The parametric solution of a linear system of inequalities Ax≤Bb, with parameter b, is considered. Fourier elimination is used to give a facial representation for the set of b-values for which the system is consistent. Some interesting applications of the problem are discussed. Although the worst case complexity of the method is an exponential function of the size of A, the computations are intuitive and very simple. © 1990 Plenum Publishing Corporation.