What is the solution (x, y) to the given system of equations?x + y = 185y = x

1. INFER the solution strategy

• Given information:
  • First equation: \(\mathrm{x + y = 18}\)
  • Second equation: \(\mathrm{5y = x}\)
• Key insight: The second equation already expresses x in terms of y, making substitution the most efficient approach.

2. SIMPLIFY by substitution

• Since \(\mathrm{x = 5y}\), substitute this into the first equation:
  • \(\mathrm{5y + y = 18}\)
  • \(\mathrm{6y = 18}\)
  • \(\mathrm{y = 3}\)

3. SIMPLIFY to find the second variable

• Substitute \(\mathrm{y = 3}\) back into \(\mathrm{x = 5y}\):
  • \(\mathrm{x = 5(3) = 15}\)

4. Verify the solution

• Check both original equations:
  • \(\mathrm{x + y = 15 + 3 = 18}\) ✓
  • \(\mathrm{5y = 5(3) = 15 = x}\) ✓

Answer: A. (15, 3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{x = 5y}\) is already solved for x, so they try to solve the system using elimination or get confused about which variable to solve for first.

They might attempt to rearrange the second equation unnecessarily or try to eliminate variables when substitution is more direct. This leads to confusion and potential arithmetic errors, causing them to guess among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the substitution strategy but make arithmetic errors when combining like terms or solving for variables.

For example, they might incorrectly solve \(\mathrm{6y = 18}\) as \(\mathrm{y = 2}\), leading to \(\mathrm{x = 5(2) = 10}\). This doesn't match any answer choice exactly, so they may select the closest option or guess.

The Bottom Line:

This problem tests whether students can recognize when one equation is already solved for a variable and execute substitution accurately. The key insight is seeing that \(\mathrm{5y = x}\) immediately gives you what to substitute, making this much simpler than it might initially appear.