Different Forms of the Equation of Line

Bookmark added to your notes.
View Notes
×

Equation of a Line in Different Forms

In this very article, we are going to discuss various forms of the equation of a line. A coordinate plane consists of an infinite number of points. If we consider a point P(x,y) in a 2d plane and a line name it as N. Then what we will determine is that the point we consider lies on the line L or it lies above or below of the line. That’s when straight-line comes into this scenario. Here we will include the important topic related to the equation of a line in different forms. 


Different Forms of the Equation of a Straight Line


A. Equation of Line Parallel to the Y-axis

Equation of a straight line which is parallel to the y-axis at a distance of ‘a’ then the equation of y-axis will be x=a (here ‘a’ is a coordinate in the plane).

Consider this  example Equation of line parallel to y-axis for coordinate (7,8) is x=8


B. Equation of Line Parallel to the X-axis

Equation of a straight line if the straight line is parallel to x-axis the equation will be y=a where ‘a’ is an arbitrary constant.

To understand one can consider this example, consider this a point (9,10) Equation of line parallel to the x-axis is x=9


C. Point- slope Form of an Equation

Let a line passing through a particular point Q(X1, Y1) and P(X, Y) be any point present in the mentioned line.

The slope of a line= Y - Y1/X – X2

And by the definition m is the slope,

Hence, m = Y - Y1/X – X2

On comparing Y – Y1 = m(X – X1) is the required point-slope form equation of a line


D. Equation of the Line in Two-point Form

Consider an arbitrary constant P(x,y) present in the line L and the Line L passes through two points A(x1,y1) and B(x2,y2). We consider ‘m’ as the slope of the line L.

m= y2-y1 / x2- x1

Then the equation of the line is

y2-y1 = m(x2-x1)

Substituting the value of m we get

y-y1={ y2- y1/ x2-x1}(x-x1)

Equation of the required line in two point form is y - y1= y2- y1/ x2 - x1(x -x1).


E. Equation of a Line in Intercept Form

Let AB line cuts intercept on the x-axis at (a, 0) and on the y-axis at (0, b)

From two-point form:

ð  y = -b/a (x – a)

ð  y = b/a ( a – x)

ð  x/ a + y/b = 1 is the required equation of line in intercept form


Example:

Consider finding the equation of a line which has made an intercept of 4 in x axis and has made a cut of y-axis in the graph


Solution

So, b = -3  and a = 4

ð  x/4 + y/-3 = 1

ð  3x – 4y = 12 hence the required equation of a line in intercept form


Slope-intercepts Form of a Line:

Consider a line L whose slope be m which cuts an intercept on the y-axis at the distance of ‘a’. hence the point is (0, a)

Hence, the required equation is:

ð  y – a = m(x – 0)

ð  y = mx + a which is the required equation of a line.


Example:

Find the equation of a line which has a  slope of -1 and has an intercept of 4 units in the positive section of the y-axis.


Solution

Here, m = -1 and a = -4

Substituting this value in y = mx + a we get:

ð  y = -x – 4

ð  x + y + 4 = 0


Solved Examples


Example

Determine the equation of a line which passes through the point (-4, -3) and it is parallel to the x-axis.


Solution

Here, m = 0, X1 = -4, Y1 = -3.

Through the above equation: Y + 3 = 0(X + 4)

ð  Y = -3 is the required equation


Example

Find the equation of the line joining by the points  (4,-2) and  (-1,3).


Solution: here the two given points are (X1,Y1) = (-1,3) and (X2,Y2)= (4,-2)

Equation of line in two point form is

ð  y – 3 = { 3 – (- 2)/ -1 – 4 }( x+1)

ð   - x – 1 = y – 3

ð  x + y – 2 = 0.

FAQ (Frequently Asked Questions)

Q1. What Does the Equation of a Line in Different Forms Mean?

Answer: Given a point, we can have an infinite number of straight lines passing through it. This suggests we have another condition for the same line to be uniquely represented on the XY plane.

Q2. What is the Example of the Equation of a Line in Different Forms?

Answer: Lets say, a straight line passing through the origin and having the slope of 90° or 270° or – 90° is the y – axis itself. So, depending upon the type of information available with us for a line, straight lines can be represented into different forms.

Q3. What are the Different Forms of the Line of Equations?

Answer: The different forms of line of equations are as follows:

  • Standard Form

  • General Form

  • Point Slope Form

  • Slope – intercept form

  • Two Point Form

Q4. How Can I Better Understand this Chapter?

Answer: In this article, you have learnt the equation of a line in different forms. To clear your concepts on various forms of the equation of a line we provided some examples in each different forms of the equation of a straight line. We hope you find this article useful.