The functions f and g are defined by \(\mathrm{f(x) = 3x - 9}\) and \(\mathrm{g(x) = -2x + 16}\). At...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 3x - 9}\)
  • \(\mathrm{g(x) = -2x + 16}\)
  • Need to find where graphs intersect

• What "graphs intersect" means mathematically: The point where both functions have the same y-value, so \(\mathrm{f(x) = g(x)}\)

2. SIMPLIFY to find the x-coordinate

• Set up the equation: \(\mathrm{3x - 9 = -2x + 16}\)
• Add 2x to both sides: \(\mathrm{5x - 9 = 16}\)
• Add 9 to both sides: \(\mathrm{5x = 25}\)
• Divide by 5: \(\mathrm{x = 5}\)

3. SIMPLIFY to find the y-coordinate

• Substitute \(\mathrm{x = 5}\) into either function
• Using f(x): \(\mathrm{f(5) = 3(5) - 9 = 15 - 9 = 6}\)
• Check with g(x): \(\mathrm{g(5) = -2(5) + 16 = -10 + 16 = 6}\) ✓

Answer: \(\mathrm{(5, 6)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not realize that "graphs intersect" means the functions are equal to each other. Instead, they might try to graph both functions or substitute answer choices randomly without understanding the underlying concept.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{f(x) = g(x)}\) but make algebraic errors when combining like terms. Common mistakes include:

• Forgetting to change signs when moving terms: getting \(\mathrm{3x - 9 = -2x + 16}\) → \(\mathrm{x - 9 = 16}\) (forgot to add 2x to left side)
• Sign errors: \(\mathrm{3x + 2x - 9 = 16}\) → \(\mathrm{5x - 9 = 16}\) → \(\mathrm{5x = 25 - 9 = 16}\) → \(\mathrm{x = 16/5}\)

This may lead them to select Choice A \(\mathrm{(3, 0)}\) or other incorrect coordinates.

The Bottom Line:

This problem tests whether students understand the geometric meaning of function intersection and can execute multi-step algebraic manipulation accurately. Success requires both conceptual understanding and careful arithmetic.