x = 49 y = sqrt(x) + 9 The graphs of the given equations intersect at the point \(\mathrm{(x, y)}\)...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{x = 49}\) (first equation)
  • \(\mathrm{y = \sqrt{x} + 9}\) (second equation)
  • The graphs intersect at point \(\mathrm{(x, y)}\)

• What this tells us: The intersection point \(\mathrm{(x, y)}\) must satisfy both equations

2. INFER the solution approach

• Since we know \(\mathrm{x = 49}\) from the first equation, we can substitute this value into the second equation to find y
• This substitution will give us the exact coordinates of the intersection point

3. SIMPLIFY by substitution and evaluation

• Substitute \(\mathrm{x = 49}\) into \(\mathrm{y = \sqrt{x} + 9}\):
  \(\mathrm{y = \sqrt{49} + 9}\)

• Evaluate the square root:
  \(\mathrm{y = 7 + 9}\)

• Perform the addition:
  \(\mathrm{y = 16}\)

Answer: A. 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding what "graphs intersect at point \(\mathrm{(x, y)}\)" means mathematically.

Students may think they need to graph both equations or find multiple intersection points, rather than recognizing that the intersection point simply satisfies both equations simultaneously. This leads to confusion about how to proceed, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Making errors with square root evaluation.

Students might incorrectly evaluate \(\mathrm{\sqrt{49}}\) as something other than 7 (perhaps confusing it with \(\mathrm{49^2}\), or miscalculating), or make simple arithmetic errors when adding \(\mathrm{7 + 9}\). These calculation mistakes could lead them to select Choice B (40), Choice C (81), or Choice D (130).

The Bottom Line:

This problem tests whether students understand that intersection points satisfy all equations in a system, combined with accurate evaluation of square roots and basic arithmetic. The key insight is recognizing that substitution is the direct path to the solution.