How to solve this linear Diophantine equation?

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$$ 61 x - 40 y = 1 $$

I know how to proceed in the addition equations, but how do you find the HCF of a negative number via Euclidean Algorithm? Please explain all steps in detail.

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You solve $61x + 40y = 1$, I got $61(-19) + 40(29) = 1$ Then you play with the signs accordingly. In this case $61(-19) - 40(-29) = 1$.