Find the values of $x,y$ and $z$ in these equations I'm stuck with these equations. Can somebody help me with solving it?
If $$(a+b) = \frac{x(y+z)}{x+y+z},\quad  (b+c) = \frac{y(x+z)}{x+y+z} ,\quad  (a+c) = \frac{z(x+y)}{x+y+z}$$ then find $x,y,z$.

On adding all the 3 we get,
$$a+b+c=\frac{xy+yz+zx}{x+y+z}$$
Subtracting from the given equations we get $$c=\frac{yz}{x+y+z}$$ $$b=\frac{xz}{x+y+z}$$ $$a=\frac{xy}{x+y+z}$$
$$\implies \frac{y}{x}=\frac{c}{b}$$
$$\implies \frac{z}{x}=\frac{c}{a}$$
$$\implies \frac{z}{y}=\frac{b}{a}$$
$$a=\frac{y}{1+\frac{y}{x}+\frac{z}{x}}$$
$$\implies y = a(1+\frac{y}{x}+\frac{z}{x})=a(1+\frac{c}{b}+\frac{c}{a})=\frac{ab+bc+ca}{b}$$
I guess you can do the same for $x$ and $z$
