find all nonnegative integer solutions of the system

Question

x+y<8
x+t<8
t+y<8
z+t<8

Answer 1

if t=0, we have
x<8
y<8
z<8
x+y<8
so, 0<x<8, y<8-x, z<8
List them all.

If t=1, change the 8's to 7's in the first 3 conditions.

And so on, up to t=7:
x<1
y<1
z<1
x+y<8
(x,y,z,t) = (0,0,0,7)

Don't see the point of the exercise, but there it is.

Answer 2

sory a mistake, the corect one problem is find all nonnegative integer solutions of the system
x+y<8
x+Z<8
t+y<8
z+t<8