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

1. TRANSLATE what the problem is asking

• Given information:
  • A graph showing both a quadratic equation (parabola) and a linear equation
  • We need to find the solution \(\mathrm{(x, y)}\) to this system

• What we need to find:
  • The point where the two graphs intersect

2. INFER the strategy

The key insight here: When solving a system of equations graphically, the solution is the point (or points) where the graphs intersect. This intersection point has coordinates that satisfy BOTH equations simultaneously.

So our approach is simple: Find where the two curves meet on the graph.

3. TRANSLATE the graph to identify the intersection point

Looking at the graph carefully:

• The parabola (U-shaped curve) opens upward
• Locate where the linear equation and quadratic equation cross each other
• The intersection appears to occur at \(\mathrm{x = 4}\) (count 4 units to the right of the origin)
• At \(\mathrm{x = 4}\), the y-coordinate is 1 (count 1 unit up from the x-axis)

Therefore, the intersection point is \(\mathrm{(4, 1)}\).

4. Verify against answer choices

Looking at the options:

• A. \(\mathrm{(0, 0)}\) - This is the origin, not where the graphs intersect
• B. \(\mathrm{(-4, 1)}\) - This has a negative x-value, outside our graph's visible range
• C. \(\mathrm{(4, -1)}\) - This has the correct x-coordinate but wrong y-coordinate (negative instead of positive)
• D. \(\mathrm{(4, 1)}\) - This matches our identified intersection point ✓

Answer: D. \(\mathrm{(4, 1)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill (misreading graph coordinates): Students may correctly identify where the graphs intersect but misread the y-coordinate. The vertex of the parabola is at \(\mathrm{y = 1}\) (one unit above the x-axis). If a student misreads this as \(\mathrm{y = -1}\) (one unit below the x-axis), they would get the wrong y-value.

This may lead them to select Choice C \(\mathrm{(4, -1)}\).

Second Most Common Error:

Weak INFER skill (conceptual confusion about what "solution" means): Some students might not understand that the solution to a system is specifically the intersection point. They might instead identify a special feature of one graph, such as the vertex of the parabola or where the parabola crosses the y-axis. If they identify where the parabola crosses the y-axis at the origin region without careful reading, they might think the origin is significant.

This may lead them to select Choice A \(\mathrm{(0, 0)}\) or causes confusion and guessing.

The Bottom Line:

This problem tests whether students can connect the visual representation of a system (intersecting graphs) to its algebraic meaning (common solution). The execution requires careful graph reading - a seemingly simple skill where small errors (like confusing positive and negative values or miscounting grid squares) lead directly to wrong answers.