Find the number of integral solutions of x+y +z = 12, where −3 ≤ x ≤ 4, 2 ≤ y ≤ 11, and z ≥ 3.

Answer 1

x y z

-3 2 13

-3 3 12

-3 4 11

-3 5 10

-3 6 9

-3 7 8

-3 8 7

-3 9 6

-3 10 5

-3 11 4

10 solutions with x =-3

-2 2 12

through

-2 11 3

10 more with x=-2

-1 2 11

through

-1 10 3

9 more with x =-1

0 2 10

through

0 9 3

8 more with x=0

1 2 9

through

1 8 3

7 more with x= 1

2 2 8

through

2 7 3

6 more with x=2

3 2 7

through

3 6 3

5 more with x=3

4 2 6

through

4 5 3

4 more with x = 4

4+5+6+7+8+9+10+10 = 59 integer solutions

when -2<x<4, 2<y<11 amd z>3

and x+y+z=12