A system of equations consists of a quadratic equation and a linear equation. The equations in this system are graphed...

1. TRANSLATE the problem information

• Given information:
  • A system consists of one quadratic equation and one linear equation
  • Both equations are graphed in the xy-plane
  • Need to find: How many solutions does this system have?

2. INFER what "solutions" means graphically

• Key concept: Solutions to a system of equations are the points \(\mathrm{(x, y)}\) that satisfy BOTH equations simultaneously
• On a graph: These solution points appear where the two curves intersect
• Strategy: Count the intersection points

3. VISUALIZE the graphs to identify the curves

• Looking at the graph, I can identify:
  • A parabola (U-shaped curve) opening upward with vertex near \(\mathrm{(0, -2)}\)
  • A straight line slanting upward from left to right
• Both curves are clearly visible on the coordinate plane

4. VISUALIZE the intersection points carefully

• Scan the graph systematically to find where the line crosses the parabola
• First intersection: On the left side, around \(\mathrm{x} = -6\), the line crosses the parabola
• Second intersection: On the right side, around \(\mathrm{x} = 5\), the line crosses the parabola again
• Check: Are there any other intersections? No - the line only touches the parabola at these two points

5. INFER the final answer

• Number of intersection points = 2
• Therefore, the system has 2 solutions

Answer: C. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak VISUALIZE skill: Hasty graph reading that miscounts intersection points

Students might rush through reading the graph and:

• Count only one intersection point (perhaps focusing only on the more obvious one on the right)
• Overlook that the line crosses the parabola twice because one intersection might be less prominent

This may lead them to select Choice B (1).

Second Most Common Error:

Conceptual confusion about intersection: Not understanding what "intersection" means in this context

Some students might:

• Count places where the curves are "close" but not actually touching
• Get confused by the grid lines and think there are multiple solution points
• Misunderstand that they need intersection points, not just points on either curve

This leads to confusion and may cause them to select Choice D (3) or guess randomly.

The Bottom Line:

This problem requires careful visual analysis. The key is to slow down and systematically scan the entire graph to count exactly where the two curves cross each other. Remember: each intersection point is one solution, and a line can intersect a parabola at most 2 times (never 3 or more).