y = x^2 - 4x + 4y = 4 - xIf the ordered pair \(\mathrm{(x, y)}\) satisfies the system of...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{y = x^2 - 4x + 4}\) (quadratic equation)
  • \(\mathrm{y = 4 - x}\) (linear equation)
• Need to find: possible values of x that satisfy both equations

2. INFER the solution strategy

• Since both expressions equal y, the solution occurs where these expressions are equal to each other
• This means we can set: \(\mathrm{x^2 - 4x + 4 = 4 - x}\)
• This will give us the x-coordinates of intersection points

3. SIMPLIFY the resulting equation

• Start with: \(\mathrm{x^2 - 4x + 4 = 4 - x}\)
• Move all terms to one side: \(\mathrm{x^2 - 4x + 4 - 4 + x = 0}\)
• Combine like terms: \(\mathrm{x^2 - 3x = 0}\)
• Factor out x: \(\mathrm{x(x - 3) = 0}\)

4. APPLY the zero product property

• Since \(\mathrm{x(x - 3) = 0}\), either factor can equal zero
• This gives us: \(\mathrm{x = 0}\) or \(\mathrm{x - 3 = 0}\)
• Therefore: \(\mathrm{x = 0}\) or \(\mathrm{x = 3}\)

5. CONSIDER ALL CASES and verify

• Both solutions are mathematically valid
• Check \(\mathrm{x = 0}\): \(\mathrm{y = 4}\) in both equations ✓
• Check \(\mathrm{x = 3}\): \(\mathrm{y = 1}\) in both equations ✓

Answer: 0, 3, or any equivalent form

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to solve each equation separately instead of recognizing they form a system. They might attempt to solve \(\mathrm{y = x^2 - 4x + 4}\) for x by setting it equal to zero, or get confused about how to handle two different equations. This leads to confusion and abandoning the systematic approach, resulting in guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{x^2 - 4x + 4 = 4 - x}\) but make algebraic errors when rearranging terms. Common mistakes include incorrect signs when moving terms or errors in combining like terms. This may lead them to get an incorrect quadratic equation and subsequently wrong values for x.

The Bottom Line:

The key insight is recognizing that a system of equations represents the intersection of two graphs, so setting the expressions equal finds where they meet. Students who miss this fundamental connection often struggle to begin the problem systematically.