The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the value of x + y?4x =...

1. TRANSLATE the problem requirements

• Given system:
  • \(\mathrm{4x = 20}\)
  • \(\mathrm{-3x + y = -7}\)
• Find: The value of \(\mathrm{x + y}\) (not individual variables)

2. INFER the most efficient strategy

• Key insight: Since we only need \(\mathrm{x + y}\), we can add the equations directly
• This eliminates the need to solve for x and y individually
• Adding equations: Left side + Left side = Right side + Right side

3. SIMPLIFY by combining the equations

• Add the equations:
  \(\mathrm{4x + (-3x + y) = 20 + (-7)}\)
• Combine like terms on the left:
  \(\mathrm{4x - 3x + y = x + y}\)
• Simplify the right side:
  \(\mathrm{20 + (-7) = 13}\)
• Result: \(\mathrm{x + y = 13}\)

Answer: C. 13

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the direct addition strategy and instead solve for individual variables using substitution or elimination, then add them together.

While this approach works, it's unnecessarily complicated and increases chances for arithmetic errors during the multi-step process. Students might solve correctly but make calculation mistakes along the way, leading to wrong final answers.

Second Most Common Error:

Poor SIMPLIFY execution: Students attempt the direct addition method but make algebraic errors when combining like terms.

For example, they might incorrectly combine \(\mathrm{4x + (-3x)}\) as \(7\mathrm{x}\) instead of x, or make sign errors when adding \(\mathrm{-3x + y}\). This leads to equations like \(\mathrm{7x + y = 13}\) or other incorrect expressions, causing them to select wrong answer choices.

The Bottom Line:

This problem rewards strategic thinking - recognizing when a direct approach (adding equations) is more efficient than the standard solve-then-substitute method. The key insight is that the question asks for a sum, not individual values.