x - y = 1x + y = x^2 - 3Which ordered pair is a solution to the system of...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{x - y = 1}\) (linear equation)
  • \(\mathrm{x + y = x^2 - 3}\) (quadratic equation)
• Find: Which ordered pair satisfies both equations

2. INFER the solving strategy

• Since we have one linear and one quadratic equation, substitution is the most efficient approach
• The linear equation is simpler to solve for one variable, so start there
• Solve the first equation for x: \(\mathrm{x = y + 1}\)

3. SIMPLIFY by substituting and expanding

• Substitute \(\mathrm{x = y + 1}\) into the second equation:
  \(\mathrm{(y + 1) + y = (y + 1)^2 - 3}\)
• Expand the left side: \(\mathrm{2y + 1}\)
• Expand the right side: \(\mathrm{(y + 1)^2 - 3 = y^2 + 2y + 1 - 3 = y^2 + 2y - 2}\)
• Set them equal: \(\mathrm{2y + 1 = y^2 + 2y - 2}\)

4. SIMPLIFY to isolate the quadratic term

• Subtract 2y from both sides: \(\mathrm{1 = y^2 - 2}\)
• Add 2 to both sides: \(\mathrm{3 = y^2}\)
• Take the square root: \(\mathrm{y = ±\sqrt{3}}\)

5. CONSIDER ALL CASES and find corresponding x-values

• If \(\mathrm{y = \sqrt{3}}\), then \(\mathrm{x = y + 1 = 1 + \sqrt{3}}\)
• If \(\mathrm{y = -\sqrt{3}}\), then \(\mathrm{x = y + 1 = 1 - \sqrt{3}}\)
• This gives us two potential solutions: \(\mathrm{(1 + \sqrt{3}, \sqrt{3})}\) and \(\mathrm{(1 - \sqrt{3}, -\sqrt{3})}\)

6. INFER which solution matches the answer choices

• Looking at the choices, \(\mathrm{(1 + \sqrt{3}, \sqrt{3})}\) matches choice A
• The other solution \(\mathrm{(1 - \sqrt{3}, -\sqrt{3})}\) is not listed

Answer: A. \(\mathrm{(1 + \sqrt{3}, \sqrt{3})}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic mistakes when expanding \(\mathrm{(y + 1)^2}\) or combining like terms. For example, they might incorrectly expand \(\mathrm{(y + 1)^2}\) as \(\mathrm{y^2 + 1}\) instead of \(\mathrm{y^2 + 2y + 1}\), or make sign errors when moving terms across the equals sign.

This leads to getting an incorrect quadratic equation, which produces wrong values for y, causing them to select an incorrect answer choice or get confused and guess.

Second Most Common Error:

Inadequate INFER reasoning: Students attempt to solve the system by elimination or try to work with the quadratic equation first, leading to unnecessarily complex algebra. Some might try to substitute from the second equation into the first, creating a more complicated expression.

This makes the problem much harder than necessary and often leads to calculation errors or abandoning the systematic solution and guessing.

The Bottom Line:

This problem tests whether students can recognize when substitution is the most efficient method and execute multi-step algebraic simplification accurately. The key insight is that the linear equation provides the simplest path to substitution.