x - 2y + z = 5 | *2
⇒ 2x - 4y+ 2z=10
3x + 3y - 2z = - 6 } I sum up these relations
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2x+3x - 4y+3y+2z-2z=10-6
5x - y = 4 (1)
3x + 3y - 2z = - 6 | *3 ⇒ 9x + 9y - 6z = - 18
2x - y + 3z = 11 | *2 ⇒ 4x - 2y + 6z= 22 I sum up these
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⇒ 9x+4x+9y-2y-6z+6z= 4
13x+ 7y= 4 (2)
I write (1) and (2)
5x - y = 4 | *7
35x - 7y= 28
13x+7y=4
48x = 32
x= 32/48=4/6 ( 32:8=4, 48:8=6)
x= 2/3
5x-y=4,
5*2/3-y=4
y=10/3 -4=10/3-12/3=-2/3
⇒ y= - 2/3
x - 2y + z = 5
2/3 - 2*(-2/3)+z=5
2/3+4/3+z=5
6/3+z=5
2+z=5
z=3
x+y+z=2/3-2/3+3=3
x+y+z=3