x^2 = 6x + yy = -6x + 36A solution to the given system of equations is \(\mathrm{(x, y)}\). Which...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{x^2 = 6x + y}\)
  • \(\mathrm{y = -6x + 36}\)
• Need to find: Possible values of \(\mathrm{xy}\) where \(\mathrm{(x, y)}\) is a solution

2. INFER the solving strategy

• Since the second equation gives y explicitly in terms of x, I can substitute this expression into the first equation
• This will eliminate y and give me an equation with only x

3. SIMPLIFY by substitution

• Substitute \(\mathrm{y = -6x + 36}\) into \(\mathrm{x^2 = 6x + y}\):
  \(\mathrm{x^2 = 6x + (-6x + 36)}\)
  \(\mathrm{x^2 = 6x - 6x + 36}\)
  \(\mathrm{x^2 = 36}\)

4. CONSIDER ALL CASES when solving for x

• Taking the square root of both sides: \(\mathrm{x = \pm6}\)
• This gives us two possible x-values: \(\mathrm{x = 6}\) and \(\mathrm{x = -6}\)

5. SIMPLIFY to find corresponding y-values

• For \(\mathrm{x = 6}\): \(\mathrm{y = -6(6) + 36 = -36 + 36 = 0}\)
• For \(\mathrm{x = -6}\): \(\mathrm{y = -6(-6) + 36 = 36 + 36 = 72}\)
• Solutions: \(\mathrm{(6, 0)}\) and \(\mathrm{(-6, 72)}\)

6. SIMPLIFY to calculate xy values

• For \(\mathrm{(6, 0)}\): \(\mathrm{xy = 6 \times 0 = 0}\)
• For \(\mathrm{(-6, 72)}\): \(\mathrm{xy = (-6) \times 72 = -432}\)

7. APPLY CONSTRAINTS to select from answer choices

• Possible \(\mathrm{xy}\) values: 0 and -432
• Only 0 appears among the given choices

Answer: A. 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the substitution strategy and instead try to solve the system by graphing or elimination, making the problem unnecessarily complex.

Without the key insight to substitute, students may attempt to manipulate both equations simultaneously or get confused about how to handle the quadratic term. This leads to confusion and guessing.

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students correctly find \(\mathrm{x^2 = 36}\) but only consider the positive solution \(\mathrm{x = 6}\), missing \(\mathrm{x = -6}\).

They find the solution \(\mathrm{(6, 0)}\) and calculate \(\mathrm{xy = 0}\), then select Choice A without realizing there's another solution. While this leads to the correct answer, it's incomplete mathematical reasoning that could cause errors on similar problems requiring all solutions.

The Bottom Line:

This problem tests whether students can strategically choose the right method for solving a system (substitution over elimination when one equation is already solved for a variable) and whether they remember to consider both positive and negative square roots.