The numbers a and b satisfy a - b = 11. In addition, \(\mathrm{3(a - b) + 4b = 85}\)....

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{a - b = 11}\)
  • Second equation: \(\mathrm{3(a - b) + 4b = 85}\)
• We need to find the value of b

2. INFER the most efficient approach

• Notice that \(\mathrm{(a - b)}\) appears as a complete expression in both equations
• Instead of solving for individual variables a and b, we can treat \(\mathrm{(a - b)}\) as a single unit
• Since we know \(\mathrm{a - b = 11}\), we can substitute this value directly into the second equation

3. SIMPLIFY by substitution

• Replace \(\mathrm{(a - b)}\) with 11 in the second equation:
  \(\mathrm{3(a - b) + 4b = 85}\) becomes \(\mathrm{3(11) + 4b = 85}\)
• Calculate: \(\mathrm{33 + 4b = 85}\)

4. SIMPLIFY to solve for b

• Subtract 33 from both sides: \(\mathrm{4b = 85 - 33 = 52}\)
• Divide by 4: \(\mathrm{b = 52 ÷ 4 = 13}\)

Answer: B (13)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the substitution opportunity and instead try to solve the system by elimination or by solving for a first, then b. They might rewrite the first equation as \(\mathrm{a = b + 11}\), then substitute this into the second equation: \(\mathrm{3((b + 11) - b) + 4b = 85}\), which leads to the same answer but with unnecessary complexity and more opportunities for error.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the substitution method but make arithmetic errors. For example, they might calculate \(\mathrm{85 - 33 = 54}\) instead of 52, leading to \(\mathrm{b = 54 ÷ 4 = 13.5}\), which isn't among the answer choices. This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can recognize when expressions (not just individual variables) can be substituted between equations. The key insight is seeing \(\mathrm{(a - b)}\) as a single unit worth 11, rather than getting caught up in solving for a and b individually.