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 impliesthe 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 $\mathcal{M}(n,c)$ denote maximum weight matching in $G(n, cn)$. We prove that when $c$ is a constant and $n→∞$, the limit $lim_{n→∞} \mathcal{M}(n,c)/n$, exists, with high probability. We further extend this result to maximum weight b-matchings also in $G(n,cn)$.