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I am trying to solve the following form of linear Diophantine equation: ax + by + cz = d

Having previously looked at two variable Diophantine equations (ax + bx = c), I have learnt to apply the Euclidean algorithm to find an initial solution and then to find other solutions through the general solution.

I have successfully applied the Euclidean algorithm to three-variable equations to find an initial solution, however I was unable to prove any general formulas I came up with to find more solutions.

Does anyone know if there is a standard general formula that can be used for three-variable Diophantine equations?

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