General solution to Diophantine equation in 3 variables

Question

I need to use Euclidean algorithm and find general solution to Diophantine equation given as

$6x+15y+20z=1$

This can be represented as

$6x+5a=1$ (1)

$15y+20z=5$ (2)

Solution to (1) is $(1+5i,-1-6i)$ and (2) is $(-1+20j,1-15j)$.

Note I can get general solution using Euler algorithm and it is

$x=a$

$y=-2b+1$

$z=3b-3a-1$

$a,b$ - integers

Answer 1

Found it.

Express (2) as

$3y+4z=1$

The solution is $(-1+4j,1-3j)$.

The multiple of 5 is $(-5+4j,5-3j)$.

The multiple of 5 and a is $(5+30i+4j,-5-30i-3j)$.

Answer 2

The solution to $3y+4z=1$ is $(-1+20j,1-15j)$; therefore the solution to $3y+4z=-1-6i$ is $([-1-6i][-1+20j],[-1-6i][1-15j])$.

Hence the general solution to $6x+15y+20z=1$ is $$\boxed{(x,y,z)\ =\ (1+5i,[-1-6i][-1+20j],[-1-6i][1-15j])}$$

Check: $$\begin{array}{rl} & 6(1+5i)+15(-1-6i)(-1+20j)+20(-1-6i)(1-15j) \\ =& 6(1+5i)+15(1+6i-20j-120ij)+20(-1-6i+15j+90ij) \\ =& 6+30i+15+90i-300j-1800ij-20-120i+300j+1800ij \\ =& 1\end{array}$$