Definition
A linear system is one that satisfies both superposition and homogeneity, also called scaling. Superposition means that the system passes the following test: $ f(x+y)=f(x)+f(y) $. Scaling means that system passes the following test: $ f(ax)=af(x) $.
Example of a Linear System
Let's take the system $ y(t)=9x(t) $.
Let's also say that $ x1(t)=3t) $ and $ x12(t)=7t) $
Now for the proof...
$ x3(t)=3t+7t $
Put it in the system ----> $ y3(t)=9(3t+7t)=90t $
Now check to see if it works the other direction...
$ y1=9(3t)=27t $
$ y1=9(7t)=63t $
$ y3=y1+y2=63t+27t=90t $
The system checks out, so it is a linear system.
