# Equation of Straight Line in Plane

## Theorem

### General Equation

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.

### Slope-Intercept Form

Let $\LL$ be the straight line defined by the general equation:

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

Then $\LL$ can be described by the equation:

- $y = m x + c$

where:

\(\ds m\) | \(=\) | \(\ds -\dfrac {\alpha_1} {\alpha_2}\) | ||||||||||||

\(\ds c\) | \(=\) | \(\ds \dfrac {\beta} {\alpha_2}\) |

such that $m$ is the slope of $\LL$ and $c$ is the $y$-intercept.

### Two-Intercept Form

Let $\LL$ be a straight line which intercepts the $x$-axis and $y$-axis respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$.

Then $\LL$ can be described by the equation:

- $\dfrac x a + \dfrac y b = 1$

### Normal Form

Let $\LL$ be a straight line such that:

- the perpendicular distance from $\LL$ to the origin is $p$
- the angle made between that perpendicular and the $x$-axis is $\alpha$.

Then $\LL$ can be defined by the equation:

- $x \cos \alpha + y \sin \alpha = p$

### Two-Point Form

Let $p_1 := \tuple {x_1, y_1}$ and $p_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.

Let $\LL$ be the straight line passing through $p_1$ and $p_2$.

Then $\LL$ can be described by the equation:

- $\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$

or:

- $\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$

### Point-Slope Form

Let $\LL$ be a straight line embedded in a cartesian plane, given in slope-intercept form as:

- $y = m x + c$

where $m$ is the slope of $\LL$.

Let $\LL$ pass through the point $\tuple {x_0, y_0}$.

Then $\LL$ can be expressed by the equation:

- $y - y_0 = m \paren {x - x_0}$

### Homogeneous Cartesian Coordinates

A straight line $\LL$ is the set of all points $P$ in $\R^2$, where $P$ is described in homogeneous Cartesian coordinates as:

- $l X + m Y + n Z = 0$

where $l, m, n \in \R$ are given, and not both $l$ and $m$ are zero.