Discrete Mathematics..

352 Systems of Linear EquationsSolution⎛⎝ 2 2 1 4 ⎞1 −1 −1 2 ⎠ (augmented matrix)3 1 0 6⎛⎞11 122 [R 1 → (R 1 ÷ 2)]⎜⎟∼ ⎝ 1 −1 −1 2 ⎠3 1 0 6⎛1⎞1 1 22⎜∼ ⎝ 0 −2 − 3 ⎟20 ⎠ [R 2 → (R 2 − R 1 )]0 −2 − 3 20 [R 3 → (R 3 − 3R 1 )]⎛1⎞1 122⎜∼3 ⎟⎝ 0 1 40 ⎠ [R 2 → (R 2 ÷ (−2))]0 −2 − 3 20⎛⎞11 1 22⎜∼3 ⎟⎝ 0 1 40 ⎠0 0 0 0 [R 3 → (R 3 + 2R 2 )].This represents the system of equationsx + y + 1 2 z = 2y + 3 4 z = 00z = 0.Writing z = t, wehavey + 3 4 t = 0(from the second equation) so thaty =− 3 4 t.Alsox − 3 4 t + 1 2 t = 2(from the first equation) so thatx = 2 + 1 4 t.In a consistent set of equations, Gaussian elimination results in a final matrix fromwhich the value of at least one of the variables can be read off directly or assigned