We open with a survey of the traditional methods for solving a system of linear equations, culminating in a very thorough description of Gauss elimination. To pique the interests of readers who have already mastered the algorithm, we insert tidbits of information that would not have occurred to them, such as operation counts and computer timing, pivoting, complex coefficients, parametrized-solution characterizations of underdetermined systems, and examples of the logical pitfalls that can arise when one fails to adhere strictly to Gauss’s instructions. Readers are guided to explore projects involving LU factorizations and the role of linear systems in various areas of application.