x + 2y +  z = 2

3x +  y - 2z = 1 

4x - 3y -  z = 3


Your augmented matrix

	X1	X2	X3	b
1	1	2	1	2
2	3	1	-2	1
3	4	-3	-1	3

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

	X1	X2	X3	b
1	1	2	1	2
2	3	1	-2	1
3	4	-3	-1	3

Multiply the 1st row by 3

	X1	X2	X3	b
1	3	6	3	6
2	3	1	-2	1
3	4	-3	-1	3

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

	X1	X2	X3	b
1	1	2	1	2
2	0	-5	-5	-5
3	4	-3	-1	3

	Multiply the 1st row by 4

	X1	X2	X3	b
1	4	8	4	8
2	0	-5	-5	-5
3	4	-3	-1	3

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

	X1	X2	X3	b
1	1	2	1	2
2	0	-5	-5	-5
3	0	-11	-5	-5

Make the pivot in the 2nd column by dividing the 2nd row by -5

	X1	X2	X3	b
1	1	2	1	2
2	0	1	1	1
3	0	-11	-5	-5

Multiply the 2nd row by 2

	X1	X2	X3	b
1	1	2	1	2
2	0	2	2	2
3	0	-11	-5	-5

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

	X1	X2	X3	b
1	1	0	-1	0
2	0	1	1	1
3	0	-11	-5	-5

Multiply the 2nd row by -11

	X1	X2	X3	b
1	1	0	-1	0
2	0	-11	-11	-11
3	0	-11	-5	-5

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

	X1	X2	X3	b
1	1	0	-1	0
2	0	1	1	1
3	0	0	6	6

Make the pivot in the 3rd column by dividing the 3rd row by 6

	X1	X2	X3	b
1	1	0	-1	0
2	0	1	1	1
3	0	0	1	1

Multiply the 3rd row by -1

	X1	X2	X3	b
1	1	0	-1	0
2	0	1	1	1
3	0	0	-1	-1

Subtract the 3rd row from the 1st row

	X1	X2	X3	b
1	1	0	0	1
2	0	1	1	1
3	0	0	-1	-1

Multiply the 3rd row by -1

	X1	X2	X3	b
1	1	0	0	1
2	0	1	1	1
3	0	0	1	1

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

	X1	X2	X3	b
1	1	0	0	1
2	0	1	0	0
3	0	0	1	1


ANSWER.  The solution is (x,y,z) = (1,0,1).