Point Slope Form: Definition and Overview

The point slope form of a linear equation is either summation, difference, multiplication, or combination of all, variables and numbers. As the name suggests, linear means straight or in a single line. This is only possible if the variables in the equation are raised to unit power ( x¹).  

If any variable in the equation has the power of more than one, i.e., 2, 3, 4,….., infinity, then that equation is discarded as a linear equation.

For instance : 

• a + 11b – 3c = 20, is a linear equation. You can confirm this by reviewing the equation. Here, variables a, b, & c are all in unit power.
• Is a² – 2 = 0 linear? No! The equation is not linear as it has the power of variable ‘a’ as 2.
• Is a + 33b – 7c + (20)² = 0 a linear equation? Yes, indeed it is! Many of us may be wondering why this is linear. All the variables a, b, & c in the equation are with unit power. The numeric term has a power of 2, which is not our concern as (20)² = (400)¹.

How to find the Equation of a line? 

We are now familiar with the fact that the equation of a straight line is known as a linear equation. But how do we find the equation of any line in the cartesian space or the ‘x’ and ‘y’ axis? Listed below are the most popular techniques or methods of determining the equation of any straight line:

1. Point Slope form 
2. Slope-Intercept form
3. Two-point form
4. Intercept form

Depending on the various conditions and the data given to us, we can use any of these methods to find the equation of any line. This article will mainly focus on the point-slope form and point-slope form examples for better comprehension of the topic.

What is Point Slope Form?

The point slope form equation is utilized to obtain the equation of a straight line that passes through a given point and is inclined at a certain angle to the x-axis. A line’s equation is a set of equations fulfilled by each point lying on the line. We utilize the point-slope formula when we know the particular slope of a line and a point on it. You will get a better understanding of what is point-slope form and how to write the point-slope form in the next part of this article.

Understanding Point Slope Form

See the figure mentioned below carefully. In the graph, we have an ‘x-axis, a ‘y’ axis, and a straight line that is inclined to the x-axis with some angle. The line lies in the first quadrant which means that the coordinates of the line will be positive.

Let us suppose this line passes through a point k with the x-coordinate of k being ‘i’ and the y-coordinate being ‘j’. This gives us the point k (i, j). Second, let us assume that the slope of the line is ‘d’. We should use the point-slope form without a second opinion when such data is given without a second opinion.

A straight line is represented in point slope-intercept form by utilizing its slope ‘d’ and a point on the line k1 (i₁, j₁). The point k1 is a random point. Now let us see how to write the point-slope form equation of any linear line.

( i – i₁) = d ( j – j₁ )

Note that i and j are the variable terms they do not have any numeric value. ‘i1’ and ‘j1’ are known values for the random point and ‘d’ is also known. You will get at least 3 values to solve such types of equations. Now you are well aware of what is point-slope form, next let us learn how to write point-slope form. 

Point Slope Formula

As mentioned above, the point-slope formula in mathematics can be written as:

( i – i_x ) = dx ( j – j_x ), where x can be any number ranging from 1, 2, 3,……, infinity.

In this formula, i and j are variables always and remain unchanged throughout the equation. ‘i_x’ and ‘j_x’ can change according to the conditions given. These can be any known points on the line whose point-slope form equation has to be determined. ‘d_x’ also depends on the points ‘i_x’ and ‘j_x’. From the equation we see:

dx = ( i – i_x ) / ( j – j_x), which clearly states that d is also a dependent quantity.

Thus, 

• (i, j) is a random point with variable values which must remain unchanged.
• (i_x, j_x) is a random point with fixed values of x and y coordinates.
• The slope of the line in the formula is ‘d_x’.

 How to derive the Point Slope Formula:

Until now, we have seen the definition of the point-slope formula, what is point-slope form, how to write the point slope form is, and how it is used in terms of mathematics. We have also noted down the formula of it but ever wondered how this method was invented? Well, let us now learn how the point slope-intercept form came into existence. This topic will cover the proof of the point-slope form formula and related concepts.

Let us assume a straight line AB in the x & y axis graph. Let us mark a point Q on this line and assume the coordinates of this point as m & n ( m being the x coordinate and y being the y coordinate). Now let us mark a point R whose coordinates are known to us, say, m1 and n1, respectively. Last, let us consider that this line is at a slope ‘d’ from the x-axis. 

From the equation of the slope mentioned above we can concur that 

Slope = d = (difference of points on y coordinates) / (difference of points on x coordinates)

d = (n – n₁) / (m – m₁)

Multiplying both the sides with ( m – m₁ )

We get d (m – m₁) = (m – m₁) x (n – n₁) / (m – m₁)

This gives us the desired formula which is d (m – m₁) = (n – n₁) 

Results and Examples of Point Slope Form

Some important notes on Point-Slope Form are listed below:

1. A line with a slope ‘d’ passing through a known set of coordinates has the formula in point-slope form equation as (y – y₁) = d (x – x₁).
2. The formula of a straight line going horizontally through a point K (a, b) is given as y = b in the point slope-intercept form.
3. The formula of a straight line passing vertically through a point K (a, b) is given as y = a, in the point-slope form equation. 

These are very important results that concurred with the concepts mentioned above. In case 2 and case 3, the straight-line slope is neglected, hence giving single-term equations.

Let us now look at some point-slope form examples to ease your understanding of the point-slope form. The data for these examples is given below.

1. Slope = 3, Known point K (5, 1). The point slope intercept form for this example would be (y – 1) = 3 (x – 5)
2. Slope = -8, Reference point R (-19, 5). The point slope form equation for this example would be (y – 5 ) = -8 (x + 19) {because x – (-19) = x +19}
3. Slope = -3/4, Reference point R ( -5/2, 1/9). The point slope intercept form for this example would be (y – 1/9 ) = -3/4 (x + 5/2) {same logic as example 2}

Add to your knowledge: You can also simplify the equation in these point-slope form examples further to achieve the form: y = mx + b. This is known as the general or standard equation of a straight line.

How to Solve Point Slope Form?

To solve the point-slope form for a given straight line for finding the equation of the given line, we can follow the steps given below.

• Step 1: Pin down the following values in your notebook, the slope ‘d’ of the line, and the coordinates of the reference point say m1 and n1.
• Step 2: Substitute these values in the point-slope form equation. 
• Step 3: Figure out the standard equation of the line by simplifying the formula obtained in step 2.

To better understand how to solve and find the equation of any straight line with given values, see one of the point-slope form examples solved for you below. Carefully examine each step done in the example.

Example: Find the line equation with a slope of -1/4 and go through point A (4, -7).

Answer: The given point A has x coordinate as 4 and y coordinate as -7, therefore 

m1 = 4

n1 = -7

Slope = d = -1/4

Equating in the formula

(n – n1) = d (m – m1)

(n – (-7)) = -(1/4) (m – 4)

(n + 7) = -(1/4) (m – 4)

4 (n + 7) = -1 (m – 4)

4n + 28 = 4 – m

4n + m = 4 – 28

4n + m = -24

Therefore the equation of the line according to the point slope form examples is 4n + m = -24 or 4n + m + 24 = 0.

Example: Find the equation of the line using (–3, 9) with slope -6. 

Solution: 

The given point is: (-3, 9) 

Slope = -6 m 

Let (-3, 9) = (x1, y1) 

We know that a line’s equation in point slope form is: 

y – y1 = m (x – x1) 

y – 9 = -6 when the values are substituted 

[x – (-3)] 

y – 9 = -6(x + 3) 

y – 9 = -6x – 18 

6x + y – 9 + 18 = 0 

6x + y + 9 = 0 

As a result, this is the line’s required equation. 

Hope you have grasped all the information necessary about what is point-slope form in mathematics, how to write point-slope form, point-slope form examples, and all the relevant concepts. Keep practicing and revising to get all the concepts at your fingertips!