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**

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