In mathematics, the slope is a widely used concept in different fields to be a supportive part or essential term. It is frequently used in the straight line equation in point-slope form and slope-intercept form to express the equation in the form of a linear equation.
The slope of the line works like a compass to find whether the lines are perpendicular, parallel, or none. The slope of the line is evaluated with the help of coordinate points of the line that are lying horizontally and vertically.
In this post, we will explore the term slope of a line along with its definition, formula, types, and examples.
What is a slope of a line?
The measure of the steepness of the line is said to be the slope of a line. It is actually a ratio of the change in the coordinate points of y with respect to the change in the coordinate points of x. The change in the coordinate points of x and y are represented by Δx & Δy respectively.
The slope of a line can be written equivalent to the tan(θ) as it is the ratio of change in the values of y over change in the values of x. The slope of a line can also be expressed as the rise over run (change in y over change in x) with the help of the coordinate points of the line.
The slope of a line is denoted by “m”.
Formula of the slope
The general formula to evaluate the slope of the line is:
The slope of a line = m = rise/run
The slope of a line = m = change in y coordinate/change in x coordinate
Slope of a line = m = y2 – y1 / x2 – x1
Where
“m” is the slope of the line
x1 & x2 are the coordinate points of the horizontal line
y1 & y2 are the coordinate points of the vertical line
Types of Slope
There are 4 types of slope depending upon the relationships among the coordinate points of y and x. below are the types of slope.
Positive Slope
Negative Slope
Zero Slope
Undefined Slope
Below is a brief introduction to the types of slopes.
Positive Slope
The positive slope is the first type of measure of steepness. In a positive slope, the line goes upward from left to right or increases when seen from left to the right. In this type of slope, the measure of the steepness is always positive.
Negative Slope
The negative slope is the second type of measure of steepness. In a negative slope, the line goes downward from left to right or decreases when seen from the left to the right. In this type of slope, the measure of the steepness is always negative.
Zero Slope
The zero slope is the third type of measure of steepness. In zero slope, the line neither goes upward nor downward from left to right or neither increases nor decreases when seen from left to the right. In this type of slope, the measure of the steepness is always zero.
The y-axis of the line is null or zero as it lies at the origin. The line is always horizontal in the zero slope.
Undefined Slope
The undefined slope is the 4th type of measure of steepness. In an undefined slope, the line goes neither goes upward nor downward or not moving from left to right or right to left. In this type of slope, the measure of the steepness is always undefined.
The x-axis of the line is null or zero as it lies at the origin. The line is always vertical in the undefined slope.
How to evaluate the slope of the line?
Below are a few examples of the slope of the line according to the types of slope.
Example 1: For positive slope
Evaluate the slope of the line with the help of the given points of the x-axis and y-axis of the line.
The points are (22, 8) and (40, 18).
Solution
Step 1: First of all, take the points of the x-axis and y-axis of the line.
x1 = 22, y1 = 8, x2 = 40, y2 = 18
Step 2: Now take the general expression of the measure of the steepness.
Slope of a line = m = 𝚫Y/𝚫X = y2 – y1 / x2 – x1
Step 3: Now evaluate the slope “m” of the line by using the points of the x-axis and y-axis of the line.
For 𝚫X
𝚫X = X2 – X1
𝚫X = 40 – 22
𝚫X = 18
For 𝚫Y
𝚫Y = Y2 – Y1
𝚫Y = 18 – 8
𝚫Y = 10
Put the points in the general expression of the slope
Slope of a line = m = y2 – y1 / x2 – x1
Slope of a line = m = 10/18
Slope of a line = m = 5/9
Slope of a line = m = 0.56
You can also try online slope calculators to find the steepness of the line with steps.
The above example is solved by https://www.slopecalculator.io/
Example 2: For negative slope
Evaluate the slope of the line with the help of the given points of the x-axis and y-axis of the line.
The points are (2, 18) and (10, 8).
Solution
Step 1: First of all, take the points of the x-axis and y-axis of the line.
x1 = 2, y1 = 18, x2 = 10, y2 = 8
Step 2: Now take the general expression of the measure of the steepness.
Slope of a line = m = 𝚫Y/𝚫X = y2 – y1 / x2 – x1
Step 3: Now evaluate the slope “m” of the line by using the points of the x-axis and y-axis of the line.
For 𝚫X
𝚫X = X2 – X1
𝚫X = 10 – 2
𝚫X = 8
For 𝚫Y
𝚫Y = Y2 – Y1
𝚫Y = 8 – 18
𝚫Y = -10
Put the points in the general expression of the slope
Slope of a line = m = y2 – y1 / x2 – x1
Slope of a line = m = -10/8
Slope of a line = m = -5/4
Slope of a line = m = -1.25
Example 3: For zero slope
Evaluate the slope of the line with the help of the given points of the x-axis and y-axis of the line.
The points are (22, 7) and (30, 7).
Solution
Step 1: First of all, take the points of the x-axis and y-axis of the line.
x1 = 22, y1 = 7, x2 = 30, y2 = 7
Step 2: Now take the general expression of the measure of the steepness.
Slope of a line = m = 𝚫Y/𝚫X = y2 – y1 / x2 – x1
Step 3: Now evaluate the slope “m” of the line by using the points of the x-axis and y-axis of the line.
For 𝚫X
𝚫X = X2 – X1
𝚫X = 30 – 22
𝚫X = 8
For 𝚫Y
𝚫Y = Y2 – Y1
𝚫Y = 7 – 7
𝚫Y = 0
Put the points in the general expression of the slope
Slope of a line = m = y2 – y1 / x2 – x1
Slope of a line = m = 0/8
Slope of a line = m = 0
Example 4: For undefined slope
Evaluate the slope of the line with the help of the given points of the x-axis and y-axis of the line.
The points are (15, 23) and (15, 33).
Solution
Step 1: First of all, take the points of the x-axis and y-axis of the line.
x1 = 15, y1 = 23, x2 = 15, y2 = 33
Step 2: Now take the general expression of the measure of the steepness.
Slope of a line = m = 𝚫Y/𝚫X = y2 – y1 / x2 – x1
Step 3: Now evaluate the slope “m” of the line by using the points of the x-axis and y-axis of the line.
For 𝚫X
𝚫X = X2 – X1
𝚫X = 15 – 15
𝚫X = 0
For 𝚫Y
𝚫Y = Y2 – Y1
𝚫Y = 33 – 23
𝚫Y = 10
Put the points in the general expression of the slope
Slope of a line = m = y2 – y1 / x2 – x1
Slope of a line = m = 10/0
Slope of a line = m = ∞
The Bottom line
The slope of the line is discussed along with the definition, formulas, types, and examples. Now you can get all the basics of the slope on one platform. Once you get the basics of the slope of the line you will be able to solve any example of the slope.