To perform Gaussian elimination, the coefficients of the terms in the system of linear equations are used to create a type of matrix called an augmented matrix. Then, elementary row operations are used to simplify the matrix. The three types of row operations used are:
t ə WORD_BOUNDARY p ə ɹ f ɔ ɹ m WORD_BOUNDARY ɡ aʊ ʃ ə n WORD_BOUNDARY ɪ l ɪ m ɪ n eɪ ʃ ə n WORD_BOUNDARY ð ə WORD_BOUNDARY k oʊ ɪ f ɪ ʃ ə n t s WORD_BOUNDARY ʌ v ð ə WORD_BOUNDARY t ɜː m z WORD_BOUNDARY ɪ n ð ə WORD_BOUNDARY s ɪ s t ə m WORD_BOUNDARY ʌ v WORD_BOUNDARY l ɪ n ɪ ɹ WORD_BOUNDARY ɪ k w eɪ ʒ ə n z WORD_BOUNDARY ɑ ɹ WORD_BOUNDARY j uː z d WORD_BOUNDARY t ə WORD_BOUNDARY k ɹ iː eɪ t WORD_BOUNDARY ʌ WORD_BOUNDARY t aɪ p WORD_BOUNDARY ʌ v WORD_BOUNDARY m eɪ t ɹ ɪ k s WORD_BOUNDARY k ɔ l d WORD_BOUNDARY ʌ n WORD_BOUNDARY ɔ ɡ m ɛ n t ɪ d WORD_BOUNDARY m eɪ t ɹ ɪ k s WORD_BOUNDARY ð ɛ n WORD_BOUNDARY ɛ l ɪ m ɛ n t ə ɹ ɹ i WORD_BOUNDARY ɹ oʊ WORD_BOUNDARY ɑ p ə ɹ ɹ eɪ ʃ ə n z WORD_BOUNDARY ɑ ɹ WORD_BOUNDARY j uː z d WORD_BOUNDARY t ə WORD_BOUNDARY s ɪ m p l ɪ f aɪ WORD_BOUNDARY ð ə WORD_BOUNDARY m eɪ t ɹ ɪ k s WORD_BOUNDARY ð ə WORD_BOUNDARY θ ɹ iː WORD_BOUNDARY t aɪ p s WORD_BOUNDARY ʌ v WORD_BOUNDARY ɹ oʊ WORD_BOUNDARY ɑ p ə ɹ ɹ eɪ ʃ ə n z WORD_BOUNDARY j uː z d WORD_BOUNDARY ɑ ɹ WORD_BOUNDARY