1. **State the problem:** We are given the equation $$\frac{P_1 V_1 T_2}{T_1} = \frac{P_2 V_2}{T_2}$$ and asked to solve for $T_2$. 2. **Write down the equation:** $$\frac{P_1 V_1 T_2}{T_1} = \frac{P_2 V_2}{T_2}$$ 3. **Multiply both sides by $T_1 T_2$ to clear denominators:** $$T_1 T_2 \times \frac{P_1 V_1 T_2}{T_1} = T_1 T_2 \times \frac{P_2 V_2}{T_2}$$ 4. **Simplify by canceling common factors:** $$\cancel{T_1} T_2 \times \frac{P_1 V_1 T_2}{\cancel{T_1}} = T_1 \times \cancel{T_2} \times \frac{P_2 V_2}{\cancel{T_2}}$$ which simplifies to $$P_1 V_1 T_2^2 = P_2 V_2 T_1$$ 5. **Isolate $T_2^2$ by dividing both sides by $P_1 V_1$:** $$T_2^2 = \frac{P_2 V_2 T_1}{P_1 V_1}$$ 6. **Take the square root of both sides to solve for $T_2$:** $$T_2 = \sqrt{\frac{P_2 V_2 T_1}{P_1 V_1}}$$ **Final answer:** $$\boxed{T_2 = \sqrt{\frac{P_2 V_2 T_1}{P_1 V_1}}}$$ This is the correct expression for $T_2$ based on the given equation.