Equation of Straight Line in Plane/General Equation

Theorem

A straight line $\LL$ 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 $\LL$.

From Line in Plane is Straight iff Slope is Constant, $\LL$ has constant slope.

Thus the derivative of $y$ with respect to $x$ will be of the form:

$y' = c$

Thus:

$\ds y$

$=$

$\ds \int c \rd x$

Fundamental Theorem of Calculus

$\ds $

$=$

$\ds c x + K$

Primitive of Constant

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 $c, K \in \R$.

$\blacksquare$

Also presented as

Some sources give the 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 can be seen by equating $a = \alpha_1, b = \alpha_2, c = -\beta$.

Similarly equivalent forms include:

$a x = b y + c$

and so on.

Also known as

Some sources refer to the as a canonical form.

Some sources refer to it as a general equation of the first degree, but that entity is usually reserved for a linear equation in $n$ variables for general $n \in \N$.

Such an entity is still the equation of a straight line, but this time in $n$-dimensional space.

Sources

1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $3$.
1958: P.J. Hilton: Differential Calculus ... (previous) ... (next): Chapter $1$: Introduction to Coordinate Geometry: $(1.2)$
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text V$: Vector Spaces: $\S 28$: Linear Transformations
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 10$: Formulas from Plane Analytic Geometry: General Equation of Line: $10.7$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): line: 2.
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): line: 2.
2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $6$: Curves and Coordinates: Cartesian coordinates
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 8$: Formulas from Plane Analytic Geometry: General Equation of Line: $8.7.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): canonical
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): straight line (in the plane)
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): canonical form (normal form)