The graph of a system of a linear equation and a nonlinear equation is shown.What is the solution \(\mathrm{(x,y)}\) to...

1. INFER what you're looking for

When solving a system of equations graphically, you need to understand this key relationship:

• The solution to a system = the point(s) where the graphs intersect
• At an intersection point, both equations are satisfied simultaneously
• This means the \(\mathrm{(x, y)}\) coordinates work in both equations

2. TRANSLATE the visual information

Now look at the graph to locate where the curves meet:

• Find where the curved line and straight line cross each other
• Trace vertically from \(\mathrm{x = 2}\) on the x-axis
• At \(\mathrm{x = 2}\), both curves pass through the same height
• Read the y-coordinate at this intersection: \(\mathrm{y = 4}\)

3. Write the solution as an ordered pair

The intersection occurs at:

• x-coordinate: 2
• y-coordinate: 4
• Ordered pair: \(\mathrm{(2, 4)}\)

Answer: C. \(\mathrm{(2, 4)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Reversing the coordinates

Students may correctly identify where the graphs intersect but reverse the x and y values when writing the ordered pair. They might see the intersection and think "the point is at 4 and 2" without carefully distinguishing which is x and which is y.

If they write \(\mathrm{(4, 2)}\) instead of \(\mathrm{(2, 4)}\), they might look at the answer choices and not see this exact pair. This causes confusion and may lead to guessing, or they might misread one of the other choices.

Second Most Common Error:

Missing conceptual knowledge: Not understanding what "solution" means graphically

Students who don't understand that the solution is the intersection point might look for other features on the graph - perhaps where a curve crosses an axis, or they might try to read individual points on one of the curves rather than looking for where they meet.

Without understanding that intersection = solution, students may select Choice A \(\mathrm{(0, 0)}\) or Choice D \(\mathrm{(4, 0)}\) if they focus on axis intercepts, or they may simply guess.

The Bottom Line:

This problem tests whether students can connect the abstract concept (solution to a system) with its visual representation (intersection point) and accurately read coordinates from a graph. The cognitive skill is straightforward once you know what to look for, but requires both conceptual understanding and careful visual interpretation.