Solve a nonlinear system of equations in 3 variables I need to solve this system of equations
$$\frac 1x+\frac{1}{y+z}=-\frac2{15}$$
$$\frac 1y+\frac{1}{x+z}=-\frac2{3}$$
$$\frac 1z+\frac{1}{x+y}=-\frac1{4}$$
I've tried to express $x$ in terms of $y$, then $y$ in terms of $z$. But this leads to nothing good. I think I should use some matrix method, but I'm not sure what exactly I have to do. Need help here.

The solution is given by
$$
(x,y,z)=(5,-1,-2)
$$
This follows by multiplying with the common denominator, which gives three polynomial equations in $x,y,z$, which can be easily solved using resultants. The first polynomial equation, for example, is $x(2y + 2z + 15) + 15(y + z)=0$. One of the resultant equations is, for example, $yz - x - y - z=0$, so that we can substitute $x=yz-y-z$.
