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!
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
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