\(\mathrm{(x - 2) - 4(y + 7) = 117}\)\(\mathrm{(x - 2) + 4(y + 7) = 442}\)The solution to the...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{(x - 2) - 4(y + 7) = 117}\)
  • \(\mathrm{(x - 2) + 4(y + 7) = 442}\)
• Find: \(\mathrm{6(x - 2)}\), not x or y individually

2. INFER the most efficient approach

• Notice the structure: both equations contain \(\mathrm{(x - 2)}\) and \(\pm 4(y + 7)\)
• Key insight: Adding the equations will eliminate the \(\mathrm{4(y + 7)}\) terms completely
• This gives us \(\mathrm{2(x - 2)}\) directly, making it easy to find \(\mathrm{6(x - 2)}\)

3. SIMPLIFY by adding the equations

• Add left sides: \(\mathrm{(x - 2) - 4(y + 7) + (x - 2) + 4(y + 7)}\)
• The \(\pm 4(y + 7)\) terms cancel: \(\mathrm{2(x - 2)}\)
• Add right sides: \(117 + 442 = 559\)
• Result: \(\mathrm{2(x - 2) = 559}\)

4. SIMPLIFY to find the target expression

• From \(\mathrm{2(x - 2) = 559}\), we get \(\mathrm{(x - 2) = 279.5}\)
• Multiply by 6: \(\mathrm{6(x - 2) = 6(279.5) = 1677}\)

Answer: 1677

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students often miss that the problem asks for \(\mathrm{6(x - 2)}\), not x itself. They solve for \(\mathrm{x = 281.5}\), then submit 281.5 as their answer instead of calculating \(\mathrm{6(x - 2) = 6(279.5) = 1677}\).

Second Most Common Error:

Weak INFER skill: Students don't recognize the elimination opportunity and instead try substitution or other complex approaches. They might solve the first equation for x in terms of y, substitute into the second equation, solve for y, then back-substitute - a much longer path that creates more opportunities for arithmetic errors and often leads to abandoning the systematic solution and guessing.

The Bottom Line:

This problem rewards students who can see the structural patterns in systems of equations. The key is recognizing that sometimes the most direct path isn't solving for individual variables, but working with the expressions the problem actually asks for.