Solve $x + y + z = xyz$ such that $x , y , z \neq0$ I came across the equation $x+y+z=xyz$ such that $x , y , z \neq 0$.

I set $x=1, y=2, z=3$ but how can i reach formal mathematical solution without " guessing " the answer ? Thank you
you can write $y+z=x(yz-1)$ and if $yz\ne1$ you will get $$x=\frac{y+z}{yz-1}$$
if $yz=1$ we get $y+z=0$ or $y^2+1=0$ which gives us complex solutions.
Solve for one variable. As the problem is symmetric, wlog solve for $z$:
$$z = \frac{x+y}{xy-1}.$$
Now you can choose any real values for $x$ and $y$, as long as $xy-1\ne0$. For your choice of $x=1$, $y=2$, you can see it gives the correct result.

If you need a solution in $\mathbb Z$, things get more complicated as the value of $z$ could be a rational for most choices of $x$ and $y$.