Algebra question

[Dr Jekyll]

The radio 4 puzzle for last Friday was easy enough through trial and error, but I can't understand the algebraic way of doing it.

https://www.bbc.co.uk/programmes/articles/3MB6TpfS...

puzzle answer said:

The annual fees for the Puzzle Challenge Club are £23 per person, with a special rate for senior citizens of £17. If the total amount of dues collected last year was £1500, what is the smallest number of senior citizens that belong to the club?

Today’s #PuzzleForToday has been set by Dr Steve Humble from Newcastle University

Click here for the answer
There are only 4 possible alternative ways of the club fees being £1500. The alternative with 3 senior citizens is the smallest.
Details of a possible solution.
23X +17Y = 1000
We can re-write this equation as
X = 12 – 17n and Y = 72 + 23n
where n can take values of 0, -1, -2, -3
Q Y senior X general Senior fees General fees
0 72 12 £1224 £276
-1 49 29 £833 £667
-2 26 46 £442 £1058
-3 3 63 £51 £1449

Where does the 1000 come from in the first equation? Or the 12 and the 72 in the next two?