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.

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