Concept 4

Connect systems to graphs and match a system to its graphical representation.

Match slopes, intercepts, and the intersection point to the graph of the system.

Core Idea

To match a system to its graph, convert each equation to slope-intercept form and compare slopes, intercepts, and the intersection point with what the graph shows.

Understanding

Match slopes, intercepts, and the intersection. A graph of two lines gives you three fast checks.

  • Read each slope.
  • Read each y-intercept.
  • Read the intersection point.
  • Convert candidate equations to 𝑦 =𝑚𝑥 +𝑏 if needed.

The correct system matches all three features, not just one of them.

Step by Step

  1. Read the slope (positive/negative, steep/shallow) of each line from the graph.
  2. Read each line's 𝑦-intercept.
  3. Read the intersection point's coordinates.
  4. Convert candidate equations to 𝑦 =𝑚𝑥 +𝑏 and see which pair matches all three features.

Misconceptions

  • Misreading the scale of the axes — assuming each grid square is 1 unit when it might be 2 or 5.
  • Matching only the slopes but ignoring intercepts, which can lead to picking a parallel but shifted pair of lines.
Question

Worked Example

A graph shows two lines intersecting at (1,4). Line A has a 𝑦-intercept of 2 and a positive slope. Line B has a 𝑦-intercept of 6 and a negative slope. Which system could represent this graph?

Select an answer to see the explanation