Since the result for X + Y in the ones column is not the same as the X + Y for the tens column, there must be a "carry", so X + Y = 10 or more. But X + Y cannot equal 10 itself, since then Z = 0, and you can't have a leading zero. You know that X + Y must be 17 or less, since X and Y are different numbers, and 8 + 9 = 17.
Since you're adding two 2-digit numbers and getting a 3-digit number, there must be an additional "carry" when you add the X + Y + 1 in the tens column. The minimum value of X + Y is 11, and the maximum is 17, so the minimum and maximum for X + Y + 1 are 12 and 18. What is Z, in either case?
Then what must be X + Y? ...and so forth.