Introduction of PointSlope Form With Definition, Derivation, and Examples
The pointslope form is used to determine the equation of a straight line. By using the slope and given point, we can easily find the equation of a straight line. Pointslope form deals with linear equations only. In a linear equation, the degree (power) of the unknown variables is one.
There are four following methods used to find the equation of a straight line by using different parameters.
 Pointslope form
 Intercept form
 Slopeintercept form
 Twopoint form
In this article, we will confine ourselves only pointslope form with its formula. We will discuss how to derive the equation of the pointslope. We will learn how to calculate the equation of a line by using pointslope form. Further, we will solve some examples with a stepbystep solution.
What is PointSlope Form?
The pointslope form is an equation of the straight line that passes through the given coordinate (x1, y1) and has a slope of line “m”. Mathematically, the equation of pointslope form is written as,
(y – y1) = m (x – x1)  Here,

We can find the equation of pointslope form when the slope and at least one point on the line are given.
Derive the equation of pointslope form
We can derive the equation of pointslope form by using the definition of the slope. Suppose the line passes through the given point (x1, y1), and (x, y) is any other point on the line. Then the slope of the line is defined as the ratio between the rate of change in y and the rate of change in x.
Slope = m = rate of change in y / rate of change in x
Slope = m = (y – y1) / (x – x1)
Multiply (x – x1) on both sides and by arranging the above equation, we get
(y – y1) = m (x – x1)
That is the equation of pointslope form.
Method to determine the equation of pointslope form
 Write down the slope and given point of the straight line.
 Put the given values in the equation of pointslope form that is (y – y1) = m (x – x1)
 Simplify the equation until it changes in a general form.
Some important Results of pointslope form
 If the line passes through the point (x1, y1), and has slope m, then the equation of pointslope form is
(y – y1) = m (x – x1)
 If the straight line passes through (a, b) horizontally, or in other words line passes through the yaxis, then it gives y1 = b, and we know that on the yaxis, x is always zero. So, x1 = 0. i.e. (x1, y1) = (0, b), and the equation of pointslope form becomes,
(y – b) = m (x – 0)
y – b = m x
y = m x + b
it is called the equation of the slopeintercept form.
 If the line passes through (a, b) vertically, or the line passes through the xaxis, then it gives x1 = a, y1 = 0 because on the xaxis y is always zero. i.e. (x1, y1) = (a, 0), then the equation of pointslope form will be,
(y – 0) = m (x – a)
y = m (x – a)
 If the straight line passes through the origin in this case (x1, y1) = (0, 0), i.e
(y – 0) = m (x – 0)
y = mx
Example of Pointsope form
To understand the pointslope form, let’s solve some examples.
Example 1.
Find the equation of a straight line when the line passes through the point (2, 3), and has slope 4.
Solution
Step 1. Note the given point and slope. Here
(x1, y1) = (2, 3)
m = 4
Step 2. Given data put in the equation of the pointslope form, i.e. (y – y1) = m (x – x1), we get
(y – 3) = 4 (x – 2)
Step 3. Simplify the above equation
y – 3 = 4x – 8
y = 4x – 8 + 3
y = 4x – 5
You can try a point slope form calculator by Allmath to evaluate the problems of line equations by using the slope and coordinate points of the line.
Example 2.
A line passes through a point (4, 7) that has a slope 2 / 3. Find the equation of a line by using the pointslope form equation.
Solution
In this example we have (x1, y1) = (4, 7) and m = 2 / 3
We know that the pointslope form equation (y – y1) = m (x – x1), now put the given value in this equation.
(y – 7) = (2 / 3) (x – 4)
3 multiply both sides, we get,
3y – 21 = 2(x – 4)
3y – 21 = 2x – 8
3y = 2x – 8 + 21
3y = 2x + 13
By dividing 3 each term,
y = (2 / 3) x + (13 / 3)
Hence, it is a required equation of a line.
Summary
In this article, we learn when we use pointslope form. We have discussed the definition of slopepoint form with its formula. All terms related to pointslope form are covered in this article. Then we derived the equation of pointslope form. We learn how to find the equation of a straight line by using pointslope form.
We discussed some important results of pointslope form in this article. Further, we solved some examples to find the equation of a straight line. After reading this article, you will be able to find the equation of the line by using the equation of point slope form.