x = 8y = x^2 + 8The graphs of the equations in the given system of equations intersect at the...

1. TRANSLATE the problem information

• Given information:
  • System: \(\mathrm{x = 8}\) and \(\mathrm{y = x^2 + 8}\)
  • The graphs intersect at point \(\mathrm{(x, y)}\)
• What this tells us: The intersection point \(\mathrm{(x, y)}\) must satisfy both equations simultaneously

2. INFER the solution approach

• Since the intersection point satisfies both equations, we can use the value from the first equation in the second equation
• The first equation directly gives us \(\mathrm{x = 8}\)
• We need to substitute this x-value into the second equation to find y

3. SIMPLIFY to find the y-coordinate

• Substitute \(\mathrm{x = 8}\) into \(\mathrm{y = x^2 + 8}\):

\(\mathrm{y = 8^2 + 8}\)

\(\mathrm{y = 64 + 8}\)

\(\mathrm{y = 72}\)

Answer: D. 72

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that intersection points satisfy both equations simultaneously. They might try to solve the system as if they need to find where \(\mathrm{x^2 + 8 = 8}\), leading to unnecessary complexity.

This confusion might cause them to get stuck and guess randomly among the choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly understand the approach but make arithmetic errors. They might calculate \(\mathrm{8^2}\) incorrectly (confusing it with \(\mathrm{8 \times 2 = 16}\)) or make addition errors.

If they compute \(\mathrm{8^2}\) as 16, they get \(\mathrm{y = 16 + 8 = 24}\), leading them to select Choice B (24).

The Bottom Line:

This problem tests whether students understand the fundamental meaning of intersection points in systems of equations—that such points satisfy all equations simultaneously. The arithmetic is straightforward once this concept is grasped.