4x + 5y = 1005x + 4y = 62If the system of equations above has solution \((\mathrm{x}, \mathrm{y})\), what is...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{4x + 5y = 100}\)
  • \(\mathrm{5x + 4y = 62}\)
• Need to find: \(\mathrm{x + y}\)

2. INFER the most efficient approach

• Instead of solving for x and y separately, notice that we only need \(\mathrm{x + y}\)
• Key insight: Adding the two equations will give us coefficients that factor nicely
• Look at what happens: \(\mathrm{(4x + 5y) + (5x + 4y) = (4+5)x + (5+4)y = 9x + 9y}\)

3. SIMPLIFY by adding the equations

• Add left sides: \(\mathrm{4x + 5y + 5x + 4y = 9x + 9y}\)
• Add right sides: \(\mathrm{100 + 62 = 162}\)
• Result: \(\mathrm{9x + 9y = 162}\)

4. SIMPLIFY to find x + y

• Factor out 9: \(\mathrm{9(x + y) = 162}\)
• Divide both sides by 9: \(\mathrm{x + y = 162 ÷ 9 = 18}\)

Answer: C. 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the shortcut of adding equations directly to find \(\mathrm{x + y}\). Instead, they attempt to solve the system completely by finding individual values for x and y first.

This approach works but is unnecessarily complicated and time-consuming. Students might use substitution or elimination to find \(\mathrm{x = -4}\) and \(\mathrm{y = 22}\), then calculate \(\mathrm{x + y = 18}\). While this gives the correct answer, it's much more work than needed and increases chances for computational errors.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize they should add the equations but make arithmetic errors in the process.

For example, they might incorrectly add coefficients (getting \(\mathrm{8x + 8y}\) instead of \(\mathrm{9x + 9y}\)) or make errors when adding \(\mathrm{100 + 62}\). This leads to wrong intermediate results and ultimately incorrect final answers that may match other answer choices.

The Bottom Line:

This problem rewards strategic thinking over computational power. The key insight is recognizing that sometimes you don't need to find individual variable values to answer the question being asked.