\(3(\mathrm{a} + 3) + 5(\mathrm{b} - 1) = 234\)\(3(\mathrm{a} + 3) - 5(\mathrm{b} - 1) = 54\)The solution to the...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{3(a + 3) + 5(b - 1) = 234}\)
  • \(\mathrm{3(a + 3) - 5(b - 1) = 54}\)
• Need to find: \(\mathrm{8(a + 3)}\)

2. INFER the most efficient approach

• Notice both equations contain the expressions \(\mathrm{(a + 3)}\) and \(\mathrm{(b - 1)}\)
• Strategic insight: Let \(\mathrm{u = a + 3}\) and \(\mathrm{v = b - 1}\) to simplify the system
• This transforms our system into: \(\mathrm{3u + 5v = 234}\) and \(\mathrm{3u - 5v = 54}\)
• Since we need \(\mathrm{8(a + 3) = 8u}\), we only need to find u

3. INFER the elimination strategy

• Looking at our simplified system:
  • \(\mathrm{3u + 5v = 234}\) ... (1)
  • \(\mathrm{3u - 5v = 54}\) ... (2)
• The coefficients of v are opposites (+5v and -5v)
• Adding these equations will eliminate v completely

4. SIMPLIFY by adding the equations

• Add equation (1) + equation (2):
  \(\mathrm{(3u + 5v) + (3u - 5v) = 234 + 54}\)
• The v terms cancel: \(\mathrm{3u + 3u = 288}\)
• Combine like terms: \(\mathrm{6u = 288}\)
• Solve for u: \(\mathrm{u = 48}\)

5. SIMPLIFY to find the final answer

• Since \(\mathrm{u = a + 3 = 48}\)
• Therefore: \(\mathrm{8(a + 3) = 8u = 8(48) = 384}\)

Answer: E) 384

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to solve for individual variables a and b first instead of recognizing the substitution opportunity.

They might expand \(\mathrm{3(a + 3) + 5(b - 1) = 234}\) to get \(\mathrm{3a + 9 + 5b - 5 = 234}\), leading to \(\mathrm{3a + 5b = 230}\). Then they get stuck trying to solve the more complex system for a and b individually, when the problem only asks for \(\mathrm{8(a + 3)}\). This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the substitution strategy but make arithmetic errors.

They might incorrectly add \(\mathrm{234 + 54 = 288}\) (getting something like 298 instead), or make errors in \(\mathrm{8 \times 48}\). For instance, if they miscalculate \(\mathrm{u = 24}\) instead of \(\mathrm{u = 48}\), they would get \(\mathrm{8(24) = 192}\), which isn't among the choices, leading them to second-guess their approach and potentially select Choice B (144) as the closest value.

The Bottom Line:

This problem rewards students who can recognize that strategic substitution eliminates unnecessary work. The key insight is seeing that the question asks for \(\mathrm{8(a + 3)}\), not for a itself, so finding \(\mathrm{u = a + 3}\) directly is the most efficient path.