Question: find two integral solutions of 13x+17y=221.
There are two (types of) solution to this problem.
The first solution is a complex one by the Diophantine Equation method.
The second one is a simple solution by observation.
I shall do the Diophantine Equation solution first and the Simple solution second.
The Diophantine Solution
There are no common factors of this equation.
Now decompose the coefficients of the x- and y-variables.
17 = 1*13 + 4
13 = 3*4 + 1
Now rearrange these equations,