Solve system of simultaneous equations in $3$ variables: $x+y+xy=19$, $y+z+yz=11$, $z+x+zx=14$ Solve the following equation system:
$$x+y+xy=19$$
$$y+z+yz=11$$
$$z+x+zx=14$$
I've tried substituting, adding, subtracting, multiplying... Nothing works. Could anyone drop me a few hints without actually solving it? Thanks!

Add $1$ to both sides of all the equations. To get
\begin{align*}
(x+1)(y+1) & = 20\\
(y+1)(z+1) & = 12\\
(z+1)(x+1) & = 15\\
\end{align*}
Now let $u=x+1,v=y+1,w=z+1$. And you have
\begin{align*}
uv&=20\\
vw&=12\\
wu&=15
\end{align*}
From this you can get
$$(uvw)=\pm 60.$$
Now use the above equations to compute $u=\pm 5$ and so on.
