What is the simplest method to solve the following system?

Question

Find all the integers$( x,y,z,t)$ satisfying: $$xz=a+uk$$ $$yt+xt-yz=-b+vk$$ $$xt+yz=c+wk$$ where $\gcd(a,b,c)=\gcd(u,v,w)=1$, and $ k\neq 0$

Answer 1

Above simultaneous equation shown below:

$xz=a+uk$

$yt+xt-yz=-b+vk$

$xt+yz=c+wk$

Above has numerical solution:

$(x,y,z,t)=(3,2,1,3)$

$(u,v,w,k)=(2,3,5,1)$

$(a,b,c)=(1,-10,6)$