x = -3y = 2 - x^2The graphs of the given equations intersect at the point \(\mathrm{(x, y)}\) in the...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{x = -3}\)
  • Second equation: \(\mathrm{y = 2 - x^2}\)
  • 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 we already know \(\mathrm{x = -3}\) from the first equation, we can find y by substituting this value into the second equation
• This is the most direct path to the answer

3. SIMPLIFY through substitution

• Substitute \(\mathrm{x = -3}\) into \(\mathrm{y = 2 - x^2}\):
  \(\mathrm{y = 2 - (-3)^2}\)

• Apply order of operations - evaluate the exponent first:
  \(\mathrm{(-3)^2 = (-3) \times (-3) = 9}\)

• Complete the calculation:
  \(\mathrm{y = 2 - 9 = -7}\)

Answer: (A) -7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Incorrectly calculating \(\mathrm{(-3)^2}\) as -9 instead of 9

Students often confuse the square of a negative number with a negative square. They might think \(\mathrm{(-3)^2 = -9}\), leading to:
\(\mathrm{y = 2 - (-9) = 2 + 9 = 11}\)

This may lead them to select Choice (C) (11)

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what "intersection point" means

Some students might think they need to solve the system more complexly, perhaps trying to set the equations equal or getting confused about what to do with \(\mathrm{x = -3}\). This confusion about the basic setup can lead to random guessing or selecting an answer without proper justification.

The Bottom Line:

This problem tests whether students understand that squaring a negative number gives a positive result and can properly substitute values into algebraic expressions. The key insight is recognizing that when x is explicitly given, finding y becomes a straightforward substitution problem.