Linearity

Given 2 functions x1 and x2, constants a and b, outputs y1 and y2, and system s
A system is linear if you take 
x1->S->y1, 
x2->S->y2 
z=a*y1+b*y2

a*x1+b*x2->S->a*y1+b*y2=w

If z==w then the system is linear
If z!=w then the system is nonlinear
This is better demonstrated by the diagram and examples below.
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Seraj Dosenbach