Math Doubts

Slope and $\large x$ intercept form of a Line

Equation

$x \,=\, m’y+c$

It is an equation of a straight line when a straight line intercepts $x$-axis at a point with some slope.

Proof

A straight line intercepts the horizontal $x$-axis at point $A$ with an intercept $c$ and it makes an angle of $\theta$ with horizontal axis. The point $A$ in the form of coordinates is $A(c, 0)$.

line x intercept

$P$ is any point on the straight line and it is $x$ and $y$ units distance from origin in horizontal and vertical axes directions respectively. So, the point $P$ in the form of coordinates is $P(x, y)$.

Therefore, the straight line is represented as $\small \overleftrightarrow{AP}$ geometrically in mathematics.

Assume, the slope of straight line is denoted by $m$ and express it in trigonometric form.

$m \,=\, \tan{\theta}$

Now, draw a perpendicular line to horizontal $x$-axis from point $P$ and it intersects the $x$-axis at point $Q$. Thus, a right triangle ($\Delta QAP$) is formed geometrically.

line x intercept right triangle

Calculate the $\tan{\theta}$ in trigonometric system and it is equal to the slope of the straight line.

$\tan{\theta} \,=\, \dfrac{PQ}{AQ}$

$\implies \tan{\theta} \,=\, \dfrac{PQ}{OQ-OA}$

$\implies \tan{\theta} \,=\, \dfrac{y}{x-c}$

$\implies m \,=\, \dfrac{y}{x-c}$

$\implies x-c \,=\, \dfrac{y}{m}$

$\implies x \,=\, \dfrac{y}{m}+c$

$\,\,\, \therefore \,\,\,\,\,\, x \,=\, \Big(\dfrac{1}{m}\Big)y+c$

It is a linear equation which represents an equation of a straight line when a straight line intercepts $x$-axis with an $x$-intercept and slope. Hence, it is called slope and $x$-intercept form equation of a straight line.

Take the reciprocal of $m$ as $m’$.

$\,\,\, \therefore \,\,\,\,\,\, x \,=\, m’y+c$

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved