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This comes from my notes. At the bottom where it says: 2*(50 - 1*35) = 3*35 - 2*50,

I am not figuring out 3 and the 2 come from in 3*35 - 2*50. I tried multiplying which gives me the 2*50 but the 3*35 I do not understand how they getting that.

This is easier with an example:

Solve 135x+ 50y= gcd(135, 50)

135 = 2·50 + 35

50 = 1·35 + 15

35 = 2·15 + 5

15 = 3·5 + 0, so gcd(135, 50) = 5

5 = 35 – 2·15

15 = 50 – 1·35,

so 5 = 35 – 2·(50 – 1·35) = 3·35 – 2·50

35 = 135 –2·50,

so 5 = 3·(135 –2·50) – 2·50 = 3·135 – 8·50 (x = 3, y = -8)