In this chapter, we deal with the emergence of linear systems of equations based on physically and practically relevant problems. The discretizations of the Poisson equation and the convection-diffusion equation presented here illustrate the occurrence of systems of equations with a sparse matrix. In addition, we explain how linear integral equations of the second kind, as well as linear least squares problems and polynomial interpolation, lead to systems of equations with a dense matrix. The sparse matrices derived in the first two examples also serve as model problems for the analysis of the methods described in the following sections.