What is the slope of a line, and how can we calculate it?

The slope of a line is one of the most important topics in geometry. This concept was first given by French mathematician Renee Descartes (1595 – 1650) and he also derived the formula for the slope of a line. An Irish mathematician Matthew O’Brien in 1844, gave the symbolical representation of the slope.

The Irish man denoted the slope by “m”. The slope is a very commonly used concept, many people use it in their daily routine like painters use it to draw professional paintings, and engineers use it to make buildings and the entering floors (ramps), especially while constructing wheelchair ramps slope plays a major role. 

The slope of a line can be zero, increasing, decreasing, or undefined. Further in this article, we will describe the slope’s basic definition and elaborate on some real-world examples. In the example section, the method of calculating the slope will be discussed. 

Definition of slope: 

In mathematics, the slope of a line is a number that describes both the direction and the steepness of the line. It is also known as the gradient of a line. 

The slope is often represented by “m”. 

Formula of slope “m”:

The slope “m” of a straight line can be calculated using the rise-over-run formula: 

Slope = m = rise / run 

In the above formula, “rise” denotes the change in the y-axis whereas “run” denotes the change in the x-axis on the Cartesian plane so the formula can be rewritten as: 

Slope = m = Δy / Δx

The change always occurs when the minimum number of points is two i.e., starting and ending points. Let x1, and x2 are the points on the x-axis, & y1 and y2 are the two points on the y-axis. The final form of the above formula will be: 

Slope = m = (y2 – y1) / (x2 – x1)

Real-world applications: 

There are some real-world applications of the term “slope”: 

1. In constructing roads one ought to parent out how steep the street will be. Architects use math to get the slope of an avenue that's direction. Architects try this to get the street secure sufficiently so as for humans and motors to skip through. They need to make journeying amusing and secure.

2. The windshields and playground slides are also designed using the concept of slope. 

Types of slopes: 

Generally, there are four types of slopes: 

• Negative slope

• Positive slope

• Undefined slope 

• Zero slope 

Example section: 

In this section, we will discuss the method of calculating the slope of a line. Generally, there are two methods to calculate the slope of a line, we will discuss both methods in the sub-sections:

Section 1: Using two points:

Example 1: 

If a line has points (1, 5) and (14, 12) then find the slope “m”. 

Solution:

Step 1: Extract the given data.

The points are x1 = 9, y1 = 5, x2 = 14 and y2 = 12

Step 2: Write down the general formula of “m”.

Slope = m = (y2 – y1) / (x2 – x1)

Step 3: Insert the given data in the above formula

Slope = m = (12 - 5) / (14 - 1)

Slope = m = (7) / (13)

Slope = m = 0.53 

The slope of the line is 0.53. As the value of the slope is positive so we can state that the slope is positive and if we look at the position of the object, it is moving in the right upward direction.

Example 2: 

Calculate the slope “m” if x1 = -3, y1 = -8, x2 = -2, and y2 = -11.

Solution:

Step 1: Extract the given data.

The points are x1 = -3, y1 = -8, x2 = -2, and y2 = -11

Step 2: Write down the general formula of “m”.

Slope = m = (y2 – y1) / (x2 – x1)

Step 3: Insert the given data in the above formula

Slope = m = (-11 – (-8)) / (-2 – (-3))

Slope = m = (-3) / (1)

Slope = m = -3 

The slope of the line is -3. As the slope is negative, if we look at the position of the object, it is moving in the left downward direction. 

You can take assistance from the slope calculator to get rid of these lengthy calculations. 

Section 2: Using y = mx + c formula 

Example 1: 

Calculate the slope of the linear equation x + 3y + 15 = 0.

Solution

Step 1: Arrange the given equation into the general form of the equation of a line.

x + 3y + 15 = 0

x + 3y = -15

3y = -x - 15

y = (-x - 15) / 3

Step 2: Simplification.

y = -x / 3 – 15 / 3 

y = -x / 3 - 5

y = - (1/3) x – 5

Step 3: Comparing the above equation with y = mx + c

y = mx + c

y = - (1/3) x – 5

Slope = m = - 1/3

Summary: 

In this article, we have studied the basic definition of the term “slope of a line”. In the introductory paragraph, we studied the basic history and some general examples. We also discussed some real-life examples and applications of slope. 

The slope is a very common term and the method of calculating it is quite simple. In the example section, we have tried to cover all the possible steps and methods for calculating the slope of a line. You have witnessed that it is not a very difficult topic now you can easily solve all the problems related to it.