Condition for Straight Lines in Plane to be Parallel/General Equation

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

Then the straight line $L'$ is parallel to $L$ iff $L'$ is the set of all $\left({x, y}\right) \in \R^2$ where:
 * $\exists \beta' \in \R: \alpha_1 x + \alpha_2 y = \beta'$