1. **State the problem:** Solve for $x$ in the equation $5^{3x} = 125$. 2. **Rewrite the right side as a power of 5:** Since $125 = 5^3$, the equation becomes: $$5^{3x} = 5^3$$ 3. **Set the exponents equal:** Because the bases are the same and nonzero, the exponents must be equal: $$3x = 3$$ 4. **Solve for $x$:** $$x = \frac{3}{3} = 1$$ --- 5. **State the second problem:** Solve for $y$ in the equation $5^{2y+1} = 2^{3-y}$. 6. **Take the natural logarithm of both sides:** $$\ln\left(5^{2y+1}\right) = \ln\left(2^{3-y}\right)$$ 7. **Use the logarithm power rule:** $$ (2y+1) \ln 5 = (3 - y) \ln 2$$ 8. **Distribute the logarithms:** $$ 2y \ln 5 + \ln 5 = 3 \ln 2 - y \ln 2$$ 9. **Group terms with $y$ on one side:** $$ 2y \ln 5 + y \ln 2 = 3 \ln 2 - \ln 5$$ 10. **Factor out $y$:** $$ y (2 \ln 5 + \ln 2) = 3 \ln 2 - \ln 5$$ 11. **Solve for $y$:** $$ y = \frac{3 \ln 2 - \ln 5}{2 \ln 5 + \ln 2}$$ 12. **Calculate numerical values:** $$ \ln 2 \approx 0.6931, \quad \ln 5 \approx 1.6094$$ $$ y = \frac{3(0.6931) - 1.6094}{2(1.6094) + 0.6931} = \frac{2.0793 - 1.6094}{3.2188 + 0.6931} = \frac{0.4699}{3.9119} \approx 0.12$$ --- 13. **Find $x + y$ to the nearest hundredth:** $$ x + y = 1 + 0.12 = 1.12$$