-x-y-z=-8
-4x+4y+5z=7
2x+2z=4

y does not appear in the third equation so:

solve the first equation for y and then substitute 
that for y in the right side of the 2nd equation,
then we'll have 2 equations in 2 unknowns:

Solve -x-y-z=-8 for y:
          -y=-8+x+z
           y=8-x-z

Substitute 8-x-z for y in -4x+4y+5z=7
                    -4x+4(8-x-z)+5z=7
                    -4x+32-4x-4z+5z=7
                            32-8x+z=7
                              -8x+z=-25
                              
                                  
Put that with the original 3rd equation.
Now we have 2 equations in 2 unknowns:

2x+2z=4
-8x+z=-25

We solve the 2nd equation just above for z 
and substitute what we get in the 1st 
equation and solve for x:

Solve -8x+z=-25 fo r z
          z=-25+8x

Substitute -25+8x for z in 2x+2z=4
                    2x+2(-25+8x)=4
                       2x-50+16x=4
                          18x-50=4
                             18x=54
                               x=3

Substitute 3 for x in z=-25+8x
                      z=-25+8(3)
                      z=-25+24
                      z=-1

Substitute 3 for x and -1 for z in y=8-x-z 
                                   y=8-3-(-1)
                                   y=8-3+1
                                   y=6

Solution: (x,y,z) = (3,6,-1)

Edwin