# Equation of Straight Line in Plane/Slope-Intercept Form

## Theorem

Let $\LL$ be the straight line in the Cartesian plane such that:

- the slope of $\LL$ is $m$
- the $y$-intercept of $\LL$ is $c$

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

- $y = m x + c$

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

## Proof 1

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

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

We have:

\(\ds \alpha_1 x + \alpha_2 y\) | \(=\) | \(\ds \beta\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \alpha_2 y\) | \(=\) | \(\ds y_1 - \alpha_1 x + \beta\) | |||||||||||

\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds -\dfrac {\alpha_1} {\alpha_2} x + \dfrac {\beta} {\alpha_2}\) |

Setting $x = 0$ we obtain:

- $y = \dfrac {\beta} {\alpha_2}$

which is the $y$-intercept.

Differentiating $(1)$ with respect to $x$ gives:

- $y' = -\dfrac {\alpha_1} {\alpha_2}$

By definition, this is the slope of $\LL$ and is seen to be constant.

The result follows by setting:

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

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

$\blacksquare$

## Proof 2

By definition, the $y$-intercept of $\LL$ is $\tuple {0, c}$.

We calculate the $x$-intercept $X = \tuple {x_0, 0}$ of $\LL$:

\(\ds m\) | \(=\) | \(\ds \dfrac {c - 0} {0 - x_0}\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds x_0\) | \(=\) | \(\ds -\dfrac c m\) |

Hence from the two-intercept form of the Equation of straight line in the plane, $\LL$ can be described as:

\(\ds \dfrac x {-c / m} + \dfrac y c\) | \(=\) | \(\ds 1\) | Definition of Slope of Straight Line | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds m x + c\) | after algebra |

$\blacksquare$

## Proof 3

Let the straight line $\LL$ make an angle $\psi$ with the $x$-axis.

The slope $m$ of $\LL$ is then given by:

- $m = \tan \psi$

Let $B := \tuple {0, c}$ be the $y$-intercept of $\LL$.

Let $P := \tuple {x, y}$ be an arbitrary point on $\LL$.

Let the perpendicular $MP$ to the $x$-axis be constructed from $P$ to intersect the $x$-axis at $M$

Let $BK$ be constructed perpendicular $MP$ from $B$ to intersect $MP$ at $K$.

Then we have:

\(\ds BK\) | \(=\) | \(\ds x\) | by construction | |||||||||||

\(\ds KP\) | \(=\) | \(\ds y - c\) | by construction | |||||||||||

\(\ds KP\) | \(=\) | \(\ds BK \tan \psi\) | Definition of Tangent of Angle | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds x \tan \psi + c\) | |||||||||||

\(\ds \) | \(=\) | \(\ds m x + c\) | Definition of $m$ |

$\blacksquare$

## Also presented as

The **slope-intercept form** of the equation of a straight line in the plane can also be seen presented as:

- $y = x \tan \psi + c$

where $\psi$ is the angle that $\LL$ makes with the $x$-axis.

## Also known as

The **slope-intercept form** of the equation of a straight line in the plane is also known as the **gradient-intercept form**.

## Examples

### Arbitrary Example

The straight line $\LL$ given by the equation:

- $y = 2 x + 4$

has:

- a slope of $2$, hence the angle between $\LL$ and the $x$-axis is $\arctan 2$
- a $y$-intercept of $\tuple {0, 4}$.

## Sources

- 1914: G.W. Caunt:
*Introduction to Infinitesimal Calculus*... (previous) ... (next): Chapter $\text I$: Functions and their Graphs: $1$. Constants and Variables - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 10$: Formulas from Plane Analytic Geometry: $10.4$: Equation of Line joining Two Points $\map {P_1} {x_1, y_1}$ and $\map {P_2} {x_2, y_2}$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**line**:**2.** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**slope-intercept form** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**line**:**2.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**slope-intercept form** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**line**(in two dimensions)**Slope-intercept form** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**straight line**(in the plane)