-6x+6z=-60 can be simplified: x-z=10, from which z=x-10.
There's a problem because 10x-4y+4z is an expression when it should be an equation, so let's make it an equation: 10x-4y+4z=A where A also has to be found (intuitively).
Substitute for z in the 1st and 2nd equations:
10x-4y+4(x-10)=A, 10x-4y+4x-40=A, 14x-4y=A+40;
8x+2y+12(x-10)=-64,
8x+2y+12x-120=-64,
20x+2y=56, 10x+y=28, y=28-10x.
14x-4(28-10x)=A+40,
14x-112+40x=A+40,
54x=A+152=A+108+44,
x=2+(A+44)/54.
Intuitively for an all-integer solution, A could be -44, 10, 64, etc. Let's make A=10, then x=2+1=3.
y=28-10x=28-30=-2, and z=x-10=3-10=-7.
Equation 1 becomes 10x-4y+4z=10.
So a solution could be (x,y,z)=(3,-2,-7).
Other solutions using A=54n-44 where n is an integer will give different values for x, y and z.