1. **State the problem:** We need to express the fraction $$\frac{5 - \sqrt{3}}{2 + \sqrt{3}}$$ in the form $$x + y\sqrt{3}$$ where $x$ and $y$ are rational numbers. 2. **Formula and rule:** To simplify expressions with surds in the denominator, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$2 + \sqrt{3}$$ is $$2 - \sqrt{3}$$. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{5 - \sqrt{3}}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{(5 - \sqrt{3})(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}$$ 4. **Calculate the denominator:** $$(2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1$$ 5. **Calculate the numerator:** $$(5 - \sqrt{3})(2 - \sqrt{3}) = 5 \times 2 - 5 \times \sqrt{3} - \sqrt{3} \times 2 + \sqrt{3} \times \sqrt{3} = 10 - 5\sqrt{3} - 2\sqrt{3} + 3 = (10 + 3) - (5\sqrt{3} + 2\sqrt{3}) = 13 - 7\sqrt{3}$$ 6. **Combine numerator and denominator:** $$\frac{13 - 7\sqrt{3}}{1} = 13 - 7\sqrt{3}$$ 7. **Identify $x$ and $y$:** $$x = 13, \quad y = -7$$ **Final answer:** $$x = 13, y = -7$$