If 2x + 4y = 16 and 3x - 6y = 6, what is the value of x - 2y?

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{2x + 4y = 16}\)
  • Second equation: \(\mathrm{3x - 6y = 6}\)
• Find: The value of \(\mathrm{x - 2y}\)

2. INFER the most efficient approach

• Rather than solving for x and y individually (which would require substitution or elimination), notice that both equations can be simplified
• This might reveal the target expression directly

3. SIMPLIFY each equation by factoring out common terms

• First equation: \(\mathrm{2x + 4y = 16}\)
  • Every term is divisible by 2
  • Divide both sides by 2: \(\mathrm{x + 2y = 8}\)

• Second equation: \(\mathrm{3x - 6y = 6}\)
  • Every term is divisible by 3
  • Divide both sides by 3: \(\mathrm{x - 2y = 2}\)

4. INFER the final answer

• The simplified second equation is exactly what we need: \(\mathrm{x - 2y = 2}\)
• No further solving required!

Answer: B) 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the simplification shortcut and instead attempt to solve the full system using substitution or elimination.

They might solve \(\mathrm{x + 2y = 8}\) and \(\mathrm{x - 2y = 2}\) simultaneously, finding \(\mathrm{x = 5}\) and \(\mathrm{y = 1.5}\), then calculating \(\mathrm{x - 2y = 5 - 2(1.5) = 2}\). While this gives the correct answer, it's unnecessarily complicated and increases chances for arithmetic errors.

This approach works but takes much longer and may lead to computational mistakes along the way.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when dividing the original equations.

For example, incorrectly simplifying \(\mathrm{3x - 6y = 6}\) as \(\mathrm{x - 3y = 2}\) instead of \(\mathrm{x - 2y = 2}\), or simplifying \(\mathrm{2x + 4y = 16}\) as \(\mathrm{x + 2y = 4}\) instead of \(\mathrm{x + 2y = 8}\).

This leads them to select Choice A) -2 or Choice C) 6 based on the incorrect simplified equations.

The Bottom Line:

The key insight is recognizing when simplification can directly reveal what you're looking for, rather than automatically defaulting to systematic solving methods. Sometimes the most elegant solution is also the shortest one.