(16 - \mathrm{k} + 5 = 3\), which simplifies to \(21 - \mathrm{k} = 3\), leading to \(\mathrm{k} = 18\).

This may lead them to select Choice D (18).

The Bottom Line:

This problem tests careful algebraic manipulation with negative signs. Students must systematically work through distribution, combining like terms, and complete variable isolation without getting tripped up by sign changes.