System with fractions

[ins-]

Solve the system:
$\frac{1}{x+y}+\frac{1}{x-y}=\frac{1}{2}$
$\frac{1}{y+z}+\frac{1}{y-z}=\frac{1}{3}$
$\frac{1}{z+x}+\frac{1}{z-x}=\frac{1}{5}$

[waxtep]

It starts nicely with $2x+3y+5z=0$, but then finishes in $1140z^2+3901z+3240=0$ (if I didn't make some stupid mistakes along the way)

[ins-]

http://artofproblemsolving.com/communit ... 88p6253499 it is the solution. More approaches are possible, but you are correct, the numbers are a little bigger and as an end result - the calculations are longer.

[waxtep]

Ho-ho, so I was accurate this time, what a relief I used a straightforward approach, $y=3\pm \sqrt {z^2+9}$, squaring everything and getting quadratic equation in $z$ (because of "magic" in coefficients; generally it could be quartic and that would immediately kill this approach).

[ins-]

I know about this. It was the reason to put "magic" coefficients. The last system I posted topic108028.html is also with "magic" coefficients. It involves quartic and is not hard to be solved. If you like you may try it.