Expressing a linear expression of a straight line in terms of slope of the line and intercept at horizontal axis is defined equation of a straight line in terms of slope and x-intercept.

A straight line often appears in geometry by passing through the horizontal axis of the Cartesian coordinate system at an $x$-intercept. It makes the standard form of equation of the straight line to transform into some other form. In this case of straight line passing through the horizontal axis at an x-intercept, the equation of the straight line is usually expressed in terms of the slope of the straight line and x-intercept.

Assume, $\overleftrightarrow{AB}$ is a straight line which passes through the horizontal $x$-axis at an $x$-intercept by making some angle with the same horizontal axis.

Assume, the angle made by the straight line is theta $(\theta )$. Also assume, the point $A$ of the straight line is intersected with the horizontal axis at a distance of $a$ units from the origin. Therefore, the coordinates of the point $A$ is $(a,0)$. The point $A$ is one of the points of the straight line and also one of the points of the horizontal axis. Hence, the point $A(a,0)$ is known $X$-intercept. Assume, the coordinates of the point $B$ is $(x,y)$.

Draw a perpendicular line from point $B$ and assume it intersects the horizontal axis at a point, which is assumed to call point $C$. Thus, a right angled triangle, known $\Delta BAC$ is formed by the straight line $\overleftrightarrow{AB}$ geometrically.

- The line segment $\stackrel{\u203e}{AB}$ becomes the hypotenuse of the right angled triangle $\Delta BAC$.
- The line segment $\stackrel{\u203e}{AC}$ becomes the adjacent side of the right angled triangle $\Delta BAC$.
- The line segment $\stackrel{\u203e}{BC}$ becomes the opposite side of the right angled triangle $\Delta BAC$.

According to the right angled triangle $\Delta BAC$,

$tan\theta =\frac{BC}{AC}$

The length of the opposite side is $BC=y$

The length of the adjacent side is $AC=OC\u2013OA=x\u2013a$

According to concept of the slope of the straight line, slope of a straight line is expressed in mathematical form as follows.

$m=tan\theta $

$\Rightarrow m=\frac{BC}{AC}$

$\Rightarrow m=\frac{y}{x\u2013a}$

It can be written as follows.

$\Rightarrow x\u2013a=\frac{y}{m}$

$\Rightarrow x=\frac{y}{m}+a$

It is an algebraic linear equation, which represents a straight line having some slope but it is passing through the horizontal axis at an $X$-intercept.

Copyright © 2012 - 2017 Math Doubts, All Rights Reserved