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

Context
Analytic Geometry

Theorem
Let $$L = \alpha_1 x_1 + \alpha_2 x_2 = \beta$$ be a line in $\mathbb{R}^2$.

Then the line $$L'$$ is parallel to $$L$$ iff $$L'$$ is the set of all $$\left({x_1, x_2}\right) \in \mathbb{R}^2$$ where $$\exists \beta' \in \reals: \alpha_1 x_1 + \alpha_2 x_2 = \beta'$$.