A line L passes through the point of intersection of the lines $3x + y - 10 = 0$ and $x - y - 2 = 0$. If the perpendicular distance of the line L from the point $(5, 1)$ is exactly $\dfrac{2}{\sqrt{5}}$ units, which of the following represents the correct equation for line L?

The point of intersection of the lines $3x + y - 10 = 0$ and $x - y - 2 = 0$ is obtained by solving the two equations.

Adding the equations gives $4x - 12 = 0 \Rightarrow x = 3$.

Substituting $x = 3$ into $x - y - 2 = 0$ gives $y = 1$. Thus, the point of intersection is $(3, 1)$.

Let the equation of the required line L passing through $(3, 1)$ be $y - 1 = m(x - 3)$, which can be rewritten as $mx - y - 3m + 1 = 0$.

The perpendicular distance of line L from the point $(5, 1)$ is given as $\dfrac{2}{\sqrt{5}}$. Using the perpendicular distance formula:

$\dfrac{|m(5) - 1 - 3m + 1|}{\sqrt{m^2 + (-1)^2}} = \dfrac{2}{\sqrt{5}}$

$\dfrac{|2m|}{\sqrt{m^2 + 1}} = \dfrac{2}{\sqrt{5}}$

Squaring both sides:

$\dfrac{4m^2}{m^2 + 1} = \dfrac{4}{5}$

$5m^2 = m^2 + 1$

$4m^2 = 1 \Rightarrow m = \pm \dfrac{1}{2}$

For $m = \dfrac{1}{2}$, the equation of the line is $y - 1 = \dfrac{1}{2}(x - 3) \Rightarrow x - 2y - 1 = 0$.

For $m = -\dfrac{1}{2}$, the equation of the line is $y - 1 = -\dfrac{1}{2}(x - 3) \Rightarrow x + 2y - 5 = 0$.

Among the given options, $x + 2y - 5 = 0$ is the correct equation.

Answer: $x + 2y - 5 = 0$