Question:x = y^2x - 2y = 3If (x, y) is a solution of the system of equations above and y...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{x = y^2}\) (first equation)
  • \(\mathrm{x - 2y = 3}\) (second equation)
  • \(\mathrm{y \gt 0}\) (constraint)
• Find: \(\mathrm{x + y}\)

2. INFER the solution approach

• Since x is already isolated in the first equation, substitution is the most direct method
• We can substitute \(\mathrm{x = y^2}\) into the second equation to get everything in terms of y

3. SIMPLIFY by substitution and algebraic manipulation

• Substitute \(\mathrm{x = y^2}\) into \(\mathrm{x - 2y = 3}\):
  \(\mathrm{y^2 - 2y = 3}\)
• Rearrange to standard form:
  \(\mathrm{y^2 - 2y - 3 = 0}\)
• Factor the quadratic:
  \(\mathrm{(y - 3)(y + 1) = 0}\)

4. CONSIDER ALL CASES for the solutions

• From \(\mathrm{(y - 3)(y + 1) = 0}\), we get:
  • \(\mathrm{y - 3 = 0}\), so \(\mathrm{y = 3}\)
  • \(\mathrm{y + 1 = 0}\), so \(\mathrm{y = -1}\)

5. APPLY CONSTRAINTS to select valid solution

• Since \(\mathrm{y \gt 0}\), we must have \(\mathrm{y = 3}\) (rejecting \(\mathrm{y = -1}\))
• Therefore: \(\mathrm{x = y^2 = 3^2 = 9}\)
• Final answer: \(\mathrm{x + y = 9 + 3 = 12}\)

Answer: D (12)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS skill: Students solve the quadratic correctly to get \(\mathrm{y = 3}\) or \(\mathrm{y = -1}\), but forget to use the constraint \(\mathrm{y \gt 0}\). They might randomly choose \(\mathrm{y = -1}\), leading to \(\mathrm{x = (-1)^2 = 1}\), so \(\mathrm{x + y = 1 + (-1) = 0}\). Since 0 isn't among the answer choices, this leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning about solution method: Students might try to solve by elimination instead of substitution, creating unnecessary complexity. They could multiply equations or rearrange incorrectly, leading to algebraic errors and potentially selecting Choice A (4) or Choice B (6) based on miscalculation.

The Bottom Line:

This problem tests whether students can efficiently choose substitution over elimination AND remember to apply given constraints. The constraint \(\mathrm{y \gt 0}\) is crucial - without it, students might select the wrong root and get confused when their answer doesn't match any choice.