x + 3 = -2y + 5x - 3 = 2y + 7The solution to the given system of equations...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{x + 3 = -2y + 5}\)
  • \(\mathrm{x - 3 = 2y + 7}\)
• What we need: The value of \(\mathrm{2x}\) (not \(\mathrm{x}\) or \(\mathrm{y}\) individually)

2. INFER the most efficient approach

• Notice that the question asks for \(\mathrm{2x}\), not \(\mathrm{x}\) and \(\mathrm{y}\) separately
• Looking at the system, if we add the left sides and right sides of both equations, the y-terms will cancel out
• This gives us \(\mathrm{2x}\) directly without needing to find \(\mathrm{x}\) and \(\mathrm{y}\) first

3. SIMPLIFY by adding the equations

• Add left sides: \(\mathrm{(x + 3) + (x - 3) = x + x + 3 - 3 = 2x}\)
• Add right sides: \(\mathrm{(-2y + 5) + (2y + 7) = -2y + 2y + 5 + 7 = 0 + 12 = 12}\)
• Result: \(\mathrm{2x = 12}\)

Answer: C. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the efficient elimination strategy and instead use substitution to find \(\mathrm{x}\) and \(\mathrm{y}\) individually. After correctly finding \(\mathrm{x = 6}\) and \(\mathrm{y = -2}\), they give the value of \(\mathrm{x}\) instead of \(\mathrm{2x}\).

This may lead them to select Choice B (6).

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly solve the system but misread what the question is asking for. They find \(\mathrm{y = -2}\) and give that as their final answer.

This may lead them to select Choice A (-2).

The Bottom Line:

This problem tests whether students can choose the most efficient solution method and pay careful attention to what quantity is actually being requested.