A solution to the given system of equations is \(\mathrm{(x, y)}\), where x gt 0. What is the value of...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{y = 4x}\) (linear equation)
  • \(\mathrm{y = x^2 - 12}\) (quadratic equation)
  • We need \(\mathrm{x \gt 0}\)

2. INFER the approach

• Since both expressions equal y, we can use substitution
• Set \(\mathrm{4x = x^2 - 12}\) to eliminate the variable y
• This will give us a quadratic equation in x only

3. SIMPLIFY to get standard quadratic form

• Starting with: \(\mathrm{4x = x^2 - 12}\)
• Rearrange: \(\mathrm{x^2 - 4x - 12 = 0}\)
• Factor: Look for two numbers that multiply to -12 and add to -4
• Those numbers are -6 and 2, so: \(\mathrm{(x - 6)(x + 2) = 0}\)

4. INFER solutions using zero product property

• If \(\mathrm{(x - 6)(x + 2) = 0}\), then either:
  • \(\mathrm{x - 6 = 0}\), which gives \(\mathrm{x = 6}\)
  • \(\mathrm{x + 2 = 0}\), which gives \(\mathrm{x = -2}\)

5. APPLY CONSTRAINTS to select final answer

• Since \(\mathrm{x \gt 0}\) is given in the problem:
• \(\mathrm{x = 6}\) satisfies this constraint
• \(\mathrm{x = -2}\) does not satisfy this constraint

Answer: 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they can set the two expressions for y equal to each other. Instead, they might try to solve each equation separately or get confused about how to handle a system with one linear and one quadratic equation.

This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly solve the quadratic and get \(\mathrm{x = 6}\) or \(\mathrm{x = -2}\), but forget to check which solution satisfies \(\mathrm{x \gt 0}\). They might randomly choose \(\mathrm{x = -2}\) as their final answer.

This may lead them to select an incorrect negative value if it were among the choices.

The Bottom Line:

The key insight is recognizing that substitution works perfectly here—since both expressions equal y, you can eliminate y by setting them equal. Missing this connection turns a straightforward problem into an impossible puzzle.