1. **State the problem:** Find the two solutions to the equation $$(3x - 1)(x - 5) = 0$$. 2. **Formula and rule:** The zero product property states that if a product of two factors equals zero, then at least one of the factors must be zero. So, we set each factor equal to zero: $$3x - 1 = 0$$ $$x - 5 = 0$$ 3. **Solve the first equation:** $$3x - 1 = 0$$ Add 1 to both sides: $$3x = 1$$ Divide both sides by 3: $$x = \frac{\cancel{3}x}{\cancel{3}} = \frac{1}{3}$$ 4. **Solve the second equation:** $$x - 5 = 0$$ Add 5 to both sides: $$x = 5$$ 5. **Final answer:** The two solutions are $$x = \frac{1}{3} \quad \text{and} \quad x = 5$$.