7x + 16y = 225
Solve using Diophantine equations.
1--7x + 16y = 225
2--Dividing through by the lowest coefficient yields x + 2y + 2y/7 = 32 + 1/7
3--(2y - 1)/7 must be an integer
4--We seek a unit coefficient for y
5--4(2y - 1)/7 = (8y - 4)/7
6--Dividing by 7 yields y + y/7 - 4/7
7--(y - 4)/y must be an integer k making y = 7k + 4
8--Substituting back into (1) yields x = 23 - 16k
9--Only k = 0 or 1 yields positive integer solutions
10--k.....0.....1
....x....23.....7
....y....4.....11
11--Checking
7(23) + 16(4) = 161 + 64 = 225
7(7) + 16(11) = 49 + 176 = 225