Find root of system (correct me)

Original system:

 

L1:      3x-3y+z=13

L2:      2x+5y-z=-5

L3:      -x+7y+3z=3

 

-3 L2:    -6x + -15y + 3z = 15

+2 L1:    3x - 6y + 2z = 26

 

-3 L2 + 2 L1 :  -21y + 5z = 41

 

- L1:     -3x + 3y - z = -13

-3 L3:    3x - 21y - 9z= -9

 

- L1 + (-3 L3): -18y - 10z = -22

 

New system:

 

L1:       3x-3y+z=13

L2:       21y + 5z = 41

L3:      -18y - 10z = -22

 

6 L2:      -126y + 30z = 246

-7 L3:      126y + 70z = 154

 

6 L2 + (-7 L3): 100z = 400

 

New system: 

 

L1:     3x-3y+z=13

L2:     -21y + 5z = 41

L3:     100z = 400

 

z=4

 

Replace z in L2:

 

-21y + 5z = 41

-21y + 5 (4) = 41

-21y +20 = 41

-21y = 21

 

y= -1

 

 

Replace y and z in L1:

 

3x - 3y + z = 13

3x - 3(-1) + 4 = 13

3x +7 = 13

3x=6

 

x=2

 

Only solution would be: (2, -1, 4)

 

Am I right?