Question

Find the solution of the following system of equations

\[

\begin{align*}

\frac{1}{x}+\frac{1}{y+z}&=\frac{1}{2}\\

\frac{1}{y}+\frac{1}{x+z}&=\frac{1}{3}\\

\frac{1}{z}+\frac{1}{x+y}&=\frac{1}{4}

\end{align*}

\]

Answer 1

Hm

Answer 2

well i spent 10 minutes typing my solution to this and my browser crashed. I'll give my result and the basic idea:

x=23/10
y=23/6
z=23/2

we can write the system as:
2(x+y+z)=x(y+z)
3(x+y+z)=y(x+z)
4(x+y+z)=z(x+y)

isolating x+y+z on the LHS of each we get the equality:
(xy+xz)/2=(xy+zy)/3=(xz+zy)/4
from here we find that:

y=z/3
x=z/5

plugging these into the first given eqution we find z=23/2 and then x and y are found easily.