1. **State the problem:** We are given two equations: $$P_A = \frac{30 + P_B}{3}$$ $$P_B = \frac{35 + P_A}{3}$$ We want to substitute the expression for $P_A$ into the equation for $P_B$ and solve for $P_B$. 2. **Substitute $P_A$ into $P_B$ equation:** Replace $P_A$ in the second equation with the expression from the first: $$P_B = \frac{35 + \frac{30 + P_B}{3}}{3}$$ 3. **Simplify the numerator:** $$35 + \frac{30 + P_B}{3} = \frac{35 \times 3}{3} + \frac{30 + P_B}{3} = \frac{105 + 30 + P_B}{3} = \frac{135 + P_B}{3}$$ 4. **Rewrite $P_B$ equation:** $$P_B = \frac{\frac{135 + P_B}{3}}{3} = \frac{135 + P_B}{9}$$ 5. **Solve for $P_B$:** Multiply both sides by 9: $$9 P_B = 135 + P_B$$ Subtract $P_B$ from both sides: $$9 P_B - P_B = 135$$ $$8 P_B = 135$$ Divide both sides by 8: $$P_B = \frac{135}{8} = 16.875$$ 6. **Find $P_A$ using the first equation:** Substitute $P_B = 16.875$ back into $P_A$: $$P_A = \frac{30 + 16.875}{3} = \frac{46.875}{3} = 15.625$$ **Final answer:** $$P_A = 15.625, \quad P_B = 16.875$$