The curve $y = \frac{5}{x} + 3$ and the line $x - 2y - 3 = 0$ intersect at the points $P$ and $Q$. Find the coordinates of $P$ and $Q$.

1. Express $x$ in terms of $y$ from the linear equation

From $x - 2y - 3 = 0$, we have:
$x = 2y + 3$

2. Substitute the expression for $x$ into the curve equation

Substitute $x = 2y + 3$ into $y = \frac{5}{x} + 3$:
$y = \frac{5}{2y + 3} + 3$

3. Simplify to form a quadratic equation in $y$

$y - 3 = \frac{5}{2y + 3}$
$(y - 3)(2y + 3) = 5$
$2y^2 + 3y - 6y - 9 = 5$
$2y^2 - 3y - 14 = 0$

4. Solve the quadratic equation for $y$

$(2y - 7)(y + 2) = 0$
$y = 3.5$ or $y = -2$

5. Find the corresponding values of $x$

When $y = 3.5$, $x = 2(3.5) + 3 = 10$
When $y = -2$, $x = 2(-2) + 3 = -1$