To be a linear graph requires a constant rate of change.
Here's an example: a phone company has a call standing charge of 0.10 currency units (pounds, dollars, etc.) and then .02 currency units per minute for the duration of the call. The total cost of a call depends on its duration and can be expressed by c=0.10+0.02t where c is the cost and t the duration in minutes (rounded up to the next minute) of the call. The graph is linear (a straight line) with t on the horizontal axis and c on the vertical axis. To use the graph you locate the number of minutes the call lasted on the t axis, go vertically upwards from that point to meet the line and from that point horizontally towards the c axis to find the cost.
A similar linear graph can be used to price car hire, taxi cab charge, etc.
Another use is to convert temperature from °C to °F or vice versa.
Another use is to price fruit or vegetables by weight, the price being directly proportional to weight.