Finding the general solutions of the linear diophantine equation in an unique way!

This video is about a linear diophantine equation.
Finding the general solutions of the diophantine equation 17x+15y=143.This means we want to find the all integral values of x and y which satisfy the given equation.
Although,we can solve this by extended Euclidean algorithm ;but in this video,we will solve it in an interesting way!
We have 17x+15y=143.Writing this equation as: 15y=143-17x ;this means 143-17x should be a multiple of 15,or it should be divided by 15.
So,we can write this whole thing as:
143-17x should be congruent to 0(mod 15),and proceed from here to find the general solutions of the linear diophantine equation.

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If you sett x=4+15a and y=5-17a into original equation (17x+15y=143) you get 143-(255-255)a=143, this means the only solution is x=4 and y=5, no more solution. So your calculation does not solve the general answer and something in your method must be corrected.