SOLUTION: Determine whether the following system of equations are consistent, independent and inconsistent t-u+2v-3w=9 4t+11v-10w=46 3t-u+8v-6w=27.

Your matrix

        X1	X2	X3	X4	b
1	1	-1	2	-3	9
2	4	0	11	-10	46
3	3	-1	8	-6	27

Find the pivot in the 1st column in the 1st row

        X1	X2	X3	X4	b
1	1	-1	2	-3	9
2	4	0	11	-10	46
3	3	-1	8	-6	27

Multiply the 1st row by 4

        X1	X2	X3	X4	b
1	4	-4	8	-12	36
2	4	0	11	-10	46
3	3	-1	8	-6	27

Subtract the 1st row from the 2nd row and restore it

        X1	X2	X3	X4	b
1	1	-1	2	-3	9
2	0	4	3	2	10
3	3	-1	8	-6	27

Multiply the 1st row by 3

        X1	X2	X3	X4	b
1	3	-3	6	-9	27
2	0	4	3	2	10
3	3	-1	8	-6	27

Subtract the 1st row from the 3rd row and restore it

        X1	X2	X3	X4	b
1	1	-1	2	-3	9
2	0	4	3	2	10
3	0	2	2	3	0

Make the pivot in the 2nd column by dividing the 2nd row by 4

        X1	X2	X3	X4	b
1	1	-1	2	-3	9
2	0	1	3/4	1/2	5/2
3	0	2	2	3	0

Multiply the 2nd row by -1

        X1	X2	X3	X4	b
1	1	-1	2	-3	9
2	0	-1	-3/4	-1/2	-5/2
3	0	2	2	3	0

Subtract the 2nd row from the 1st row and restore it

        X1	X2	X3	X4	b
1	1	0	11/4	-5/2	23/2
2	0	1	3/4	1/2	5/2
3	0	2	2	3	0

Multiply the 2nd row by 2

        X1	X2	X3	X4	b
1	1	0	11/4	-5/2	23/2
2	0	2	3/2	1	5
3	0	2	2	3	0

Subtract the 2nd row from the 3rd row and restore it

        X1	X2	X3	X4	b
1	1	0	11/4	-5/2	23/2
2	0	1	3/4	1/2	5/2
3	0	0	1/2	2	-5

Make the pivot in the 3rd column by dividing the 3rd row by 1/2

        X1	X2	X3	X4	b
1	1	0	11/4	-5/2	23/2
2	0	1	3/4	1/2	5/2
3	0	0	1	4	-10

Multiply the 3rd row by 11/4

        X1	X2	X3	X4	b
1	1	0	11/4	-5/2	23/2
2	0	1	3/4	1/2	5/2
3	0	0	11/4	11	-55/2

Subtract the 3rd row from the 1st row and restore it

        X1	X2	X3	X4	b
1	1	0	0	-27/2	39
2	0	1	3/4	1/2	5/2
3	0	0	1	4	-10

Multiply the 3rd row by 3/4

        X1	X2	X3	X4	b
1	1	0	0	-27/2	39
2	0	1	3/4	1/2	5/2
3	0	0	3/4	3	-15/2

Subtract the 3rd row from the 2nd row and restore it

        X1	X2	X3	X4	b
1	1	0	0	-27/2	39
2	0	1	0	-5/2	10
3	0	0	1	4	-10


Solution set:

x1 = 39 + (27/2)x4
x2 = 10 + (5/2)x4
x3 = -10 - 4x4

x4 - free