This work provides a tutorial to model a nonlinear system by means of a set of linear parameter varying (LPV) systems, linearized around equilibrium points in which the control design will be applied. The proposed approach is developed in the coupled tanks system, a nonlinear system that is widely used for educational purposes. Two distinct settings of coupled tanks are presented: the fist one with two tanks, and the second one with four tanks. During the simulations, it is demonstrated that the proposed model manages to pursue multiple equilibrium points, providing theoretical guarantees of stability and performance in their neighborhood.