Question:y = x^2 - 8x + 13y = -2x + 8A solution to the given system of equations is \(\mathrm{(x,...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{y = x^2 - 8x + 13}\) (quadratic)
  • Second equation: \(\mathrm{y = -2x + 8}\) (linear)
  • Need to find possible values of \(\mathrm{x + y}\) for solutions

2. INFER the solution strategy

• Since both equations equal y, we can set their right sides equal to each other
• This will give us the x-coordinates where the parabola and line intersect
• Key insight: The intersection points are the solutions to our system

3. SIMPLIFY by setting up and solving the equation

• Set the expressions equal: \(\mathrm{x^2 - 8x + 13 = -2x + 8}\)
• Move all terms to one side: \(\mathrm{x^2 - 8x + 13 + 2x - 8 = 0}\)
• Combine like terms: \(\mathrm{x^2 - 6x + 5 = 0}\)
• Factor the quadratic: We need two numbers that multiply to 5 and add to -6
• Those numbers are -1 and -5: \(\mathrm{(x - 1)(x - 5) = 0}\)
• Therefore: \(\mathrm{x = 1}\) or \(\mathrm{x = 5}\)

4. CONSIDER ALL CASES by finding corresponding y-values

• For \(\mathrm{x = 1}\): Using \(\mathrm{y = -2x + 8}\)
  \(\mathrm{y = -2(1) + 8 = 6}\)
  Solution: \(\mathrm{(1, 6)}\), so \(\mathrm{x + y = 1 + 6 = 7}\)
• For \(\mathrm{x = 5}\): Using \(\mathrm{y = -2x + 8}\)
  \(\mathrm{y = -2(5) + 8 = -2}\)
  Solution: \(\mathrm{(5, -2)}\), so \(\mathrm{x + y = 5 + (-2) = 3}\)

5. APPLY CONSTRAINTS to select the final answer

• Our calculations show \(\mathrm{x + y}\) can be either 7 or 3
• Checking the answer choices: (A) 3, (B) 5, (C) 6, (D) 8
• Only the value 3 appears among the choices

Answer: A (3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they should set the two expressions for y equal to each other. Instead, they might try to substitute one equation into the other incorrectly, or attempt to solve each equation separately without finding their intersection points. This leads to confusion about how to find the actual solutions to the system, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{x^2 - 8x + 13 = -2x + 8}\) but make algebraic mistakes when rearranging terms or factoring the resulting quadratic. They might get \(\mathrm{x^2 - 6x + 5 = 0}\) but factor it incorrectly (like \(\mathrm{(x - 2)(x - 3) = 0}\)), leading to wrong x-values and ultimately wrong \(\mathrm{x + y}\) calculations. This may lead them to select Choice B (5) or another incorrect option.

The Bottom Line:

This problem tests whether students understand that solutions to a system are intersection points, and whether they can execute the algebraic steps accurately. The key insight is recognizing that "setting y-expressions equal" is the path forward, followed by careful quadratic solving.