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

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 $L_2$ :


 * $\mu_1 = \mu_2$

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

Then:


 * the slope of $L_1$ is $\mu_1 = -\dfrac {l_1} {m_1}$


 * the slope of $L_2$ is $\mu_2 = -\dfrac {l_2} {m_2}$.

From Condition for Straight Lines in Plane to be Parallel: General Equation:


 * $L_1$ and $L_2$ are parallel $L_2$ 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.