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:
 * $$\exists \beta' \in \R: \alpha_1 x + \alpha_2 y = \beta'$$