If $$\frac{{xy}}{{x + y}} = a,$$   $$\frac{{xz}}{{x + z}} = b$$   and $$\frac{{yz}}{{y + z}} = c{\text{,}}$$   where a, b, c are all non - zero numbers, then x equals to?

Solution (By Examveda Team)

$$\eqalign{ & \frac{{xy}}{{x + y}} = a,\,\frac{{xz}}{{x + z}} = b,\,\frac{{yz}}{{y + z}} = c{\text{ }} \cr & {\text{Now,}} \cr & \frac{{x + y}}{{xy}} = \frac{1}{a} \cr & \frac{{x + z}}{{xz}} = \frac{1}{b} \cr & \frac{{y + z}}{{yz}} = \frac{1}{c} \cr & \Rightarrow \frac{1}{y} + \frac{1}{x} = \frac{1}{a},\frac{1}{z} + \frac{1}{x} = \frac{1}{b},\frac{1}{x} + \frac{1}{y} = \frac{1}{c} \cr & {\text{Now we have to find the value of }}x \cr & \therefore \frac{1}{a} + \frac{1}{b} - \frac{1}{c} = \frac{1}{y} + \frac{1}{x} + \frac{1}{z} + \frac{1}{x} - \frac{1}{y} - \frac{1}{z} \cr & \therefore \frac{1}{a} + \frac{1}{b} - \frac{1}{c} = \frac{2}{x} \cr & \Rightarrow \frac{{bc + ac - ab}}{{abc}} = \frac{2}{x} \cr & \Rightarrow x = \frac{{2abc}}{{bc + ac - ab}} \cr} $$