How do you solve using gaussian elimination or gauss-jordan elimination, $x_{1} + 2 x_{2} - x_{3} + 3 x_{4} = 2$ $x_1 +2x_2 − x_3 +3x_4 =2$ , $2 x_{1} + x_{2} + x_{3} + 3 x_{4} = 1$ $2x_1 + x_2 + x_3 +3x_4 =1$ , $3 x_{1} + 5 x_{2} - 2 x_{3} + 7 x_{4} = 3$ $3x_1 +5x_2 − 2x_3 +7x_4 =3$ , $2 x_{1} + 6 x_{2} - 4 x_{3} + 9 x_{4} = 8$ $2x_1 +6x_2 − 4x_3 +9x_4 =8$ ?

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Answer 1 · 2017-08-07T12:42:55

The solutions are $x_{4} = 2$ $x_4=2$ , $x_{3}$ $x_3$ free, $x_{1} = - 2 - x_{3}, x_{2} = - 1 + x_{3}$ $x_1=-2-x_3, x_2=-1+x_3$

We perform the Gauss Jordan elimination with the augmented matrix

$((1,2,-1,3,:,2),(2,1,1,3,:,1),(3,5,-2,7,:,3),(2,6,-4,9,:,8))$

$R4larrR4-2R1$ , $=>$ , $((1,2,-1,3,:,2),(2,1,1,3,:,1),(3,5,-2,7,:,3),(0,2,-2,3,:,4))$

$R3larrR3-3R1$ , $=>$ , $((1,2,-1,3,:,2),(2,1,1,3,:,1),(0,-1,1,-2,:,-3),(0,2,-2,3,:,4))$

$R2larrR2-2R1$ , $=>$ , $((1,2,-1,3,:,2),(0,-3,3,-3,:,-3),(0,-1,1,-2,:,-3),(0,2,-2,3,:,4))$

$R2larrR1-R4$ , $=>$ , $((1,0,1,0,:,-2),(0,-3,3,-3,:,-3),(0,-1,1,-2,:,-3),(0,2,-2,3,:,4))$

$R4larrR4+2R3$ , $=>$ , $((1,0,1,0,:,-2),(0,-3,3,-3,:,-3),(0,-1,1,-2,:,-3),(0,0,0,-1,:,-2))$

$R4larr(R4)/(-1)$ , $=>$ , $((1,0,1,0,:,-2),(0,-3,3,-3,:,-3),(0,-1,1,-2,:,-3),(0,0,0,1,:,2))$

$R2larr(R2)/(-3)$ , $=>$ , $((1,0,1,0,:,-2),(0,1,-1,1,:,1),(0,-1,1,-2,:,-3),(0,0,0,1,:,2))$

$R3larrR3+R2$ , $=>$ , $((1,0,1,0,:,-2),(0,1,-1,1,:,1),(0,0,0,-1,:,-2),(0,0,0,1,:,2))$

$R3larr(R3)/(-1)$ , $=>$ , $((1,0,1,0,:,-2),(0,1,-1,1,:,1),(0,0,0,1,:,2),(0,0,0,1,:,2))$

$R4larrR4-R3$ , $=>$ , $((1,0,1,0,:,-2),(0,1,-1,1,:,1),(0,0,0,1,:,2),(0,0,0,0,:,0))$

$R2larrR2-R3$ , $=>$ , $((1,0,1,0,:,-2),(0,1,-1,0,:,-1),(0,0,0,1,:,2),(0,0,0,0,:,0))$

The solutions are $x_4=2$ , $x_3$ free, $x_1=-2-x_3, x_2=-1+x_3$