Question 1122545
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I used the online free of charge matrix solver  
https://matrix.reshish.com/gaussSolution.php


which, in particular, makes the Gauss-Jordan elimination.


So, I inputted the augmented matrix and pressed the "Solve" button.


The solver does not select a most efficient way, but, in any case, makes his work from the begginning to the very end.


Below see the results:


<pre>
Your matrix

      X1	X2	X3	b
1	0	-2	3	1
2	3	6	-3	-2
3	6	6	3	5


Make the pivot in the 1st column by dividing the 2nd row by 3 and swap the 2nd and the 1st rows

      X1	X2	X3	b
1	1	2	-1	-2/3
2	0	-2	3	1
3	6	6	3	5


Multiply the 1st row by 6

      X1	X2	X3	b
1	6	12	-6	-4
2	0	-2	3	1
3	6	6	3	5


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

      X1	X2	X3	b
1	1	2	-1	-2/3
2	0	-2	3	1
3	0	-6	9	9


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

      X1	X2	X3	b
1	1	2	-1	-2/3
2	0	1	-3/2	-1/2
3	0	-6	9	9


Multiply the 2nd row by 2

      X1	X2	X3	b
1	1	2	-1	-2/3
2	0	2	-3	-1
3	0	-6	9	9


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

      X1	X2	X3	b
1	1	0	2	1/3
2	0	1	-3/2	-1/2
3	0	-6	9	9


Multiply the 2nd row by -6

      X1	X2	X3	b
1	1	0	2	1/3
2	0	-6	9	3
3	0	-6	9	9


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

      X1	X2	X3	b
1	1	0	2	1/3
2	0	1	-3/2	-1/2
3	0	0	0	6
</pre>

Do you see this last line of the transformed matrix which consists of zero coefficients with non-zero right side ?


It means that the system is INCONSISTENT.