3x + 4y = 23 2x + 3y = 16 The solution to the given system of equations is \((\mathrm{x},...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{3x + 4y = 23}\) (equation 1)
  • \(\mathrm{2x + 3y = 16}\) (equation 2)
• Need to find: \(\mathrm{x + y}\) (not individual values)

2. INFER the most efficient approach

• Key insight: The question asks for \(\mathrm{x + y}\), not individual values
• Notice the coefficients: x terms are 3 and 2 (differ by 1), y terms are 4 and 3 (differ by 1)
• Strategic decision: Subtract equation 2 from equation 1 to directly get \(\mathrm{x + y}\)

3. SIMPLIFY using elimination

• Subtract equation 2 from equation 1:

\(\mathrm{(3x + 4y) - (2x + 3y) = 23 - 16}\)

• Distribute the negative sign carefully:

\(\mathrm{3x + 4y - 2x - 3y = 7}\)

• Combine like terms:

\(\mathrm{(3x - 2x) + (4y - 3y) = 7}\)

\(\mathrm{x + y = 7}\)

Answer: C (7)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the elegant shortcut and immediately jump into solving for x and y individually using substitution or elimination. While this approach works, it's time-consuming and creates more opportunities for calculation errors. Students may correctly solve the system but make arithmetic mistakes in the longer process, potentially selecting wrong answers or running out of time.

Second Most Common Error:

Poor SIMPLIFY execution: When distributing the negative sign in \(\mathrm{(3x + 4y) - (2x + 3y)}\), students forget to apply it to both terms, writing \(\mathrm{3x + 4y - 2x + 3y = 7}\) instead of \(\mathrm{3x + 4y - 2x - 3y = 7}\). This leads to \(\mathrm{x + 7y = 7}\) instead of \(\mathrm{x + y = 7}\), causing confusion and potentially leading them to select Choice D (39) if they substitute incorrect values.

The Bottom Line:

This problem rewards strategic thinking over brute force calculation. Students who recognize patterns in the coefficients and consider what the question actually asks for can solve it in seconds, while those who default to standard procedures face unnecessary complexity and error opportunities.