Gaussan Metod

Neciš metodas tazostusiden sistem andase koumesaumaižen nägun ekvivalentišt sistemad elementarižiden toižetamižiden abul, vajehtujiden jäl'gendajanke erindanke.

Hot' nece metod kuctas Gaussan metodaks (Karl Fridrih Gaussan oiktastuseks), se oli tetaban kitajalaižil völ, «Matematik ühesas kirjas»-traktatas.

Augotižsistem:
${\displaystyle \left\{{\begin{array}{lcr}a_{11}x_{1}+\ldots +a_{1n}x_{n}&=&b_{1}\\\ldots &&\\a_{m1}x_{1}+\ldots +a_{mn}x_{n}&=&b_{m}\\\end{array}}\right.\iff Ax=b,\quad A=\left({\begin{array}{ccc}a_{11}&\ldots &a_{1n}\\\ldots &&\\a_{m1}&\ldots &a_{mn}\end{array}}\right),\quad x=\left({\begin{array}{c}x_{1}\\\vdots \\x_{n}\end{array}}\right),\quad b=\left({\begin{array}{c}b_{1}\\\vdots \\b_{m}\end{array}}\right).\quad (1)}$ 

Jäl'ges elementarišt toižetamišt:
${\displaystyle \left\{{\begin{array}{rcl}\alpha _{1j_{1}}x_{j_{1}}+\alpha _{1j_{2}}x_{j_{2}}+\ldots +\alpha _{1j_{r}}x_{j_{r}}+\ldots +\alpha _{1j_{n}}x_{j_{n}}&=&\beta _{1}\\\alpha _{2j_{2}}x_{j_{2}}+\ldots +\alpha _{2j_{r}}x_{j_{r}}+\ldots +\alpha _{2j_{n}}x_{j_{n}}&=&\beta _{2}\\&\ldots &\\\alpha _{rj_{r}}x_{j_{r}}+\ldots +\alpha _{rj_{n}}x_{j_{n}}&=&\beta _{r}\\0&=&\beta _{r+1}\\&\ldots &\\0&=&\beta _{m}\end{array}}\right.,\qquad \alpha _{1j_{1}},\ldots ,\alpha _{rj_{r}}\neq 0.}$ 

Jäl'gused:

1. Ku ühthižes sistemas kaik vajehtujad oma pävajehtujad, znamoičeb, ningoi sistem om märhapandud.
2. Ku vajehtujiden verd sistemas om enamba mi tazostusiden verd, znamoičeb, ningoi sistem om märhapanmatoi vai ühthitoi.