  Systems of random linear equations may or may not have solutions with all
components being non-negative. The question is, e.g., of relevance when the
unknowns are concentrations or population sizes. In the present paper we show
that if such systems are large the transition between these two possibilities
occurs at a sharp value of the ratio between the number of unknowns and the
number of equations. We analytically determine this threshold as a function of
the statistical properties of the random parameters and show its agreement with
numerical simulations. We also make contact with two special cases that have
been studied before: the storage problem of a perceptron and the resource
competition model of MacArthur.
