DETERMINE EQUATION OF LINE FROM TWO POINTS

Example 1 :

Find the equation of the line which is passing through the points (1, 4)  and (3, -2).

Solution :

Here (x₁, y₁) = (1, 4) and (x₂, y₂) = (3, -2).

Equation of a line :

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

(y - 4)/(-2 - 4)  =  (x - 1)/(3 - 1)

(y - 4)/(-6)  =  (x - 1)/2

2(y - 4)  =  (x - 1)(-6)

2y - 8  =  - 6 x + 6

6x + 2y - 8 - 6 = 0

6x + 2y - 14 = 0

Dividing the entire equation by 2,

3x + y - 7 = 0  

Example 2 :

Find the equation of the line which is passing through the points (0, 4)  and (3, 1).

Solution :

Here (x₁, y₁) = (0, 4) and (x₂, y₂) = (3, 1).

Equation of a line :

(y - 4)/(1 - 4) = (x - 0)/(3 - 0)

(y - 4)/(-3) = x/3

3(y - 4) = x(-3)

3y - 12 = - 3 x

3x + 3y - 12 = 0

 ÷ by 3 => x + y - 4 = 0  

Example 3 :

Find the equation of the line which is passing through the points (3, 1)  and (-3, 2).

Solution :

Here (x₁, y₁) = (3, 1) and (x₂, y₂) = (-3, 2).

Equation of a line :

(y - 1)/(2 - 1) =  (x - 3)/(-3 - 3)

(y -  1)/1 =   (x - 3)/(-6)

(-6)(y - 1) = 1(x - 3)

-6y + 6 = x - 3

x + 6y - 3 - 6 = 0

x + 6y - 9 = 0

Example 4 :

Find the equation of the line which is passing through the points (2, 6)  and (1, 2). 

Solution :

Here (x₁, y₁) = (2, 6) and (x₂, y₂) = (1, 2).

Equation of a line:

(y - 6)/(2 - 6) = (x - 2)/(1 - 2)

(y - 6)/(-4) = (x - 2)/(-1)

(-1)(y - 6) = (-4)(x - 2)

-y + 6 = -4x + 8

4x - y + 6 - 8 = 0

4x - y - 2 = 0

Example 5 :

Find the equation of the line which is passing through the points (5, -5)  and (2, 7)

Solution :

Here (x₁, y₁) = (5, -5) and (x₂, y₂) = (2, 7).

Equation of a line :

(y - (-5))/(7 - (-5)) =  (x - 5)/(2 - 5)

(y + 5)/(7 + 5) =   (x - 5)/(-3)

(y + 5)/12 =   (x - 5)/(-3)

-3(y + 5) = 12(x - 5)

-3y - 15 = 12 x - 60

12 x + 3y - 60 + 15 = 0

12 x + 3y - 45 = 0

 ÷ by 3 =>        4 x + y - 15 = 0 

Example 6 :

Find the equation of the line which is passing through the points (1, 3)  and (0, 7). 

Solution :

Here (x₁, y₁) = (1, 3) and (x₂, y₂) = (0, 7).

Equation of a line :

(y - 3)/(7 - 3) =  (x - 1)/(0 - 1)

(y - 3)/4 =   (x - 1)/(-1)

-1(y - 3) = 4(x - 1)

-y + 3 = 4 x - 4

4 x + y - 4 - 3 = 0

4 x + y - 7 = 0

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