x^2 + y^2 = 68x - y = 6The graphs of the equations in the given system of equations intersect...

1. INFER the best solution strategy

• We have one linear equation (\(\mathrm{x - y = 6}\)) and one quadratic equation (\(\mathrm{x^2 + y^2 = 68}\))
• Since the linear equation can be easily solved for one variable, substitution is the most efficient approach

2. SIMPLIFY by solving for one variable

• From \(\mathrm{x - y = 6}\), solve for y:
• \(\mathrm{y = x - 6}\)

3. SIMPLIFY by substituting into the quadratic equation

• Substitute \(\mathrm{y = x - 6}\) into \(\mathrm{x^2 + y^2 = 68}\):
• \(\mathrm{x^2 + (x - 6)^2 = 68}\)

4. SIMPLIFY by expanding and combining terms

• Expand \(\mathrm{(x - 6)^2}\):
  \(\mathrm{x^2 + x^2 - 12x + 36 = 68}\)
• Combine like terms:
  \(\mathrm{2x^2 - 12x + 36 = 68}\)
• Move everything to one side:
  \(\mathrm{2x^2 - 12x - 32 = 0}\)
• Divide by 2:
  \(\mathrm{x^2 - 6x - 16 = 0}\)

5. SIMPLIFY by factoring

• Look for two numbers that multiply to -16 and add to -6
• Those numbers are -8 and +2
• Factor: \(\mathrm{(x - 8)(x + 2) = 0}\)

6. CONSIDER ALL CASES to find all solutions

• \(\mathrm{x - 8 = 0}\) gives \(\mathrm{x = 8}\)
• \(\mathrm{x + 2 = 0}\) gives \(\mathrm{x = -2}\)
• Both solutions are mathematically valid

7. APPLY CONSTRAINTS based on answer choices

• Among the given choices, only \(\mathrm{x = -2}\) appears as option (B)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x - 6)^2}\) or combining terms.

A typical error is expanding \(\mathrm{(x - 6)^2}\) as \(\mathrm{x^2 - 36}\) instead of \(\mathrm{x^2 - 12x + 36}\), forgetting the middle term. This leads to an incorrect quadratic equation and wrong solutions. Students might end up with \(\mathrm{x^2 + x^2 - 36 = 68}\), giving \(\mathrm{2x^2 = 104}\), so \(\mathrm{x^2 = 52}\), leading them to calculate \(\mathrm{x = \pm\sqrt{52} \approx \pm7.2}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Incomplete CONSIDER ALL CASES reasoning: Students find only one solution instead of both, or find both but don't check which appears in the answer choices.

Some students might correctly factor to get \(\mathrm{(x - 8)(x + 2) = 0}\) but only solve \(\mathrm{x - 8 = 0}\) to get \(\mathrm{x = 8}\). Since 8 isn't among the choices, they get confused. Others might find both solutions but pick the wrong one without carefully checking the answer choices. This may lead them to select Choice D (6) by misreading or guessing.

The Bottom Line:

This problem tests your ability to execute a multi-step algebraic process without making calculation errors, while also requiring you to find all solutions and match them to the given choices. The key is methodical expansion and careful factoring.