Equation of a straight line in terms of slope and X-intercept

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.

Geometrical Explanation

Assume, $\stackrel{↔}{AB}$ 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 $\left(\theta \right)$. 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 $\left(a,0\right)$. 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 $A\left(a,0\right)$ is known $X$-intercept. Assume, the coordinates of the point $B$ is $\left(x,y\right)$.

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 $\stackrel{↔}{AB}$ geometrically.

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

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

$m=tan\theta$

$⇒m=\frac{BC}{AC}$

It can be written as follows.

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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