Solving for PQR from Algebraic Equations

We are given the following algebraic equations:

• Equation 1: $P + \frac{1}{Q} = 1$
• Equation 2: $Q + \frac{1}{R} = 1$

The objective is to determine the value of the product $PQR$.

Deriving Expressions for P and R

From Equation 1, we can isolate $P$:

$P = 1 - \frac{1}{Q}$

Combine the terms on the right side:

$P = \frac{Q - 1}{Q}$

From Equation 2, we can isolate $\frac{1}{R}$:

$\frac{1}{R} = 1 - Q$

Assuming $Q \neq 1$ (which must be true, otherwise $1/R = 0$, which is impossible), we find $R$:

$R = \frac{1}{1 - Q}$

Calculating the Product PQR

Substitute the derived expressions for $P$ and $R$ into the product $PQR$:

$PQR = \left(\frac{Q - 1}{Q}\right) \times Q \times \left(\frac{1}{1 - Q}\right)$

Cancel the variable $Q$ present in the numerator and denominator:

$PQR = (Q - 1) \times \frac{1}{1 - Q}$

Factor out $-1$ from $(Q - 1)$:

$PQR = -(1 - Q) \times \frac{1}{1 - Q}$

Cancel the $(1 - Q)$ terms:

$PQR = -1 \times 1$

$PQR = -1$

Conclusion

The value of $PQR$ calculated from the given equations is -1.