As you've probably noticed, an unsystematic approach to the elimination procedure can be extremely tedious, very time consuming and prone to error.
The Gauss and Gauss-Jordan methods organize the elimination process into a systematic procedure (or algorithm) which can be applied relatively easily to a fairly large linear ( »< 100 unknowns)
Systematic procedure ie. anable to programming...
there is a well-defined algoriyhm
S = 1 + 2 + 3 + ... + 100
S = 100 + 99 + 98 + ... + 1
2S = 101 + 101 + 101 + ... + 101
2S = (101) 100 = 10100
\ S = 5050
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