Equation of a Line
Different Forms of Equation of a Straight Line
There are different forms of an equation of a line. Some of them are explained below: –

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

Equation For Vertical Line: A vertical line is parallel the yaxis. Hence, all the points on this line need to be equidistant from the yaxis. The abscissa defines the distance of a point from the yaxis. Since all the points of the vertical line are equidistant from the yaxis, 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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Point – Slope Equation: The pointslope form of equation can be used for a nonhorizontal, nonvertical 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 pointslope 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.

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

Slope Intercept Form: Consider the following image. Line L cuts the yaxis at point P(0,a). Here, a is the distance from the origin at which the line L cuts the yaxis. This distance is referred to as the yintercept. 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 pointslope 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 xaxis at Q(b,0). Here, b is the distance from the origin, at which the line M cuts the xaxis. This distance is called the xintercept. 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 pointslope equation:
m (x − x1)= (y − y1)
m (x − b)= (y − 0)
m (x – b) = y

Intercept Form: In this case, we are given a line with xintercept (b,0) and yintercept (0,a). Consider a random point I (x, y) on the line. Now, we lug in the values in the twopoint form equation.
\[(y – y_{1})\] =\[(y_{2} – y_{1})\] x \[\frac{(x – x_{1})}{(x_{2} – x_{1})}\]
(y0) = (a0) x \[\frac{(xb)}{0b}\]
y= \[(\frac{a}{b})\] (xb)
y= \[(\frac{a}{b})\] (bx)
\[(\frac{x}{b}) + (\frac{y}{a})\] = 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 xintercept is 3, and the yintercept 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.
1. What is a Cartesian Plane?
A Cartesian plane, also called a coordinate plane, is a 2dimensional plane created by two perpendicular lines defined as the xaxis (horizontal axis) and the yaxis (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 Xaxis value is the abscissa value, while the longitudinal value, i.e. the Yaxis value, is the ordinate value. In the ordered pair (2, 3), 2 is abscissa , and 3 is the ordinate.