1. **State the problem:** We are given the expression $$S = \frac{(P - 1)}{(P - 2)} \times \frac{(P - 3)}{(P - \alpha)}$$ and asked to solve it by linear equations. 2. **Understand the expression:** This is a product of two rational expressions. To solve for $P$, we typically set $S$ equal to some value and solve the resulting equation. 3. **Set up the equation:** Assuming we want to solve for $P$ given a value of $S$, write: $$S = \frac{(P - 1)(P - 3)}{(P - 2)(P - \alpha)}$$ 4. **Cross-multiply to clear denominators:** $$S (P - 2)(P - \alpha) = (P - 1)(P - 3)$$ 5. **Expand both sides:** Left side: $$S (P^2 - (2 + \alpha)P + 2\alpha) = S P^2 - S(2 + \alpha)P + 2 S \alpha$$ Right side: $$P^2 - 4P + 3$$ 6. **Bring all terms to one side:** $$S P^2 - S(2 + \alpha)P + 2 S \alpha - P^2 + 4P - 3 = 0$$ 7. **Group like terms:** $$ (S - 1) P^2 + (-S(2 + \alpha) + 4) P + (2 S \alpha - 3) = 0$$ 8. **Solve the quadratic equation for $P$:** Use the quadratic formula: $$P = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}$$ where $$a = S - 1$$ $$b = -S(2 + \alpha) + 4$$ $$c = 2 S \alpha - 3$$ 9. **Interpretation:** The solutions for $P$ depend on the values of $S$ and $\alpha$. This formula gives the roots of the quadratic equation derived from the original expression. **Final answer:** $$P = \frac{-\left(-S(2 + \alpha) + 4\right) \pm \sqrt{\left(-S(2 + \alpha) + 4\right)^2 - 4 (S - 1)(2 S \alpha - 3)}}{2 (S - 1)}$$