Condition for Straight Lines in Plane to be Parallel

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:

$\mu_1 = \mu_2$

Examples

Arbitrary Example $1$

Let $\LL_1$ be the straight line whose equation in general form is given as:

$3 x - 4 y = 7$

Let $\LL_2$ be the straight line parallel to $\LL_1$ which passes through the point $\tuple {1, 2}$.

The equation for $\LL_2$ is:

$3 x - 4 y = -5$