If \(x+\displaystyle\frac{1}{y}=1\) and \(y+\displaystyle\frac{1}{z}=1\), find the value of \(z+\displaystyle\frac{1}{x}\).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1
Both \(x\) and \(z\) can be written in terms of \(y\) alone; then \(z+\frac{1}{x}\) becomes a single fraction in \(y\).
From \(x+\frac{1}{y}=1\): \(x\displaystyle=1-\frac{1}{y}=\frac{y-1}{y}\), so \(\displaystyle\frac{1}{x}=\frac{y}{y-1}\)
From \(y+\frac{1}{z}=1\): \(\displaystyle\frac{1}{z}=1-y\), so \(z\displaystyle=\frac{1}{1-y}=-\frac{1}{y-1}\)
\(z+\displaystyle\frac{1}{x}=-\frac{1}{y-1}+\frac{y}{y-1}=\frac{y-1}{y-1}=1\)