Straight Lines: two-point form

When the graph of a linear function passes through the points A (x1, y1) and B (x2, y2), then the equation can be written as:


It can also be written as:



For verifying this equation let P(x,y) be any point on the given line that passes through A(x$_{1}$,y$_{1}$) and B(x$_{2}$,y$_{2}$), as shown in the graph above. From A and B, we draw AL and BN that is perpendicular to the X-axis. From point P, we draw PM that is also perpendicular to the X-axis. Also, from A draw a line that is perpendicular to AD on BN.

Consider the similar triangles ADB and ACP constructed on the graph above. By the definition of a slope and similar triangles we can deduce that:



PC = PM -CM = y- y1

BD=BN-DN = y2 – y1

AC=LM=OM-ON = x-x1


Substitute the point values: $\frac{PC}{BD}=\frac{AC}{AD}$


This equation can also be written as:


There is another way of deriving the two-point form of the equation of a line. Recall that the slope of the line AB is the ratio of the change in y to the change in x as shown in the equation below:


$\Delta y=m\Delta x$


Substitute the equation for m:


The two- form is the starting step to derive the standard equation of a line when two points are given.

Example 1. Find a general form equation and point-slope form equation for the line through the pair of points (-1,2) and (5,/-4)?

Let ordered pair (x$_{1}$, y$_{1}$) be (1, 2) and (x$_{2}$, y$_{2}$) be (5, -4). Substitute these values in the two-point form equation







$4y+6x-2=0$ The point intercept form can be written from the two-point form as:



Make y as the subject of the equation





Here, m = -3/2 and b=7/2

Example 2. Write the two-point form and the general equation of the linear function y whose graph passes through the given pairs of points: (-1, 1) and (2, -4)

Plugging (x₁ , y₁) = (-1, 1) and (x₂, y₂) = (2, -4), we get:


Simplify the equation to get the general equation:





The two-point method to derive a general equation can not be used for the vertical line.

graph vertical line

Step 1. Find the slope:



We can convert the two-form equation as a function of x to obtain the general equation.


$y-y_{1}\frac{x_{2}-x_{1}}{ y_{2}-y_{1}}=x-x_{1}$






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