Equation of Straight Line through Intersection of Two Straight Lines

Theorem
Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed using the general equations:

Let $\LL_3$ be a third straight lines embedded in $\CC$, passing through the point of intersection of $\LL_1$ and $\LL_2$.

$\LL_3$ can be expressed using the general equation:


 * $(1): \quad \paren {l_1 x + m_1 y + n_1} - k \paren {l_2 x + m_2 y + n_2} = 0$

Proof
Let $P = \tuple {x, y}$ be the point of intersection of $\LL_1$ and $\LL_2$.

We have that:

and so equation $(1)$:
 * $\paren {l_1 x + m_1 y + n_1} - k \paren {l_2 x + m_2 y + n_2} = 0$

is satisfied by the point $P$.

Then:

Each of $l_1 - k l_2$, $m_1 - k m_2$ and $n_1 - k n_2$ is a real number.

Hence $(1)$ is the equation of a straight line.

Also presented as
This result can also be conveniently presented as follows: