Solution Of Diophantine Equation

 Solve: 3x - 4y = 29  

There are no immediate integral solutions since neither 3 or 4 divide evenly into 29. So we write, using the Euclidean algorithm:   

4 = 1*3 + 1   and 3 + 1*2 + 1   and 1 = 4- 3 

So that (3, 4) = 1  

=>  3(3) - 4(2) = 1  

=> 3(11) - 4(1) = 29 

 So that x = 11, and y = 1  

Other solutions (for r = integer) can be obtained using:  

x = 11 + 4r   and y = 1 + 3r  

Check for r =2 :  x = 11 + 4(2) = 19  and y = 1 + 3(2) = 7 = 7 

 Subst. into the equation: 3x - 4y = 29 to get:   3(19) - 4(7) = 57 - 28 = 29  

Other values of r can also be tried by the reader, just ensure they're integers!