SOLUTION: solve the linear system by Gauss–Jordan elimination. − 2b + 3c = 1 3a + 6b − 3c = −2 6a + 6b + 3c = 5

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Question 1122545: solve the linear system by Gauss–Jordan
elimination.
− 2b + 3c = 1
3a + 6b − 3c = −2
6a + 6b + 3c = 5
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
-2b + 3c = 1
3a + 6b - 3c = -2
6a + 6b + 3c = 5
-------------------- Add row 2 & 3
9a + 12b = 3 --> 3a + 4b = 1
=========
0a - 2b + 3c = 1
3a + 6b - 3c = -2
-------------------- Add row 1 & 2
3a + 4b = -1
-------------------
Inconsistent.
===============
GJE is not a good choice for this one, IMO.
If needed, look it up on Wikipedia.

Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.
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: 

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

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.