Equation of Straight Line in Plane/Slope-Intercept Form

From ProofWiki
Jump to navigation Jump to search

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


This page may be the result of a refactoring operation.
As such, the following source works, along with any process flow, will need to be reviewed.
When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering.
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{SourceReview}} from the code.


Retrieved from "https://proofwiki.org/w/index.php?title=Equation_of_Straight_Line_in_Plane/Slope-Intercept_Form&oldid=688619"