Linear systems arise in virtually every area of science and engineering—some are as big as 1,000,000 equations in as many unknowns. Such huge systems require more sophisticated treatment than the methods introduced here. They will allow you to solve systems with several thousand equations without a problem. This chapter explains the basic ideas underlying linear systems, starting with Gauss elimination with pivoting, incorporating the Gauss matrix. We revisit the idea that the forward elimination step is a shear, represented as elementary matrix. Existence and uniqueness questions are addressed. LU decomposition optimizes the Gauss elimination steps. Linear least squares approximation is described, and the method is expanded upon with a companion application of sphere fitting. Other applications of linear systems in this chapter are polynomial interpolation and data smoothing. Sketches and figures illustrate the concepts.