Integer Linear Combinations: Solving the 7 and 13 Florin Puzzle

Question

Person A has an unlimited number of 7-florin coins, person B has an unlimited number of 13-florin coins.
How A can pay 5 florins to B?

---

7x - 13y = 5

2*A - B = D

2*7 - 13 = 1

So let's find a multiple to get our goal and it's 5 because 5 x D or (5x1) is 5.
Now let's multiply each value.

14 - 13 = 1

14*5 - 13*5 = 5

70 - 65 = 5

x = 70 / 7 = 10

y = 65 / 13 = 5

∴  x = 10, y = 5