A linear relationship is any relationship between two quantities where one quantity has a constant rate of change with respect to the other. For example, Andre babysits and charges a fee for traveling to and from the job, and then a set amount for every additional hour he works. Since the total amount he charges with respect to the number of hours he works changes at a constant rate, this is a linear relationship. But since Andre charges a fee for traveling, and the graph does not go through the point $(0,0)$, this is not a proportional relationship. Here is a graph of how much Andre charges based on how many hours he works.

Graph, horizontal axis, time in hours, scale 0 to 9, by 1's. vertical axis, amount earned in dollars, scale 0 to 140, by 20's. line starting at 0 comma 10, passing through 2 comma 40 and 60 comma 100.

The rate of change can be calculated using the graph. Since the rate of change is constant, we can take any two points on the graph and divide the amount of vertical change by the amount of horizontal change. For example, the points $(2,40)$ and $(6,100)$ mean that Andre earns 40 dollars for working 2 hours and 100 dollars for working 6 hours. The rate of change is $\frac{100-40}{6-2}=15$ dollars per hour. Andre's earnings go up 15 dollars for each hour of babysitting.

Notice that this is the same way we calculate the slope of the line. That's why the graph is a line and why we call this a “linear relationship.” The rate of change of a linear relationship is the same as the slope of its graph.