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 there is a $\beta' \in \R^2$ such that:


 * $L' = \left\{{ \left({x, y}\right) \in \R^2 : \alpha_1 x + \alpha_2 y = \beta' }\right\}$

Sufficient Condition
Let $L' \ne L$ be a straight line given by the equation:


 * $\alpha_1 x + \alpha_2 y = \beta'$

Suppose we have a point $\vec x = \left({x, y}\right) \in L \cap L'$.

Then, as $\vec x \in L$, it also satisfies:


 * $\alpha_1 x + \alpha_2 y = \beta$

It follows that $\beta = \beta'$, so $L = L'$.

This contradiction shows that $L \cap L' = \varnothing$, i.e., $L$ and $L'$ are parallel.

The remaining case is when $L' = L$. By definition, $L$ is parallel to itself.

The result follows.

Also See

 * Condition for Planes being Parallel