Different Forms of the Equation of Line

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.