The graph of a system of a linear equation and an exponential equation is shown. What is the solution \((x,y)\)...

1. TRANSLATE the problem requirement

The question asks for "the solution (x, y) to this system."

What this means:

• We need to find the point where BOTH equations are true at the same time
• Graphically, this is where the two curves intersect

2. INFER the solution strategy

Key insight: When equations are already graphed, the solution to the system is simply the intersection point of the graphs.

Why? At the intersection point, the x and y values satisfy both equations simultaneously—that's what makes it the solution.

3. TRANSLATE the visual information to coordinates

Looking at the graph:

• The solid line (linear function \(\mathrm{y = x + 2}\)) and the dashed curve (exponential function \(\mathrm{y = 2^x}\)) meet at one point
• The legend helpfully labels this as "Intersection (2, 4)"
• Reading from the graph:
  • The x-coordinate (horizontal position) is 2
  • The y-coordinate (vertical position) is 4

4. Verify the answer (optional but recommended)

We can check our reading by substituting \(\mathrm{x = 2}\) into both equations:

• Linear: \(\mathrm{y = 2 + 2 = 4}\) ✓
• Exponential: \(\mathrm{y = 2^2 = 4}\) ✓

Both give y = 4, confirming our intersection point is correct.

Answer: (2, 4) - Choice B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing x and y coordinates when reading from the graph

Students might read the coordinates in reverse order, thinking the point is (4, 2) instead of (2, 4). This happens when they don't carefully distinguish between:

• x-coordinate = horizontal position (left-right) = 2
• y-coordinate = vertical position (up-down) = 4

This may lead them to select Choice A: (1, 2) if they're also approximating incorrectly, or causes confusion since (4, 2) isn't an option, leading to guessing.

Second Most Common Error:

Missing conceptual knowledge: Not understanding what "solution to a system" means

Students might think they need to solve the equations algebraically (setting \(\mathrm{x + 2 = 2^x}\)), which is very difficult without graphing technology. Or they might think they need to find ALL points on both curves, not just the intersection.

This confusion leads students to abandon a systematic approach and guess among the answer choices.

The Bottom Line:

This problem tests whether students understand that a system's solution is the intersection point, and whether they can accurately read coordinates from a graph. The actual mathematics is simple—the challenge is in the interpretation and visual reading skills. Always remember: x comes first in an ordered pair (x, y), and corresponds to horizontal position on the graph.