<p>This paper describes new approaches to the solution of a sequence of large linear systems of equations or large linear least squares problems with the same matrix and several right-hand side vectors that represent data. We consider both the situations when the matrix of the systems to be solved is fairly well-conditioned and when the matrix is very ill-conditioned. In the latter case regularization is applied. We are concerned with the situation when the matrix is too large to make the application of direct solution methods possible or attractive. Our solution methods apply flexible Arnoldi or flexible Golub-Kahan decompositions. These decompositions allow the solution subspace computed during the solution of a seed system to be expanded by residual vectors that are computed during the solution of subsequent systems. Computed examples illustrate the competitiveness of the proposed methods.</p>