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You can use this calculator to obtain pair of Bézout's coefficients and the general form of Bézout's coefficients as well. Some theory can be found below the calculator
Bézout's identity and Bézout's coefficients
To recap, the Bézout's identity (aka Bézout's lemma) is the following statement:
Let a and b be integers with greatest common divisor d. Then, there exist integers x and y such that ax + by = d. More generally, the integers of the form ax + by are exactly the multiples of d.
If d is the greatest common divisor of integers a and b, and x, y is any pair of Bézout's coefficients, the general form of Bézout's coefficients is
and the general form of Bézout's identity is