Solutions To Diophantine Algebra Problems

1)     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!

2) Solve: 11x + 12y = 58  

There are no immediate integral solutions since neither 11 or 12 divide evenly into 29. So we write, using the Euclidean algorithm:  

1 1 = 10*1 + 1  and 12 = 11*1+ 1 with 1=  12 – 11 

so that (11,12) = 1 (a, b)

                      =>   12(2) - 11(2) = 2  

=>   12(3) - 11(3) = 3  

 

Then x= 2, y = 3

 

Viz.  11(2)   +  12 (3) =  22 +  36  = 58

     For other solns. Use x = 2 + 12r,  and y = 3 - 11 r

                 Then, for r = 1:  x = 14,  y =  3 - 11(1) = 3 - 11 = -8

                    And:   11(14)  + 12(-8) =   154 -  96 =   58