x + y = 17 xy = 72 If one solution to the system of equations above is \(\mathrm{(x, y)}\),...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{x + y = 17}\) (linear equation)
  • \(\mathrm{xy = 72}\) (quadratic relationship)
• Need: One possible value of x

2. INFER the solution strategy

• Since we have one linear and one quadratic equation, substitution will work well
• Solve the simpler (linear) equation for one variable, then substitute

3. SIMPLIFY using substitution

• From first equation: \(\mathrm{y = 17 - x}\)
• Substitute into second equation: \(\mathrm{x(17 - x) = 72}\)
• Apply distributive property: \(\mathrm{17x - x^2 = 72}\)
• Rearrange to standard form: \(\mathrm{x^2 - 17x + 72 = 0}\)

4. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to 72 and add to 17
• \(\mathrm{8 \times 9 = 72}\) and \(\mathrm{8 + 9 = 17}\) ✓
• Factor: \(\mathrm{(x - 8)(x - 9) = 0}\)

5. INFER the solutions using Zero Product Property

• If \(\mathrm{(x - 8)(x - 9) = 0}\), then either:
  • \(\mathrm{x - 8 = 0}\) → \(\mathrm{x = 8}\), or
  • \(\mathrm{x - 9 = 0}\) → \(\mathrm{x = 9}\)

Answer: 8 or 9 (either value is acceptable)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors when expanding or rearranging the quadratic equation

Students might incorrectly expand \(\mathrm{x(17 - x) = 72}\) as \(\mathrm{17x + x^2 = 72}\), leading to \(\mathrm{x^2 + 17x - 72 = 0}\). This gives completely different factors and wrong solutions. This leads to confusion when their factored solutions don't check against the original equations.

Second Most Common Error:

Poor factoring execution within SIMPLIFY: Incorrect factor pairs for the quadratic

Students correctly reach \(\mathrm{x^2 - 17x + 72 = 0}\) but struggle to find the right factor pair. They might try combinations like \(\mathrm{(x - 6)(x - 12)}\) because \(\mathrm{6 \times 12 = 72}\), forgetting that \(\mathrm{6 + 12 \neq 17}\). This leads to getting stuck and potentially guessing random values.

The Bottom Line:

This problem tests whether students can handle the transition from a system with mixed equation types to a single quadratic equation, requiring solid algebraic manipulation skills throughout multiple steps.