Augmented matrices are a topic of Linear Algebra that is commonly taught in Precalculus classes for some reason. It formalizes the concept of a linear system, and while that sounds wonderful to a mathematician it can actually be a royal pain when it comes right down to running numbers around.

Even as student in college level math courses, Gaussian elimination (the method used to solve these matrices) is a task that causes frequent errors, so I decided to take a very slow, steady approach to the topic to better inform confused juniors and seniors across America [that sounded less cheesy in my head].