If $x + \frac{1}{y} = 1$ and $y + \frac{1}{z} = 1$ are, then the value of $z + \frac{1}{x}$ is -

Explanation

Given that, $x + \frac{1}{y} = 1$ $x = 1 - \frac{1}{y} = \frac{y-1}{y}$ $\therefore \frac{1}{x} = \frac{y}{y-1}$ ................ (i) and $y + \frac{1}{z} = 1$ $\frac{1}{z} = 1 - y$ $\therefore z = \frac{1}{1-y}$ ................ (ii) On adding eq. (i) and (ii), we get $z + \frac{1}{x} = \frac{y}{y-1} + \frac{1}{1-y}$ $= \frac{y}{y-1} - \frac{1}{y-1}$ $= \frac{1}{y-1}(y-1) = 1$ ## Page 32