Find z in the unique solution of the systemx + 2y + 3z =1-x - y + 3z = 2-6x + y + z = -2

We'll add 1st and the 2nd equations to eliminate
x:

x + 2y + 3z - x - y + 3z = 1 +
2

We'll combine like terms:

y
+ 6z = 3 (4)

We'll multiply the 1st equation by
6:

6x + 12y + 18z = 6
(5)

We'll add (5) and (3) to eliminate
x:

6x + 12y + 18z - 6x + y + z = 6 -
2

13y + 19z = 4(6)

We'll
eliminate y from (4) and (6)

We'll multiply (4) by -13 and
(6) by 6:

13y - 78z = -39
(7)

We'll add (7) and
(6):

-13y - 78z  + 13y +
19z= -39+4

We'll eliminate and combine like
terms:

-59z =
-35

z =
35/59