1. **State the problem:** We are given the equation $$\frac{1}{u} = \frac{1}{v} + \frac{1}{w}$$ with complex numbers $$v = 1 - 2i$$ and $$w = 3 + i$$. We need to find $$u$$ in the form $$x + yi$$ where $$x, y \in \mathbb{R}$$. 2. **Write the formula and explain:** The equation relates the reciprocals of complex numbers. To find $$u$$, we first find $$\frac{1}{v} + \frac{1}{w}$$ and then take the reciprocal of the result. 3. **Calculate $$\frac{1}{v}$$:** $$\frac{1}{v} = \frac{1}{1 - 2i}$$ Multiply numerator and denominator by the conjugate $$1 + 2i$$: $$\frac{1}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{1 + 2i}{(1)^2 - (2i)^2} = \frac{1 + 2i}{1 - (-4)} = \frac{1 + 2i}{5} = \frac{1}{5} + \frac{2}{5}i$$ 4. **Calculate $$\frac{1}{w}$$:** $$\frac{1}{w} = \frac{1}{3 + i}$$ Multiply numerator and denominator by the conjugate $$3 - i$$: $$\frac{1}{3 + i} \times \frac{3 - i}{3 - i} = \frac{3 - i}{9 - (-1)} = \frac{3 - i}{10} = \frac{3}{10} - \frac{1}{10}i$$ 5. **Add $$\frac{1}{v} + \frac{1}{w}$$:** $$\left(\frac{1}{5} + \frac{2}{5}i\right) + \left(\frac{3}{10} - \frac{1}{10}i\right) = \left(\frac{2}{10} + \frac{3}{10}\right) + \left(\frac{4}{10} - \frac{1}{10}\right)i = \frac{5}{10} + \frac{3}{10}i = \frac{1}{2} + \frac{3}{10}i$$ 6. **Find $$u$$ by taking reciprocal:** $$u = \frac{1}{\frac{1}{2} + \frac{3}{10}i}$$ Multiply numerator and denominator by the conjugate $$\frac{1}{2} - \frac{3}{10}i$$: $$u = \frac{\frac{1}{2} - \frac{3}{10}i}{\left(\frac{1}{2}\right)^2 + \left(\frac{3}{10}\right)^2} = \frac{\frac{1}{2} - \frac{3}{10}i}{\frac{1}{4} + \frac{9}{100}} = \frac{\frac{1}{2} - \frac{3}{10}i}{\frac{25}{100} + \frac{9}{100}} = \frac{\frac{1}{2} - \frac{3}{10}i}{\frac{34}{100}} = \left(\frac{1}{2} - \frac{3}{10}i\right) \times \frac{100}{34}$$ 7. **Simplify:** $$u = \frac{100}{34} \times \frac{1}{2} - \frac{100}{34} \times \frac{3}{10}i = \frac{100}{68} - \frac{300}{340}i = \frac{50}{34} - \frac{30}{34}i = \frac{25}{17} - \frac{15}{17}i$$ **Final answer:** $$u = \frac{25}{17} - \frac{15}{17}i$$