1. **State the problem:** We are given the equation $$\frac{\mu_X}{P_X} = \frac{\mu_Y}{P_Y}$$ with $\mu_X = 3X^{1/2}Y^{1/2}$, $P_X = 9$, $\mu_Y = X^{3/2}Y^{-1/2}$, and $P_Y = 3$. 2. **Write the equation substituting the given expressions:** $$\frac{3X^{1/2}Y^{1/2}}{9} = \frac{X^{3/2}Y^{-1/2}}{3}$$ 3. **Simplify both sides:** Left side: $$\frac{3}{9}X^{1/2}Y^{1/2} = \frac{1}{3}X^{1/2}Y^{1/2}$$ Right side: $$\frac{1}{3}X^{3/2}Y^{-1/2}$$ So the equation becomes: $$\frac{1}{3}X^{1/2}Y^{1/2} = \frac{1}{3}X^{3/2}Y^{-1/2}$$ 4. **Multiply both sides by 3 to clear denominators:** $$X^{1/2}Y^{1/2} = X^{3/2}Y^{-1/2}$$ 5. **Rewrite powers to isolate variables:** $$X^{1/2}Y^{1/2} = X^{3/2}Y^{-1/2}$$ Divide both sides by $$X^{1/2}Y^{-1/2}$$: $$\frac{X^{1/2}Y^{1/2}}{X^{1/2}Y^{-1/2}} = \frac{X^{3/2}Y^{-1/2}}{X^{1/2}Y^{-1/2}}$$ Simplify exponents: Left side: $$Y^{1/2 - (-1/2)} = Y^{1} = Y$$ Right side: $$X^{3/2 - 1/2} = X^{1} = X$$ So we get: $$Y = X$$ 6. **Interpretation:** The solution to the equation is the line $$Y = X$$. This means for the given functions and prices, the ratio equality holds when $$Y$$ equals $$X$$. **Final answer:** $$\boxed{Y = X}$$