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Module 8: Linear Equations and Linear Systems

Introduction and Terminology

Imagine a subscription to a service charges a one-time activation fee of $5 on top of $12 a month. This is one of the many examples that can be modelled on a graph. This particular subscription will follow a linear relation as the monthly rate stays at $12/month.

This is the essence of linear functions. When there is constant rate, there is a straight line on the graph.

These graphs are illustrated on a Cartesian plane. The Cartesian plane has 4 quadrants, which are divided by the x and y axis. As we move up and to the right, the x and y values increase. As we move down and to the left, the x and y values decrease. As an example, the equation y = 2 x + 1 {"version":"1.1","math":"y=2x+1 "} is pictured below.


Term Definition Example
Vertex (Point)
  • Any point on the line having a defined x and y value.
  • These are denoted as (x,y).
  • The rate of change of the line.
  • Can be positive (upward) or negative (downward) from left → right.
    • Defined between two points by r i s e r u n = y 2 y 1 x 2 x 1 {"version":"1.1","math":"\dfrac{rise}{run}= \dfrac{y_2-y_1}{x_2-x_1} "}
The slope in this case is 2.
  • Point at which the line intersects the y axis
  • Where x = 0 {"version":"1.1","math":"x = 0"}.
(0, 1)
  • Point at which the line intersects the x axis.
  • Where y = 0 {"version":"1.1","math":"y = 0"}

(-0.5, 0)