I show that sharp identified sets in a large class of econometric models can be characterized by solving linear systems of equations. These linear systems determine whether, for a given value of a parameter of interest, there exists an admissible joint distribution of unobservables that can generate the distribution of the observed variables. The parameter of interest can be a structural function, but it can also be a more complicated feature of the model primitives, such as an average treatment effect. The joint distribution of unobservables is not required to satisfy any parametric restrictions, but can (if desired) be assumed to satisfy a variety of location, shape and/or conditional independence restrictions. To prove that this characterization is sharp, I generalize a classic lemma in copula theory concerning the extendibility of subcopulas to show that related objects, termed subdistributions, can be extended to proper distribution functions. This result is then used to reduce the characterization of the identified set to the determination of the existence or non-existence of suitably-constrained subdistributions, which in turn is often equivalent to solving a linear system of equations. I describe this argument as partial identification by extending subdistributions, or PIES. I apply PIES to an ordered discrete response model and a two-sector Roy model. One product of the first application is a tractable characterization of the sharp identified set for the average treatment effect in the semiparametric binary response model considered by Manski (1975, 1985, 1988).