\((3\mathrm{x} - 1) - 2(5\mathrm{y} + 4) = 137\)\(2(3\mathrm{x} - 1) + 4(5\mathrm{y} + 4) = 910\)The ordered pair \((\mathrm{x},...

1. TRANSLATE the problem information

• Given system:
  • \((3x - 1) - 2(5y + 4) = 137\)
  • \(2(3x - 1) + 4(5y + 4) = 910\)
• Need to find: \(8(3x - 1)\)

2. INFER the best approach

• These expressions \((3x - 1)\) and \((5y + 4)\) appear multiple times in both equations
• Strategic insight: Use substitution variables to simplify before solving
• Let \(A = 3x - 1\) and \(B = 5y + 4\)

3. SIMPLIFY the system using substitution variables

• The system becomes:
  • \(A - 2B = 137\) ... (1)
  • \(2A + 4B = 910\) ... (2)
• This is much cleaner to work with!

4. SIMPLIFY by solving for A

• From equation (1): \(A = 137 + 2B\)
• Substitute into equation (2): \(2(137 + 2B) + 4B = 910\)
• Distribute: \(274 + 4B + 4B = 910\)
• Combine: \(274 + 8B = 910\)
• Solve: \(8B = 636\), so \(B = 79.5\)

5. SIMPLIFY to find A

• \(A = 137 + 2(79.5) = 137 + 159 = 296\)

6. TRANSLATE back to answer the question

• We need \(8(3x - 1) = 8A = 8(296) = 2368\) (use calculator)

Answer: 2368

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to expand and solve the original system directly instead of using substitution variables.

They distribute \((3x - 1) - 2(5y + 4) = 3x - 1 - 10y - 8 = 3x - 10y - 9 = 137\), leading to \(3x - 10y = 146\). Similarly for the second equation. This creates a messier system that's harder to solve and more prone to arithmetic errors. This approach often leads to confusion and abandoning systematic solution.

Second Most Common Error:

Poor TRANSLATE reasoning: Students solve for A correctly but then forget what the question is asking for.

They find \(A = 296\) and think this is the final answer, not realizing they need to calculate \(8(3x - 1) = 8A\). This leads them to enter 296 as their answer instead of 2368.

The Bottom Line:

This problem rewards recognizing patterns and using substitution strategically. The key insight is seeing that \((3x - 1)\) and \((5y + 4)\) are repeated expressions that can be treated as single variables to simplify the system dramatically.