Crazy system of equations

[ins-]

Solve the system:
$\frac{1}{x}+\frac{2}{y-z}=\frac{1}{2}$
$\frac{3}{y}+\frac{1}{z-x}=\frac{1}{3}$
$\frac{1}{z}+\frac{4}{x-y}=\frac{1}{4}$

[arqady]

Легко получить $x-y=23z$ и добить.

[ins-]

If we put 3, 5, 8 instead of 2,3,4 it is a little more interesting, but still solvable. If we put $a$, $b$, $c$ instead - I suppose in the common case it is not solvable.

[ins-]

For $a=u$, $b=u+1$, $c=u+2$ and $c=a+b$ or $a-b=1$ where $a$, $b$, $c$, $u$ are real numbers the system:
$\frac{1}{x}+\frac{a}{y-z}=\frac{1}{a}$
$\frac{b}{y}+\frac{1}{z-x}=\frac{1}{b}$
$\frac{1}{z}+\frac{c}{x-y}=\frac{1}{c}$
is solvable in real numbers, but the answers are 1 km long. Even more common systems can be created and solved.