Equation of Straight Line in Plane/General Equation

Theorem
A straight line $\mathcal L$ is the set of all $\tuple {x, y} \in \R^2$, where:
 * $\alpha_1 x + \alpha_2 y = \beta$

where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero.

Proof
Let $y = \map f x$ be the equation of a straight line $\mathcal L$.

From Line in Plane is Straight iff Gradient is Constant, $\mathcal L$ has constant slope.

Thus the derivative of $y$ $x$ will be of the form:
 * $y' = c$

Thus:

where $K$ is arbitrary.

Taking the equation:
 * $\alpha_1 x + \alpha_2 y = \beta$

it can be seen that this can be expressed as:
 * $y = - \dfrac {\alpha_1} {\alpha_2} x + \dfrac {\beta} {\alpha_2}$

thus demonstrating that $\alpha_1 x + \alpha_2 y = \beta$ is of the form $y = c x + K$ for some $c, K \in \R$.

Also presented as
Some sources give this as:


 * $a x + b y + c = 0$

where $a, b, c \in \R$ are given, and not both $a, b$ are zero.

Its equivalence to the given form can be seen by equating $a = \alpha_1, b = \alpha_2, c = -\beta$.