Intersection of Straight Lines in General 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:


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

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

Determinant Form
This result can also be expressed in the form:

Proof
First note that by the parallel postulate $\LL_1$ and $\LL_2$ have a unique point of intersection they are not parallel.

From Condition for Straight Lines in Plane to be Parallel, $\LL_1$ and $\LL_2$ are parallel $l_1 m_2 = l_2 m_1$.

Let the equations for $\LL_1$ and $\LL_2$ be given.

Let $\tuple {x, y}$ be the point on both $\LL_1$ and $\LL_2$.

We have:

Similarly: