Our model is a generalized linear programming relaxation of a much studied random K-SAT problem. Specifically, a set of linear constraints C on K variables is fixed. From a pool of n variables, K variables are chosen uniformly at random and a constraint is chosen from C also uniformly at random. This procedure is repeated m times independently. We are interested in whether the resulting linear programming problem is feasible. We prove that the feasibility property experiences a linear phase transition, when n→∞ and m=cn for a constant c. Namely, there exists a critical value c* such that, when c < c*, the problem is feasible or is asymptotically almost feasible, as n→∞, but, when c>c*, the "distance" to feasibility is at least a positive constant independent of n. Our result is obtained using the combination of a powerful local weak convergence method developed in Aldous [1992, 2000], Aldous and Steele [2003], Steele [2002] and martingale techniques. By exploiting a linear programming duality, our theorem implies the following result in the context of sparse random graphs G(n, cn) on n nodes with cn edges, where edges are equipped with randomly generated weights. Let M(n,c) denote maximum weight matching in G(n, cn). We prove that when c is a constant and n→∞, the limit limn→∞ M(n,c)/n, exists, with high probability. We further extend this result to maximum weight b-matchings also in G(n,cn).