Equation of a Line

Coordinate geometry is a branch of geometry, in which a point on a cartesian plane is defined by using 2 points. These two points are abscissa (x-coordinate) and ordinate (y-coordinate). The abscissa and ordinate define the distance of the particular point from the y-axis and x-axis, respectively. For the point represented in the figure, x=4, and y=2


A line on a graph is always represented in the form of an equation in terms of x and y. Every point that satisfies a line’s equation is a part of that line. 


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The general form of an equation of a line is represented as Ax + By + C, where A is the coefficient of x, B is the coefficient of y and C is the content term. A, B, and Can take the value of any rational number, including zero.


Different Forms of Equation of a Straight Line

There are different forms of an equation of a line. Some of them are explained below: -


  1. Equation For Horizontal Line: A horizontal line is parallel the x-axis. Hence, all the points on this line need to be equidistant from the x-axis. The ordinate defines the distance of a point from the x-axis. Since all the points of the horizontal line are equidistant from the x-axis, the ordinates for all of them will be the same, i.e. y=k for all the points in a particular horizontal line.


  1. Equation For Vertical Line: A vertical line is parallel the y-axis. Hence, all the points on this line need to be equidistant from the y-axis. The abscissa defines the distance of a point from the y-axis. Since all the points of the vertical line are equidistant from the y-axis, the abscissa for all of them will be the same i.e. x=k for all the points in a particular vertical line.


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  1. Point - Slope Equation: The point-slope form of equation can be used for a non-horizontal, non-vertical line, when we know the values for the slope, and a point on the line. Let us consider I(x, y) as an imaginary point on this line, and P(x1, y1) as the defined point on the line. Then the point-slope equation can be represented as: 

m=\[\frac{(y − y_{1})}{(x − x_{1})}\]


\[m(x - x_{1})\] = \[(y - y_{1})\]


By plugging in the values of the slope and the defined point, you will get an equation in terms of x and y, which is the equation of the line. 

  1. Equation of Line in Two Point Form: In the two point form of equation, we take a line and consider an imaginary point I (x, y). Now, take two defined points P (x1, y1) and Q (x2, y2) , which are collinear to point I. Since these points are collinear, the slope of PI= slope of PQ. Putting this in the form of an equation:


\[\frac{(y - y_{1})}{(x - x_{1})}\] = \[\frac{(y_{2} - y_{1})}{(x_{2} - x_{1})}\]


\[(y - y_{1})\] = \[(y_{2} - y_{1}) \times \frac{(x - x_{1}}{(x_{2} - x_{1}})\]


  1. Slope Intercept Form: Consider the following image. Line L cuts the y-axis at point P(0,a). Here, a is the distance from the origin at which the line L cuts the y-axis. This distance is referred to as the y-intercept. Here, let us take I(x, y) as some imaginary point on this line and P(0,a) as the defined point. We will plug in values in the point-slope equation:


\[m(x - x_{1}) = (y - y_{1})\]


m(x − 0)= (y − a)


mx - 0 = y - a


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mx - y + a = 0  (or) y = mx + a


Similarly, let us take a line M that cuts the x-axis at Q(b,0). Here, b is the distance from the origin, at which the line M cuts the x-axis. This distance is called the x-intercept. Here, let us take I(x, y) as some imaginary point on this line and Q(b,0) as the defined point. We will plug in values in the point-slope equation:


m (x − x1)= (y − y1)


m (x − b)= (y − 0)


m (x - b) = y


  1. Intercept Form: In this case, we are given a line with x-intercept (b,0) and y-intercept (0,a). Consider a random point I (x, y) on the line. Now, we lug in the values in the two-point form equation.


\[(y - y_{1})\] =\[(y_{2} - y_{1})\] x \[\frac{(x - x_{1})}{(x_{2} - x_{1})}\]


(y-0) = (a-0) x \[\frac{(x-b)}{0-b}\]  


y= \[(\frac{-a}{b})\] (x-b)


y= \[(\frac{a}{b})\] (b-x)


\[(\frac{x}{b}) + (\frac{y}{a})\] = 1


Solved Examples

Example 1: P(3,4) and Q(6, 4) are two collinear points. Represent it in the form of an equation. 


Solution:

The ordinates are equal; hence, this is a horizontal line. The slope of a horizontal line is zero, and it is represented as y = k. Therefore, the equation is y=4.


Example 2: In a particular line, the x-intercept is 3, and the y-intercept is 4. Find the value of the line. 


Solution:

Since we have both the intercepts, we can use the intercept form of equation and substitute values. 


\[(\frac{x}{b}) + (\frac{y}{a})\] = 1


\[(\frac{x}{3}) + (\frac{y}{4})\] = 1


\[\frac{(4x + 3)}{12}\] = 1


4x + 3y = 12


4x + 3y - 12 = 0


Hence, the equation for this line is 4x + 3y - 12 = 0.

FAQ (Frequently Asked Questions)

1. What is a Cartesian Plane?

A Cartesian plane, also called a coordinate plane, is a 2-dimensional plane created by two perpendicular lines defined as the x-axis (horizontal axis) and the y-axis (vertical axis). The precise location of the point on the Cartesian plane can be calculated using coordinates, that are written in the form of an ordered pair (x, y).

2. What are the Coordinates?

Coordinates are a set of values that allows us to indicate the precise location of a point in the coordinate plane. The abscissa and the ordinate are used to represent the location of the point on the graph. The horizontal value or the X-axis value is the abscissa value, while the longitudinal value, i.e. the Y-axis value, is the ordinate value. In the ordered pair (2, 3), 2 is abscissa , and 3 is the ordinate.