A Beginner's Guide to Point-Slope Form in Mathematics

A Beginner's Guide to Point-Slope Form in Mathematics


In mathematics, the point slope form is a well-known method and is one of the ways to compute the straight line equation. All points along the line are satisfied by the equation of the straight line. There is a linear equation in two variables, x & y, that represents the line.


According to the given information, the straight line equation has various techniques for the representation of the line equation such as slope intercept form, point slope form, two point form, and intercept form. Let’s explore one of the methods for representing the equation of the line say point slope form with examples. 

What is a Point Slope Form?

The well-known method for representing the straight line equation is said to be the point slope form. This method can be applied when using a line's slope and points. For repressing point slope form, we use the following equation:

y – y1 = m (x – x1)

Where x & y are the arbitrary points of the line, x1 & y1 are the given points of the line, and “m” represents the slope of the line. In point slope form, the above formula (equation) only works if we know the slope "m" of the line and the point (x1, y1)

You must have sound knowledge about the slope of the line if the slope is not given, you must have some knowledge about finding the slope with the help of the x and y coordinate points of the line.

What is a Slope?

A slope is an inclined plane that measures the steepness/sharpness of the x & y-axis line. The slope “m” of the line is a ratio/quotient of change in the values of y “y2 – y1” and the change in the values of x “x2 – x1”. The rate of change of the coordinate points will help us to find the slope of the line. 

The general equation to find the slope of a line is:

Sharpness of a line = m = (change in the values of y) / (change in the values of x)

Sharpness of a line = m = (y2 – y1)/(x2 – x1)

Deriving the Point-Slope Formula

Let’s derive the point slope form equation with the help of the equation of the slope intercept form “y = mx + b” where “m” represents the slope of the line and the y-intercept of the line is represented by “b”.

First of all, choose the points from the line say x1 & y1

Substitute the points (x1, y1) in place of (x, y) in the equation of slope intercept form.

y1 = mx1 + b … (A)

b = y1 – mx1 … (B)

Now isolate the equation of slope intercept form in terms of y and subtract the y-intercept “b” from both sides.

y = mx + b

y – b = mx 

Substitute (B) in the above equation.

y – (y1 – mx1) = mx

y – y1 + mx1 = mx

y – y1 = mx – mx1

y – y1 = m(x – x1)

Hence, the equation of slope intercept form has been transformed into the equation of point slope form.

Ways to Find Point Slope Form

There are several ways to find point slope form to represent a straight line equation. Such as:

  • Two points form

  • 1 Point and Slope form

If two points are given, then you have to find the slope of the line and evaluate it to the general expression of the point slope form along with a point of the line. While the one point and slope form method is a way when the slope and a point are given. 

Let’s take a few examples to understand how to compute the point slope form equation.

Example A: For Two Points

Compute the linear equation of the straight line according to the point slope form if the given coordinate points of x are: (x1, x2) = (5, 15) and y are: (y1, y2) = (10, 50)

Solution

Step 1: Write the given points of the x-axis and y-axis of the line.

Given Components

Point of x (say x1)

5

Point of x (say x2)

15

Point of y (say y1)

10

Point of y (say y2)

50

Step 2: Now find the slope of the line by using the rise over run formula as the slope of the line is not given.

Formula 

Calculation

m = (y2 – y1) / (x2 – x1)

Slope = m = (50 – 10) / (15 – 5) 

Slope = m = (40) / (10) 

Slope = m = 4/ 1

 Slope = m = 4

Step 3: Now take the above derived equation of the point slope form.

y – y1 = m (x – x1)

Step 4: Now find the straight line equation by putting slope “m” and coordinate point of x & y in the general formula of point slope form.

y – y1 = m (x – x1)

y – (10) = 4 * (x – 5)

A line's linear equation can be expressed using the above equation, calculated as a point slope form. We can simplify this equation further by putting it in standard form.

(y – 10) = 4 * (x – 5)

y – 10 = 4x + 20

y – 10 – 4x – 20 = 0

y – 4x – 30 = 0

4x – y + 30 = 0 or y = 4x + 30

Example B: For 1 point and slope 

Compute the linear equation of the line according to the point slope form 10 is the slope of the line and the point of the line is: (x1, y1) = (-1, -6) 

Solution

Step 1: Write the given slope and a point of the line.

Given Components

Slope (say m)

10

Point of x (say x1)

-1

Point of y (say y1)

-6

Step 2: Now take the above derived equation of the point slope form.

y – y1 = m (x – x1)

Step 3: Now find the straight line equation by putting slope “m” and coordinate point of x & y in the general formula of point slope form.

y – y1 = m (x – x1)

y – (-6) = 10 * (x – (-1))

y + 6 = 10 * (x + 1)

A line's linear equation can be expressed using the above equation, calculated as a point slope form. We can simplify this equation further by putting it in standard form.

y + 6 = 10 * x + 10 * 1

y + 6 = 10x + 10

y + 6 - 10x – 10 = 0

y – 10x – 4 = 0

10x – y + 4 = 0

or

y = 10x + 4

The problems of finding equation of the line using point slope form can also be solved by using online tools as these tools provide the solution with steps in a matter of seconds. We personally use the below tools to solve the above discussed problems.

Final Words

The point slope form is a well-known way of representing the equation of the straight line along with the slope intercept form. The basics of this method have been discussed above to understand this topic precisely. You can have a look at the examples to learn how to solve the problems of point slope form.

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