Solve the third equation for x.

-2x - 2z = 62

-2x = 62 + 2z

x = (62 + 2z) / -2

x = -31 - z

Substitute this expression in for x in the first equation.

-x - 5y - z = 211

-(-31 - z) - 5y - z = 211

31 + z - 5y - z = 211

31 - 5y = 211

-5y = 180

y = 180 / -5

y = -36

Now substitute (-31 - z) in for x and -36 in for y in the second equation and solve for z.

11x + 8y - 5z = -213

11(-31 - z) + 8(-36) - 5z = -213

-341 - 11z - 288 - 5z = -213

-629 - 16z = -213

-16z = 416

z = -26

We know from the third equation that x = -31 - z.  Substitute -26 in for z and solve for x.

x = -31 - z

x = -31 - (-26)

x = -5

Solution:  {x = -5, y = -36, z = -26}

Check these answers using substitution.

-x - 5y - z = 211
-(-5) - 5(-36) - (-26) = 211
211 = 211

11x + 8y - 5z = -213
11(-5) + 8(-36) - 5(-26) = -213
-213 = -213

-2x - 2z = 62
-2(-5) - 2(-26) = 62
62 = 62

The solutions work in all three equations.