1. **State the problem:** Evaluate the expression $$\frac{5i}{3 - i}$$ where $i$ is the imaginary unit with the property $i^2 = -1$. 2. **Formula and rules:** To simplify a complex fraction, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $3 - i$ is $3 + i$. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{5i}{3 - i} \times \frac{3 + i}{3 + i} = \frac{5i(3 + i)}{(3 - i)(3 + i)}$$ 4. **Expand numerator:** $$5i \times 3 = 15i$$ $$5i \times i = 5i^2 = 5(-1) = -5$$ So numerator is $$15i - 5$$ 5. **Expand denominator using difference of squares:** $$(3)^2 - (i)^2 = 9 - (-1) = 9 + 1 = 10$$ 6. **Write the fraction:** $$\frac{15i - 5}{10}$$ 7. **Separate real and imaginary parts:** $$\frac{-5}{10} + \frac{15i}{10} = -\frac{1}{2} + \frac{3}{2}i$$ **Final answer:** $$-\frac{1}{2} + \frac{3}{2}i$$