Multicomponent solution calculations can be complicated for students and practiced chemists alike. This article describes how to simplify the calculations by representing a solution's composition as a point in a "concentration space," whose axes are the concentrations of each solute. The graphical representation of mixing processes in a concentration space diagram enables qualitative insights such as whether a mixture can be made from a set of stock solutions and the relative volume proportions needed, solely by visual inspection and without detailed calculation. The diagrams correspond to elementary linear algebra equations which facilitates the specification of quantitative calculations. The relationship of the solution mixing problem to convex analysis is discussed. Example applications of this approach to problems of calculating the composition following mixing, determining the volumes to use to create a specified mixture, selecting a set of stock solutions maximizing the number of possible unique mixtures that can be made, random experiment sampling, and common-ion effect problems are demonstrated.