How to Find the Equation of a Line?

Equation of a line is defined as the algebraic equation which can be described as
Y = mx + d, Where value of ‘m’ is the given Slope and the y- intercept is given by ‘d’. Here we will see how to find the equation of a line.

There are various methods to find the equation of a line such as:

When Point and slope are given.

When two points are given.

When a point and a parallel line are given.

When a point and a perpendicular line are given.
 
Mathematically, to find the equation of a line through two different co Ordinate points (a₁, b₁) and (a₂, b₂) is given by

(i)         b = b₁ + [(b₂ – b₁) / (a₂ – a₁)] · (a – a₁),

where a₁ and a₂ are supposed to be distinct. If two points are equal, then equation can be written as a = a₁ and it does not need a second point.
 Above equation of line can also be written as:

   b – b₁ = [(b₂- b₁) / (a₂ – a₁)] · (a – a₁),

or it can be written as:

(a₂ – a₁)·(b – b₁) = (b₂ – b₁)·(a – a₁),

Whereas, the easiest formula for line to remember is:

(b – b₁)/(b₂ – b₁) = (a – a₁)/(a₂ – a₁),

Above equation is the simplest formula to find the equation of line.

To find the equation of a line through Point-slope equation is given by:

The equation of a line through point (p, q) with a given slope of m is y = m(x - a) + b
 or y - b = m(x - a).
 
Example: Find the equation of a line if the Slope of a line is 1/3 and the point is (3, - 4)?

Solution: Here m = 1/3, a= 3 and b= -4 given, therefore according to the formula y = m(x – a) + b

So, y = 1/3(x – 3) + 4.

Equation is given as: 3y = x + 9.