Question

3x+4y=24. What is the smallest integer solution for x+y?

Answer 1

lets solve for x, and then we'll find an integer solution:
3x + 4y = 24
3x = 24 - 4y
x = (24 - 4y)/3
so the right side has to be an integer, or 24 - 4y has to be a multiple of 3:
24 - 4y
then, y has to be 3, see:
x = (24 - 4y)/3
x = (24 - 4(3))/3
x = (24 - 12)/3
x = (12)/3
x = 4
then, we have x = 4 and y = 3

Answer 2

Since we are looking for integer solutions, x can get really small, as long as it remains a multiple of 12. This ensures that y can get it back to 24. Therefore, the answer is -infinity.