# Condition for Straight Lines in Plane to be Parallel/Slope Form/Proof 1

## Theorem

Let $L_1$ and $L_2$ be straight lines in the Cartesian plane.

Let the slope of $L_1$ and $L_2$ be $\mu_1$ and $\mu_2$ respectively.

Then $L_1$ is parallel to $L_2$ if and only if:

$\mu_1 = \mu_2$

## Proof

Let $L_1$ and $L_2$ be described by the general equation:

 $\ds L_1: \,$ $\ds l_1 x + m_1 y + n_1$ $=$ $\ds 0$ $\ds L_2: \,$ $\ds l_2 x + m_2 y + n_2$ $=$ $\ds 0$

Then:

the slope of $L_1$ is $\mu_1 = -\dfrac {l_1} {m_1}$
the slope of $L_2$ is $\mu_2 = -\dfrac {l_2} {m_2}$.
$L_1$ and $L_2$ are parallel if and only if $L_2$ is given by the equation:
$m_1 x + m_2 y = n'$
for some $n'$.

But then the slope of $L_2$ is $-\dfrac {l_1} {m_1}$.

That is:

$-\dfrac {l_1} {m_1} = -\dfrac {l_2} {m_2}$

and the result follows.

$\blacksquare$