Question: If \(3(\mathrm{n} - 4) = \mathrm{n} + 8\), what is the value of n?

1. INFER the solution strategy

• Given equation: \(\mathrm{3(n - 4) = n + 8}\)
• Strategy: Distribute first, then collect like terms, then isolate the variable

2. SIMPLIFY by applying the distributive property

• Distribute the 3: \(\mathrm{3(n - 4) = 3 \times n - 3 \times 4 = 3n - 12}\)
• New equation: \(\mathrm{3n - 12 = n + 8}\)

3. SIMPLIFY by collecting like terms

• Get all n terms on one side by subtracting n from both sides:
  \(\mathrm{3n - n - 12 = n - n + 8}\)
  \(\mathrm{2n - 12 = 8}\)

4. SIMPLIFY by isolating the variable term

• Add 12 to both sides to eliminate the constant on the left:
  \(\mathrm{2n - 12 + 12 = 8 + 12}\)
  \(\mathrm{2n = 20}\)

5. SIMPLIFY to find the final answer

• Divide both sides by 2:
  \(\mathrm{n = 20/2 = 10}\)

Answer: 10 (also acceptable as n = 10)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make distribution errors in the first step
Instead of correctly getting \(\mathrm{3n - 12}\), they might write \(\mathrm{3n - 4}\) (forgetting to multiply -4 by 3). This leads to the wrong equation \(\mathrm{3n - 4 = n + 8}\), which when solved gives \(\mathrm{2n = 12}\), so \(\mathrm{n = 6}\). This causes them to select an incorrect answer or get confused when verification fails.

Second Most Common Error:

Poor SIMPLIFY technique: Students make sign errors when collecting like terms
After correctly distributing to get \(\mathrm{3n - 12 = n + 8}\), they might incorrectly subtract n from both sides, getting \(\mathrm{3n - n - 12 = 8 - n}\) instead of \(\mathrm{3n - n - 12 = 8}\). This algebraic mistake leads to the wrong equation and incorrect final answer, causing confusion and potentially random guessing.

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires careful attention to distribution rules and consistent application of equality properties through multiple steps.