I hadn't read the whole thing, I thought you were having problems with the bit that I did.
Yes I can see the set up really is tricky!
I did mine without thinking about Heureka's and it looks a bit different. (But it is really the same)
Anyway see if it makes sense. I have put numbers next to many of the lines so you can say which one you don't understand. 
The digits of a two-digit number differ by 3 . If the digits are interchanged and the resulting number is added to the original number, we get 143 . What can be he original number ?
Heureka let the first digit be a and the second digit be b.
So the original number is 10a+b (1)
The digits are different by 3 That is |a-b|=3 I put the absolute signs in because I don't know which digit is the biggest one (2)
If the digits are interchanged then the new number is 10b+a (3)
When we add these 2 numbers the answer is 143 SO 10a+b+10b+a=143 (4)
11a+11b=143 (5)
Lets assume for the moment that a is the biggest digit so a-b=3
We have 2 equations that have to be solved simultaneously. (6)
a-b=3 (i) this can be rewritten as a=3+b (iii)
11a+11b=143 (ii) (7)
Sub (iii) into (ii)
11(3+b)+11b=143 (8)
33+11b+11b=143
33+22b=143
22b=110
b= 5 (9)
sub into (i)
a-5=3
a=8 (10)
So the original number may have been 85
Now if you can follow all this then you can see what happens if you let b be the big one. 
Ok
