[compumike]

For a gentler introduction to LU factorization and how useful the decomposition is for efficiently re-solving the same system multiple times, I'd offer up the "Systems of Equations" section of the "Ultimate Electronics" book I've been working on: https://www.circuitlab.com/textbook/systems-of-equations/

I tried to go back and forth between the 5x+2y=3 style equations that most people are familiar with and the matrix forms.