Condition for Straight Lines in Plane to be Parallel
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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:
- $3 x - 4 y = -5$