\(\mathrm{y = (x - 2)(x + 4)}\) y = 6x - 12 Which ordered pair \(\mathrm{(x, y)}\) is the solution...

1. INFER the solution strategy

• Given: Two equations both equal to y
• Strategy: Since both expressions equal y, we can set them equal to each other (substitution method)
• This gives us: \(\mathrm{(x - 2)(x + 4) = 6x - 12}\)

2. SIMPLIFY by expanding and solving

• Expand left side: \(\mathrm{(x - 2)(x + 4) = x^2 + 4x - 2x - 8 = x^2 + 2x - 8}\)
• Set equal: \(\mathrm{x^2 + 2x - 8 = 6x - 12}\)
• Collect all terms: \(\mathrm{x^2 + 2x - 8 - 6x + 12 = 0}\)
• Combine like terms: \(\mathrm{x^2 - 4x + 4 = 0}\)
• Factor: \(\mathrm{(x - 2)^2 = 0}\)
• Solve: \(\mathrm{x = 2}\)

3. Find the y-coordinate

• Substitute \(\mathrm{x = 2}\) into either original equation
• Using \(\mathrm{y = 6x - 12}\): \(\mathrm{y = 6(2) - 12 = 0}\)

4. Verify the solution

• Check in both equations:
  • \(\mathrm{y = (x - 2)(x + 4)}\): \(\mathrm{y = (2-2)(2+4) = 0 \times 6 = 0}\) ✓
  • \(\mathrm{y = 6x - 12}\): \(\mathrm{y = 6(2) - 12 = 0}\) ✓

Answer: C. (2, 0)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when expanding \(\mathrm{(x - 2)(x + 4)}\) or when collecting like terms.

For example, they might incorrectly expand to get \(\mathrm{x^2 - 2x - 8}\) instead of \(\mathrm{x^2 + 2x - 8}\), or make errors when moving terms from one side to the other. These algebraic mistakes lead to a different quadratic equation and therefore wrong values for x and y.

This leads to confusion and incorrect answer selection among choices B or D.

Second Most Common Error:

Coordinate confusion: Students solve correctly for \(\mathrm{x = 2}\) and \(\mathrm{y = 0}\) but then mix up the order when writing the ordered pair.

They might think the answer is \(\mathrm{(0, 2)}\) instead of \(\mathrm{(2, 0)}\), forgetting that ordered pairs are written as \(\mathrm{(x, y)}\).

This may lead them to select Choice A (0, 2).

The Bottom Line:

This problem tests whether students can systematically apply the substitution method while maintaining accuracy through multiple algebraic steps. The key challenge is keeping track of signs and terms during the expansion and simplification process.