So why linear systems?
1. Linear systems are well-defined and well understood. On the other hand, nonlinear systems are not concretely defined. They can be quadratic, cubic, polynomial, exponential, harmonic, blah blah...
Simplicity is not the only reason though. Here's the more important one.
2. Any smooth function can be approximated locally by a linear function. In layman terms, most functions, when zoomed in, look like a linear function.
Linear approximation is the essence of Newton's method (for nonlinear optimization), finite difference (approximating the derivative of a function), etc.
Do you get the picture above?