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!
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