Question:y - x = 4y + 2x = x^2If \(\mathrm{(x,y)}\) is a solution to the system of equations above, which...

1. TRANSLATE the problem information

• Given system:
  • Equation 1: \(\mathrm{y - x = 4}\)
  • Equation 2: \(\mathrm{y + 2x = x^2}\)
• We need to find which answer choice could be the value of x

2. INFER the solution approach

• Since we have two equations with two unknowns, we can solve this system
• The substitution method works well here since the first equation can easily be solved for y
• We'll substitute the expression for y into the second equation to create a single-variable equation

3. SIMPLIFY using substitution

• From equation 1: \(\mathrm{y - x = 4 \rightarrow y = x + 4}\)
• Substitute this into equation 2:
  \(\mathrm{(x + 4) + 2x = x^2}\)
  \(\mathrm{3x + 4 = x^2}\)

4. SIMPLIFY to standard quadratic form

• Rearrange: \(\mathrm{x^2 - 3x - 4 = 0}\)
• Factor the quadratic (find two numbers that multiply to -4 and add to -3):
  The numbers are -4 and 1, so: \(\mathrm{(x - 4)(x + 1) = 0}\)

5. SIMPLIFY using zero product property

• From \(\mathrm{(x - 4)(x + 1) = 0}\):
  \(\mathrm{x - 4 = 0 \rightarrow x = 4}\)
  \(\mathrm{x + 1 = 0 \rightarrow x = -1}\)

6. APPLY CONSTRAINTS to select from answer choices

• The complete mathematical solution gives \(\mathrm{x = 4}\) or \(\mathrm{x = -1}\)
• Looking at the choices: (A) -1, (B) 0, (C) 1, (D) 2
• Only \(\mathrm{x = -1}\) appears in the answer choices

Answer: (A) -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may attempt to solve both equations simultaneously without recognizing that substitution is the most efficient approach, or they might try elimination method which becomes unwieldy with the quadratic term.

This leads to confusion with multiple variables and may cause them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the substitution but make algebraic errors when expanding \(\mathrm{(x + 4) + 2x = x^2}\), perhaps getting \(\mathrm{3x + 4 = x^2}\) wrong, or making errors in factoring the resulting quadratic equation.

This may lead them to select Choice (B) (0) or Choice (C) (1) based on incorrect factors.

The Bottom Line:

This problem combines system-solving skills with quadratic algebra, requiring students to maintain accuracy through multiple algebraic steps while recognizing that not all mathematical solutions may appear in the answer choices.