Find a Cartesian equation for a plane through a point containing a given line

Let $L$ be the line through the point $(1,2,3)$ and parallel to the vector $(1,1,1)$ and let $(2,3,5)$ be a point not on $L$ . Find a Cartesian equation for the plane $M$ which passes through $(2,3,5)$ and entirely contains $L$ .

The line $L$ is the set of points

$L = \{ (1,2,3) + t(1,1,1) \} \quad \implies \quad P = (1,2,3), \ Q = (2,3,4) \in L.$

Then, the plane $M$ is the set of points

$\begin{align*} M &= \{ (2,3,5) + s((1,2,3) - (2,3,5)) + t((2,3,4) - (2,3,5)) \} \\ &= \{ (2,3,5) + s(-1,-1,-2) + t(0,0,-1) \}. \end{align*}$

Then, to get the Cartesian equation, we have

$\begin{align*} x &= 2-s \\ y &= 3-s \\ z &= 5 - 2s - t. \end{align*}$

The first two equations give $s = 2-x$ and so $y = 3-2 + x$ . This implies $x - y = -1$ , and $z$ is arbitrary. So, the plane is described by the equation $x-y = -1$ .