The idea here is to combine the equations into one in only one variable. The last equation is in only x and z so lets sub out y in the second.
-x - y - z = -8
8 - x - z = y
Substitute y with (8 - x - z) into the next equation.
-4x + 4(8 - x - z) + 5z = 7
-4x + 32 - 4x - 4z + 5z = 7
-8x + 32 + z = 7
The variable z is easiest to get alone; use that as substitution in the last equation.
z = 7 + 8x - 32
z = 8x - 25
Substitute z with (8x - 25)
2x + 2(8x - 25) = 4
2x + 16x - 50 = 4
18x = 4 + 50
18x = 54
x = 3
So use x = 3 in that last equation to find z.
2(3) + 2z = 4
6 + 2z = 4
2z = 4 - 6
2z = -2
z = -1
Now, use both x = 3 and z = -1 to find the final variable y.
-3 - y - (-1) = -8
-3 - y + 1 = -8
-2 - y = -8
-y = -8 + 2
-y = -6
y = 6
Solutions x,y,z = ( 3, 6, -1)