1. **State the problem:** We are given the system of equations: $$xy = x^x$$ and $$y = 9$$ We need to find the value(s) of $x$ that satisfy both equations. 2. **Substitute $y=9$ into the first equation:** $$x \cdot 9 = x^x$$ which simplifies to $$9x = x^x$$ 3. **Rewrite the equation:** $$x^x = 9x$$ 4. **Analyze the equation:** We want to find $x$ such that $x^x = 9x$. 5. **Check for $x=0$:** Not valid since $x^x$ is undefined at $x=0$. 6. **Check for $x=1$:** $$1^1 = 1$$ $$9 \times 1 = 9$$ Not equal. 7. **Check for $x=3$:** $$3^3 = 27$$ $$9 \times 3 = 27$$ Equal, so $x=3$ is a solution. 8. **Check for $x=9$:** $$9^9$$ is very large, while $$9 \times 9 = 81$$ Not equal. 9. **Check for $x=0.5$ (for completeness):** $$0.5^{0.5} = \sqrt{0.5} \approx 0.707$$ $$9 \times 0.5 = 4.5$$ Not equal. 10. **Conclusion:** The only solution is $x=3$. **Final answer:** $$x=3, y=9$$