General techniques are developed to obtain: (1) the completion of a system of nonlinear first-order partial differential equations (PDEs) which is an independent set of further PDEs derivable from the system differentiation and elimination; and (2) simplifications of the system by choosing appropriate new independent and dependent variables using a result from Lie group theory. The number of dependent and independent variables is reduced to the minimum. The theory specializes to the classical theory of a single nonlinear PDE with one unknown and can be combined with the methods of Olver (1989), Edelen (1989) and Estabrook and Wahlquist (1986). Most of the methods appear to be sufficiently well defined for automation as are the techniques in Olver, A second-order nonlinear equation in n dimensions is given which is related to a functional differential equation in statistical mechanics. It is reducible to two dimensions for any value of n>or=2.