If x and y are numbers such that 3x + 2y = 42 and 2x + 3y = 38, what...

1. TRANSLATE the problem information

• Given system:
  • Equation 1: \(3\mathrm{x} + 2\mathrm{y} = 42\)
  • Equation 2: \(2\mathrm{x} + 3\mathrm{y} = 38\)
• Target: Find the value of \(\mathrm{x} - \mathrm{y}\) (not individual values of x and y)

2. INFER the most efficient approach

• Notice the target expression is \(\mathrm{x} - \mathrm{y}\)
• Look at the coefficients: equation 1 has \(3\mathrm{x}\) and \(2\mathrm{y}\), equation 2 has \(2\mathrm{x}\) and \(3\mathrm{y}\)
• Key insight: If I subtract equation 2 from equation 1, the coefficients will give me exactly \(\mathrm{x} - \mathrm{y}\)!

3. SIMPLIFY by subtracting equations

• Set up the subtraction: \((3\mathrm{x} + 2\mathrm{y}) - (2\mathrm{x} + 3\mathrm{y}) = 42 - 38\)

• Distribute the negative sign carefully:
  \(3\mathrm{x} + 2\mathrm{y} - 2\mathrm{x} - 3\mathrm{y} = 4\)

• Combine like terms:
  • x terms: \(3\mathrm{x} - 2\mathrm{x} = \mathrm{x}\)
  • y terms: \(2\mathrm{y} - 3\mathrm{y} = -\mathrm{y}\)
  • Result: \(\mathrm{x} - \mathrm{y} = 4\)

Answer: C) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the direct path and instead try to solve for x and y individually using elimination or substitution.

While this approach works, it's much more time-consuming and creates more opportunities for arithmetic errors. Students who use elimination might multiply equations incorrectly or make mistakes with the larger numbers involved (like working with \(6\mathrm{x} + 4\mathrm{y} = 84\) and \(6\mathrm{x} + 9\mathrm{y} = 114\)).

This approach still leads to the correct answer but wastes valuable test time.

Second Most Common Error:

Poor SIMPLIFY execution: Students attempt the direct subtraction method but make sign errors when distributing the negative.

Common mistake: Writing \((3\mathrm{x} + 2\mathrm{y}) - (2\mathrm{x} + 3\mathrm{y})\) as \(3\mathrm{x} + 2\mathrm{y} - 2\mathrm{x} + 3\mathrm{y} = 4\), forgetting to distribute the negative to the \(3\mathrm{y}\) term. This gives \(\mathrm{x} + 5\mathrm{y} = 4\) instead of \(\mathrm{x} - \mathrm{y} = 4\), leading to confusion and potentially guessing among the answer choices.

The Bottom Line:

Success on this problem requires recognizing that sometimes the most elegant solution comes from strategic manipulation of the given equations rather than following the standard "solve for each variable" approach.