Finding a linear equation Ax + By + C = 0 given 2 points

If you have 2 points, for example:

P( 2, 1 ); Q( 5, 7 )

You can find the linear equation of the line that passes through those points in the form:

Ax + By + C = 0

in one step by simply using the formula:

(y₁ – y₂)x + (x₂ – x₁)y + (x₁y₂ – x₂y₁) = 0

OR, we can write this as (easier to read!)
(p_y – q_y)x + (q_x – p_x)y + (p_xq_y – q_xp_y) = 0

Let’s try it:

Take Point1=( 2, 1 )
Take Point2=( 5, 7 )

Find the LINEAR EQUATION of the line that passes through the points (2,1) and (5,7). Your answer must be in the form of Ax + By + C = 0.

Using the equation:
(y₁ – y₂)x + (x₂ – x₁)y + (x₁y₂ – x₂y₁) = 0

We’ll just plug numbers in:
(1 – 7)x + (5 – 2)y + ( (2 x 7) – (5 x 1) ) = 0
-6x + 3y + (14 – 5) = 0
-6x + 3y + 9 = 0

Factoring a -3 out:
-3( 2x – y – 3 ) = 0

Dividing both sides by -3:
2x – y – 3 = 0

And that’s the answer.

That’s hella shorter than having to go through and do it using y = mx + b, and its a systematic way to do it in a computer in case m is infinity.

Formula from Peter Shirley’s CG book