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 |
$\blacksquare$
Also presented as
This equation 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.
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)
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- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $4$. Special forms of the equation of a straight line: $(1)$ Gradient forms