y = x^2 + 2x + 1x + y + 1 = 0If \(\mathrm{(x_1,y_1)}\) and \(\mathrm{(x_2,y_2)}\) are the two solutions...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y = x^2 + 2x + 1}\) (quadratic equation)
  • \(\mathrm{x + y + 1 = 0}\) (linear equation)
• Need to find: \(\mathrm{y_1 + y_2}\) where \(\mathrm{(x_1, y_1)}\) and \(\mathrm{(x_2, y_2)}\) are the two solutions

2. INFER the solution strategy

• Since we have one linear and one quadratic equation, substitution method is most efficient
• The linear equation is easier to solve for one variable, so solve \(\mathrm{x + y + 1 = 0}\) for y first

3. SIMPLIFY to isolate y in the linear equation

• From \(\mathrm{x + y + 1 = 0}\):
• \(\mathrm{y = -x - 1}\)

4. SIMPLIFY by substituting into the quadratic equation

• Substitute \(\mathrm{y = -x - 1}\) into \(\mathrm{y = x^2 + 2x + 1}\):
• \(\mathrm{-x - 1 = x^2 + 2x + 1}\)
• Move everything to one side: \(\mathrm{0 = x^2 + 2x + 1 + x + 1}\)
• Combine like terms: \(\mathrm{x^2 + 3x + 2 = 0}\)

5. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to 2 and add to 3: that's 1 and 2
• \(\mathrm{x^2 + 3x + 2 = (x + 1)(x + 2) = 0}\)
• By zero product property: \(\mathrm{x + 1 = 0}\) or \(\mathrm{x + 2 = 0}\)
• Therefore: \(\mathrm{x = -1}\) or \(\mathrm{x = -2}\)

6. SIMPLIFY to find corresponding y-values

• For \(\mathrm{x = -1}\): \(\mathrm{y = -(-1) - 1 = 1 - 1 = 0}\)
• For \(\mathrm{x = -2}\): \(\mathrm{y = -(-2) - 1 = 2 - 1 = 1}\)
• Solutions are \(\mathrm{(-1, 0)}\) and \(\mathrm{(-2, 1)}\)

7. SIMPLIFY to find the final answer

• \(\mathrm{y_1 + y_2 = 0 + 1 = 1}\)

Answer: D. 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when substituting and rearranging the equation. A common mistake is incorrectly combining terms when going from \(\mathrm{-x - 1 = x^2 + 2x + 1}\) to standard form. They might get \(\mathrm{x^2 + 2x + 2 = 0}\) instead of \(\mathrm{x^2 + 3x + 2 = 0}\), leading to completely different solutions.

This may lead them to select Choice A (-3) or cause confusion and guessing.

Second Most Common Error:

Poor INFER reasoning about problem requirements: Students might find the correct x-values (-1 and -2) but then calculate \(\mathrm{x_1 + x_2 = -1 + (-2) = -3}\) instead of finding the corresponding y-values and calculating \(\mathrm{y_1 + y_2}\).

This leads them to select Choice A (-3).

The Bottom Line:

This problem tests both systematic algebraic manipulation and careful attention to what the question is actually asking for. Success requires methodical substitution work and reading comprehension to sum the correct coordinates.