5x + 2y = 193x + 4y = 17The solution to the given system of equations is \((\mathrm{x}, \mathrm{y})\). What...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{5x + 2y = 19}\) (equation 1)
  • \(\mathrm{3x + 4y = 17}\) (equation 2)
  • Need to find: \(\mathrm{x - y}\)

2. INFER the best approach

• Key insight: The question asks specifically for \(\mathrm{x - y}\), not the individual values
• Two strategic options:
  • Option A: Solve for x and y individually, then calculate \(\mathrm{x - y}\)
  • Option B: Manipulate the equations directly to get \(\mathrm{x - y}\)

3. SIMPLIFY using direct manipulation (most efficient)

• Subtract equation 2 from equation 1:

\(\mathrm{(5x + 2y) - (3x + 4y) = 19 - 17}\)

• SIMPLIFY the left side:

\(\mathrm{5x - 3x + 2y - 4y = 2x - 2y}\)

• This gives us:

\(\mathrm{2x - 2y = 2}\)

• SIMPLIFY by dividing everything by 2:

\(\mathrm{x - y = 1}\)

Answer: A (1)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the direct approach and instead solve for x and y individually, increasing chances for arithmetic errors.

They might correctly find \(\mathrm{x = 3}\) and \(\mathrm{y = 2}\), but then make a simple subtraction error when computing \(\mathrm{3 - 2}\), or they might solve incorrectly for one of the variables due to arithmetic mistakes in the longer elimination process. This often leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students attempt the direct method but make sign errors when subtracting equations.

When subtracting \(\mathrm{(3x + 4y = 17)}\) from \(\mathrm{(5x + 2y = 19)}\), they might incorrectly get \(\mathrm{2x + 6y = 2}\) instead of \(\mathrm{2x - 2y = 2}\), leading them to an impossible situation or incorrect final answer. This may lead them to select Choice B (2) or causes them to abandon the systematic approach and guess.

The Bottom Line:

This problem rewards students who can recognize that the question asks for a specific combination \(\mathrm{(x - y)}\) that can be found directly through equation manipulation, rather than requiring individual variable solutions. The key is strategic thinking about what the problem actually needs.