Equations of straight lines

The equations of a straight lines describes a relationship between two variables. This variables are usually described as x which is independent variable and y which is dependant variable. The value of x determines the value of y in the line.

consider two arbitrary points on a Cartesian plane shown below.

Any two points joined together can make a straight line. Let the points be A(x₁ y₁) and B(x₂, y₂). From our previous lessons on gradient of a line; we can find gradient from the equation:

and so the gradient of any arbitrary line on a cartesian plane can be given by:

let the gradient be m:

we will make y to be the subject;

we express the general equation of the line as :

where c is the point at which the line cuts the y -axis.

Example problem on equations of straight lines

What is the equation of the line passing through A(-2, 3) and B(5, 8) on a cartesian plane?

solution:

y₁ = 3, y₂=8

x₁ = -2 , x₂=5

substituting in the equation above;

we now take an arbitrary (x, y) and one point we know about line point (5,8) point on the line and calculate gradient again. That is

we see that 7(y-8)= 5(x-5)

7y – 56 = 5x -25

7y = 5x -25 +56

7y = 5x +31

hence the equation of the line is :

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• Equations of straight lines
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