If $y + \frac{1}{z} = 1$ and $x + \frac{1}{y} = 1$, then what is the value of $xyz$?

### Concept & Formula This is another cyclic substitution problem. To find the product $xyz$, you must express two of the variables in terms of the third variable using the provided equations. By doing so, the variables will cleanly cancel out when multiplied together. $$ \text{Goal: Convert } x \text{ and } z \text{ into expressions of } y. $$ ### Step-by-Step Solution * **Given:** * Equation 1: $y + \frac{1}{z} = 1$ * Equation 2: $x + \frac{1}{y} = 1$ * Target expression: $xyz$ * **Calculation:** 1. From Equation 1, isolate $z$ in terms of $y$. $\frac{1}{z} = 1 - y$. Take the reciprocal: $z = \frac{1}{1 - y}$. 2. From Equation 2, isolate $x$ in terms of $y$. $x = 1 - \frac{1}{y}$. Find a common denominator: $x = \frac{y - 1}{y}$. 3. Now substitute the expressions for $x$ and $z$ into the target product $xyz$. $xyz = \left(\frac{y - 1}{y}\right) \times (y) \times \left(\frac{1}{1 - y}\right)$. 4. Observe the terms. The variable $y$ in the numerator and denominator cancels out. $xyz = \frac{y - 1}{1 - y}$. 5. Notice that $(y - 1)$ is the exact negative of $(1 - y)$. Factor out a $-1$ from the numerator. $y - 1 = -1(1 - y)$. 6. Substitute this back into the fraction: $\frac{-1(1 - y)}{(1 - y)}$. 7. Cancel the $(1 - y)$ terms. The result is $-1$. ### Exam Strategy & Shortcut Use Value Assumption. Pick a simple value for $y$, avoiding $y=1$ or $y=0$ which would cause division by zero. Let $y = 2$. If $y = 2$, then from Eq 1: $2 + \frac{1}{z} = 1 \implies \frac{1}{z} = -1 \implies z = -1$. If $y = 2$, then from Eq 2: $x + \frac{1}{2} = 1 \implies x = \frac{1}{2}$. Now multiply them together: $xyz = (\frac{1}{2}) \times (2) \times (-1) = 1 \times -1 = -1$. This guarantees the correct answer in seconds. ### Common Pitfall A frequent mistake during the final simplification step is assuming that $\frac{y - 1}{1 - y}$ equals $1$ instead of $-1$. Be extremely careful with signs when variables are subtracted in reverse order in numerators and denominators. ### Final Answer Therefore, the correct answer is -1.