y = 76 y = x^2 - 5 The graphs of the given equations in the xy-plane intersect at the...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{y = 76}\)
  • Second equation: \(\mathrm{y = x^2 - 5}\)
  • The graphs intersect at point (x, y)
• What this tells us: The intersection point must satisfy both equations simultaneously

2. INFER the solution approach

• Since both equations are set equal to y, we can use substitution
• The first equation already gives us \(\mathrm{y = 76}\), so we can substitute this value into the second equation
• This will give us a single equation in terms of x only

3. SIMPLIFY by substitution and solving

• Substitute \(\mathrm{y = 76}\) into the second equation:
  \(\mathrm{76 = x^2 - 5}\)
• Add 5 to both sides:
  \(\mathrm{x^2 = 81}\)
• Take the square root of both sides:
  \(\mathrm{x = ±9}\)

4. CONSIDER ALL CASES for the solutions

• We have two possible x-values: \(\mathrm{x = 9}\) and \(\mathrm{x = -9}\)
• Both are mathematically valid solutions to our system

5. APPLY CONSTRAINTS from the answer choices

• Looking at the given options: A. -76/5, B. -9, C. 5, D. 76
• Only -9 appears as choice B
• Therefore, the answer is B

Answer: B. -9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "intersection point" means the same (x, y) values must work in both equations. Instead, they might try to solve each equation separately or get confused about what the problem is asking for.

This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students solve correctly to get \(\mathrm{x^2 = 81}\) but only consider the positive square root, finding \(\mathrm{x = 9}\). When they don't see 9 in the answer choices, they get stuck or guess randomly instead of recognizing that \(\mathrm{x = -9}\) is also a valid solution.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students understand that intersection points must satisfy all equations in a system, and whether they remember that quadratic equations typically have two solutions. The key insight is recognizing that both mathematical solutions are valid, but only one appears in the answer choices.