Consider a large linear system with random underlying matrix: x_n = 1_n + 1/(α_n √β_n) M_n x_n, where x_n is the unknown, 1_n is a vector of ones, M_n is a random matrix and α_n, β_n are scaling parameters to be specified. We investigate the componentwise positivity of the solution x n depending on the scaling factors, as the dimensions of the system grow to infinity. We consider 2 models of interest: The case where matrix M_n has independent and identically distributed standard Gaussian random variables, and a sparse case with a growing number of vanishing entries.In each case, there exists a phase transition for the scaling parameters below which there is no positive solution to the system with growing probability and above which there is a positive solution with growing probability.These questions arise from feasibility and stability issues for large biological communities with interactions.