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, 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.

Straight line passes through x-axis at an x-intercept

Assume, the angle made by the straight line is theta (θ). 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).

Straight line passes through x-axis at an x-intercept forms a right angled triangle

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 ΔBAC is formed by the straight line AB geometrically.

  • The line segment AB becomes the hypotenuse of the right angled triangle ΔBAC.
  • The line segment AC becomes the adjacent side of the right angled triangle ΔBAC.
  • The line segment BC becomes the opposite side of the right angled triangle ΔBAC.

According to the right angled triangle ΔBAC,

tanθ = BCAC

The length of the opposite side is BC = y

The length of the adjacent side is AC = OC   –   OA = x  –  a

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

m = tanθ

m = BCAC

m = yx  –  a

It can be written as follows.

x  –  a = ym

x = ym  +  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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