Point Slope Form: An Introduction with Examples

(Last Updated On: June 20, 2022)

In mathematics, the point-slope form is used in algebra to determine the straight-line equation. The linear equation of the straight line is frequently used in differential, integral, and many other branches of mathematics.

The equation of the line can be determined by using various methods and techniques like intercept form, x & y-intercept form, two points form, point-slope form, and slope-intercept form. In this article, we’ll study all the basics of the point-slope form along with a lot of examples.

What is the point-slope form?

In algebra, the point-slope form is a well-known technique to find the lines’ equations. According to the point-slope form, the linear equation of the line is in the form of (y – y1) = m * (x – x1), where x & y are the fixed variables of the line.

X1 & y1 are the given coordinate points of the line, and “m” is the slope of the line. By the name of the point-slope form, it is clear that you have to use the slope and coordinate points of the line to find the straight line’s equation.

The slope is usually denoted by “m” and is the steepness of the line in which the terms can be positive, negative, zero, or undefined. You can calculate the slope of the line either by using two points or a line equation.

It is in the fractional form like change in the terms of y with respect to change in the terms of x. The slope of the line is a fraction of the rise and run by using the two coordinate points of the line. A slope calculator can also be used to find the slope of the line in a fraction of seconds.

Point slope form

 The slope of the line = m = rise / run

The slope of the line = m = (y2 – y1) / (x2 – x1)

In the equation of the point-slope form, if the slope “m” is not determined then you have to calculate it first. You can use the above-mentioned calculator or formula to determine the slope of the line in a fraction of seconds.

How to calculate the straight line’s equation by using the point-slope form?

The linear equation of the straight line can be determined easily with the help of the general equation of the point-slope form. You must have sound knowledge about the calculation of the slope of the line because it is the key factor of the line’s equation.

You can follow the below-mentioned steps to learn how to determine the line’s equation.

  1. First of all, identify the slope and coordinate points. If the slope is not given calculate it.
  2. Take the slope and a pair of points.
  3. Take the general equation of the point-slope form.
  4. Substitute the slope and points in the equation of the point-slope form.
  5. The output must be the equation of the straight line.

Let’s understand this concept by taking examples.

Example-I: For positive integers

Determine the straight line’s equation with the help of the point slope form if (x1, y1) = (12, 23) and (x2, y2) = (16, 33).

Solution

Step-I: Identify the given coordinate points of the line.

x1 = 12, x2 = 16, y1 = 23, y2 = 33

Step-II: Since the slope is not given, take the general equation of the slope of a line.

The slope of the line = m = [y2 – y1] / [x2 – x1]

Step-III: Substitute the given coordinate points of the line in the above equation.

The slope of the line = m = [y2 – y1] / [x2 – x1]

                                             = [33 – 23] / [16 – 12]

                                             = [10] / [4]

                                             = 10 / 4

                                             = 5 / 2

                                             = 2.5

Step-IV: Now take the general equation of the point-slope form.

(y – y1) = m * (x – x1)

Step-V: Now substitute the first pair of points (x1, y1) = (12, 23) and the slope m = 2.5 of the line in the above equation to determine the straight line’s equation.  

(y – y1) = m * (x – x1)

(y – 23) = 2.5 * (x – 12)

(y – 23) = 2.5 * x – 2.5 * 12

(y – 23) = 2.5x – 30

y – 23 – 2.5x + 30 = 0

y – 2.5x + 7 = 0

2.5x – y – 7 = 0

To ease up the calculation of determining the line’s equation, use a point-slope form calculator to get the solution with steps in a couple of seconds. To find the line’s equation using a calculator follow the below steps.

1: Input the pair of coordinate points and the slope “m” of the line.

2: Click the calculate button

3: The result will show below the calculate button.

4: Press the show more button to view the solution with steps.

Example 2: For negative integers

Determine the straight line’s equation with the help of the point slope form if (x1, y1) = (-2, -13) and (x2, y2) = (-6, -25).

Solution

Step-I: Identify the given coordinate points of the line.

x1 = -2, x2 = -6, y1 = -13, y2 = -25

Step-II: Since the slope is not given, take the general equation of the slope of the line.

The slope of the line = m = [y2 – y1] / [x2 – x1]

Step-III: Substitute the given coordinate points of the line in the above equation.

The slope of the line = m = [y2 – y1] / [x2 – x1]

                                             = [-25 – (-13)] / [-6 – (-2)]

                                             = [-25 + 13] / [-6 + 2]

                                             = [-12] / [-4]

                                             = -12 / -4

                                             = 6 / 2

                                             = 3

Step-IV: Now take the general equation of the point-slope form.

(y – y1) = m * (x – x1)

Step-V: Now substitute the first pair of points (x1, y1) = (-2, -13) and the slope m = 3 of the line in the above equation to determine the straight line’s equation.

(y – y1) = m * (x – x1)

(y – (-13)) = 3 * (x – (-2))

(y + 13) = 3 * (x + 2)

(y + 13) = 3 * x + 3 * 2

(y + 13) = 3x + 6

y + 13 – 3x – 6 = 0

y – 3x + 7 = 0

3x – y – 7 = 0

Summary

In this post, we have learned all the basics of the point-slope form along with explanations and examples. Now after reading the above post, you can easily solve any problem on this topic and grab all the basics of the point-slope form.

Author: Ruby Isobel

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