The sum of three numbers is 11. the first number minus the second plus the third is 5. the first minus the third is 1 more than the second. find the numbers.

X+y+z=11, x-y+z=5, x-z=y+1 and solve it by any of the methods: isolation, elimination or determinants. isolation (or back substitution, would be for instance: z = 11-x-y, z=5-x+y, z = x-y-1 and then impose z=z:
11-x-y=5-x+y,
11-x-y=x-y-1,

The first takes you to: 6=2y or y=3, then the second to 12=2x or x=6, so that:

x=6, y=3, z=2.

You can see that it checks out

What do you mean by that?