Unification and Gaussian Elimination
I don't consider this an answer. I'm abusing the answer box to pretty print a comment.
There is a strict sense in which Euclid's GCD algorithm, Gaussian elimination, Buchberger's algorithm and Knuth-Bendix form a strict sequence of generalisations and are all instances of what is called a completion algorithm. There is also a close relationship between these algorithms and resolution in logic. I do not know a good reference for this but I have seen the fact mentioned very often. These might help.

History and basic features of the critical-pair/completion procedure, Bruno Buchberger, 1987
Canonical Reduction Systems in Symbolic Mathematics, Franz Winkler. Springer Link

Let me know if you find better references.