Intersection of Straight Lines in General Form/Determinant Form

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

The point of intersection of $\LL_1$ and $\LL_2$ has coordinates given by:
 * $\dfrac x {\begin {vmatrix} m_1 & n_1 \\ m_2 & n_2 \end {vmatrix} } = \dfrac y {\begin {vmatrix} n_1 & l_1 \\ n_2 & l_2 \end {vmatrix} } = \dfrac 1 {\begin {vmatrix} l_1 & m_1 \\ l_2 & m_2 \end {vmatrix} }$

where $\begin {vmatrix} \cdot \end {vmatrix}$ denotes a determinant.

This point exists and is unique $\begin {vmatrix} l_1 & m_1 \\ l_2 & m_2 \end {vmatrix} \ne 0$.

Proof
From Intersection of Straight Lines in General Form, the point of intersection of $\LL_1$ and $\LL_2$ has coordinates given by:


 * $\dfrac x {m_1 n_2 - m_2 n_1} = \dfrac y {n_1 l_2 - n_2 l_1} = \dfrac 1 {l_1 m_2 - l_2 m_1}$

which exists and is unique $l_1 m_2 \ne l_2 m_1$.

The result follows by definition of determinant of order $2$.