y = x + 1 y = x^2 + x If \(\mathrm{(x,y)}\) is a solution to the system of equations...

1. INFER the solving strategy

• Given information:
  • \(\mathrm{y = x + 1}\) (equation 1)
  • \(\mathrm{y = x^2 + x}\) (equation 2)

• Key insight: Since both expressions equal y, we can set them equal to each other to eliminate y and solve for x.

2. SIMPLIFY by setting the equations equal

• Set the right sides equal: \(\mathrm{x + 1 = x^2 + x}\)

• Subtract x from both sides: \(\mathrm{x + 1 - x = x^2 + x - x}\)

• This gives us: \(\mathrm{1 = x^2}\)

3. SIMPLIFY to find all solutions

• Take the square root of both sides: \(\mathrm{x = ±\sqrt{1} = ±1}\)

• So x can be either 1 or -1

4. CONSIDER ALL CASES and match to answer choices

• We found two mathematical solutions: \(\mathrm{x = 1}\) and \(\mathrm{x = -1}\)

• Checking the given choices: A. -1, B. 0, C. 2, D. 3

• Only \(\mathrm{x = -1}\) appears among the choices

Answer: A. -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students try to substitute each answer choice back into both equations instead of solving the system algebraically.

For example, they might test \(\mathrm{x = 0}\): First equation gives \(\mathrm{y = 0 + 1 = 1}\), second equation gives \(\mathrm{y = 0^2 + 0 = 0}\). Since \(\mathrm{1 ≠ 0}\), they eliminate choice B. They continue this process for all choices, which is inefficient and prone to arithmetic errors.

This approach works but takes much longer and increases chances of calculation mistakes.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when manipulating \(\mathrm{x + 1 = x^2 + x}\).

A common mistake is incorrectly canceling terms, like thinking \(\mathrm{x + 1 = x^2 + x}\) becomes \(\mathrm{1 = x^2 + 1}\), leading to \(\mathrm{x^2 = 0}\) and \(\mathrm{x = 0}\). This would lead them to select Choice B (0), but this answer doesn't actually satisfy the original system.

The Bottom Line:

The key insight is recognizing that systems of equations can often be solved by elimination - in this case, eliminating y by setting the two expressions for y equal to each other. Students who jump straight to testing answer choices miss this elegant algebraic approach.