x^2 + y^2 = 10x - y = 2If \(\mathrm{(x_1, y_1)}\) and \(\mathrm{(x_2, y_2)}\) are the two solutions to the...

1. TRANSLATE the system for strategic solving

• Given system:
  • \(\mathrm{x^2 + y^2 = 10}\) (circle equation)
  • \(\mathrm{x - y = 2}\) (linear equation)
• The linear equation is simpler to manipulate, so solve for x: \(\mathrm{x = y + 2}\)

2. INFER the substitution approach

• Since we have x expressed in terms of y, substitute this into the circle equation
• This will give us a quadratic equation in y only

3. SIMPLIFY by substituting and expanding

• Substitute \(\mathrm{x = y + 2}\) into \(\mathrm{x^2 + y^2 = 10}\):
  \(\mathrm{(y + 2)^2 + y^2 = 10}\)
• Expand the binomial:
  \(\mathrm{y^2 + 4y + 4 + y^2 = 10}\)
• Combine like terms:
  \(\mathrm{2y^2 + 4y + 4 = 10}\)
  \(\mathrm{2y^2 + 4y - 6 = 0}\)
  \(\mathrm{y^2 + 2y - 3 = 0}\)

4. SIMPLIFY by factoring the quadratic

• Look for two numbers that multiply to -3 and add to 2
• Those numbers are 3 and -1: \(\mathrm{(y + 3)(y - 1) = 0}\)
• Therefore: \(\mathrm{y = -3}\) or \(\mathrm{y = 1}\)

5. INFER the final answer strategy

• The question asks for \(\mathrm{y_1 + y_2}\), not the individual coordinates
• Simply add the two y-values: \(\mathrm{(-3) + (1) = -2}\)

Answer: B (-2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students solve for both x and y coordinates of each solution point, then get confused about what to do with all four values. They might calculate \(\mathrm{x_1 + x_2 + y_1 + y_2}\) instead of just \(\mathrm{y_1 + y_2}\), or spend unnecessary time finding the x-coordinates when the question only needs the sum of y-coordinates.

This leads to confusion and potentially selecting the wrong answer choice or running out of time.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(y + 2)^2}\) or combining terms in the quadratic equation. A common mistake is writing \(\mathrm{(y + 2)^2 = y^2 + 4}\) (forgetting the middle term), which leads to the wrong quadratic equation and incorrect y-values.

This may lead them to select Choice A (-4) or another incorrect option.

The Bottom Line:

This problem tests whether students can efficiently combine substitution technique with quadratic solving while staying focused on exactly what the question asks for. Success requires clean algebraic manipulation and reading comprehension to avoid solving for more than necessary.