1. **State the problem:** Find the values of $x$ and $y$ such that $$(3x - y) + (x - 3y)i = 5 - i.$$\n\n2. **Identify real and imaginary parts:** For two complex numbers to be equal, their real parts must be equal and their imaginary parts must be equal.\nThis gives us the system:\n$$3x - y = 5$$\n$$x - 3y = -1.$$\n\n3. **Solve the system:** From the first equation, express $y$ in terms of $x$:\n$$y = 3x - 5.$$\nSubstitute into the second equation:\n$$x - 3(3x - 5) = -1,$$\nwhich simplifies to\n$$x - 9x + 15 = -1,$$\n$$-8x + 15 = -1,$$\n$$-8x = -16,$$\n$$x = 2.$$\n\n4. **Find $y$:** Substitute $x=2$ into $y=3x - 5$:\n$$y = 3(2) - 5 = 6 - 5 = 1.$$\n\n5. **Conclusion:** The values of $x$ and $y$ that satisfy the equation are $\boxed{x=2}$ and $\boxed{y=1}$.