One of the major applications of Implicit Function Theorem is the lesson it teaches:
 
                          Locally, Manifold Theory = Linear Algebra. 

That is, locally, we can perform our calculus as if it is linear algebra. Solving simultaneous equations, discussing about linear independence of coordinates, basis set and mapping from one manifold to another can be viewed as linear transformations. Discuss the invertibility of functions as if they are linear transformations. In fact by [Darboux's theorem][1], in Symplectic manifold theory the linear algebra aspects are more prominent.  




 


  [1]: https://en.wikipedia.org/wiki/Darboux's_theorem