Find the values of x and y for the equation (4+i)x + (5−3i)y = 13+i.

This gives a system of two linear equations:

1. Real parts: $4x + 5y = 13$
2. Imaginary parts: $x - 3y = 1$

From equation (2), we can express $x$ in terms of $y$:

$$x = 1 + 3y \quad (*)$$

$$4(1 + 3y) + 5y = 13$$ $$4 + 12y + 5y = 13$$ $$4 + 17y = 13$$ $$17y = 13 - 4$$ $$17y = 9$$ $$y = \frac{9}{17}$$

Your calculation for $y$ is correct.

$$x = 1 + 3\left(\frac{9}{17}\right)$$ $$x = 1 + \frac{27}{17}$$

To add these, find a common denominator:

$$x = \frac{17}{17} + \frac{27}{17}$$ $$x = \frac{17 + 27}{17}$$ $$x = \frac{44}{17}$$

The values are $\boxed{x = \frac{44}{17}}$ and $\boxed{y = \frac{9}{17}}$.