Abstract: In this paper a parallel algorithm to solve linear equation systems is presented. This method, known as Neville elimination, is appropriate especially for the case of a totally positive matrix (all its minors are non-negative). We prove that this algorithm is cost-optimal for a given parallel implementation of Neville elimination, in which the coefficient matrix is rowwise stripe-partitioned among the processors. In case of Gaussian elimination it is necessary a pipelined version to obtain the optimal cost. Furthermore, experimental results obtained on an IBM SP2 multicomputer using MPI corroborate the theoretic estimation about the algorithm efficiency.