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

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We have three equations to solve for z.

x + 2y + 3z =1 ...(1)

-x - y + 3z = 2 ...(2)

-6x + y + z = -2 ...(3)

x + 2y + 3z =1

=> x = 1 - 2y - 3z ...(4)

substitute in (2)

-x - y + 3z = 2

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

=> y + 6z = 3

=> y = 3 - 6z

Substitute in (4)

=> x = 1 - 6 + 12z - 3z

=> x = -5 + 9z ... (5)

Substitute x and y from (5) and (4) resp. in (3)

-6(-5 + 9z) + 3 - 6z + z = -2

=> 30 - 54z + 3 - 6z + z = -2

=> -59z = -35

=>** z = 35/59**

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**