Condition for Straight Lines in Plane to be Parallel

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Theorem

General Equation

Let $L: \alpha_1 x + \alpha_2 y = \beta$ be a straight line in $\R^2$.


Then the straight line $L'$ is parallel to $L$ if and only if there is a $\beta' \in \R^2$ such that:

$L' = \set {\tuple {x, y} \in \R^2: \alpha_1 x + \alpha_2 y = \beta'}$


Slope Form

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:

$\mu_1 = \mu_2$