Your matrix

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

Find the pivot in the 1st column and swap the 2nd and the 1st rows

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

Eliminate the 1st column

	X1	X2	X3	b
1	1	-3	1	4
2	0	5	-1	-5
3	0	-15	3	23

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

	X1	X2	X3	b
1	1	-3	1	4
2	0	1	-1/5	-1
3	0	-15	3	23

Eliminate the 2nd column

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

Solution set:

The system is inconsistent.