Balancing chemical equations is a basic problem in chemistry. An easy and commonly employed method is to convert the task to a linear algebra problem. We use the row-reduced echelon form of the coefficient matrix which gives directly a basis for its null space. The dimension of the null space indicates how many basic mathematically balanced chemical equations are necessary to generate all balanced reactions. If we impose the reactants and the products, the algebraic structure of the solution set is not a subspace but a cone. The problem is now to determine all extreme rays of a cone, or all extreme points of a convex polyhedral set, it is a vertex enumeration problem, which is a hard one in algorithmic.