The value of $x$ that satisfies the equations $\frac{1}{x} + \frac{1}{y} = 8, \frac{1}{y} + \frac{1}{z} = 12, \frac{1}{z} + \frac{1}{x} = 10$ is -

Explanation

$$ \frac{1}{x} + \frac{1}{y} = 8 \quad ..........(i) $$ $$ \frac{1}{y} + \frac{1}{z} = 12 \quad ..........(ii) $$ $$ \frac{1}{z} + \frac{1}{x} = 10 \quad ..........(iii) $$ Adding equations (i), (ii) and (iii) $$ 2\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) = 8 + 12 + 10 = 30 $$ $$ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 15 \quad ..........(iv) $$ Subtracting equation (ii) from equation (iv) $$ \left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) - \left(\frac{1}{y} + \frac{1}{z}\right) = 15 - 12 = 3 $$ $$ \frac{1}{x} = 3 $$ $\therefore x = \frac{1}{3}$