What Is Slope? Definition, Formula, Types, and Examples
The slope is one of the most important terms in geometry, and it’s used in all sorts of different ways in our everyday lives. Below, we’ll take a look at the definition, formula, types, and examples of the slope.
Slope Definition:
The slope is the measure of the steepness and inclination of a line. It is usually represented by the letter m. In other words, Slope is (change in y- coordinate)/(change in x- coordinate), where y is the change in height (in units) and x is the change in width (in units).
The slope can be linear, quadratic, or cubic. Linear slopes are linear in shape and quadratic slopes are quadratic in shape. Cubic slopes are cubical in shape and have three corners at (0,-1), (0,-2), and (1,-2).
Slopes can be found everywhere in our daily lives. For example, when you’re hiking up a mountain or looking at a road downhill from your car – you’re seeing a slope every step of the way. Similarly, when you’re building something like a roof or road surface, the slope is critical to ensuring that water will flow downhill evenly and that your structure will be sturdy.
Finally, using the slope to calculate other things isn’t just limited to physical objects. In engineering and architecture, a slope can also be used to determine how steep a road should be or how much water will flow down a hillside during rainstorms.
Formula of slope:
Slope can be expressed as:
| m = (y2 – y1)/(x2 – x1) |
Where,
- m is slope
- y is the y-intercept on the graph
- x is the x-intercept on the graph
This equation tells you how steep a line is based on its data points. Try a slope calculator to get the result of the slope of a line according to the above formula with a single click.
Types of Slope
There are four types of slopes:
- Positive slope
- Negative slope
- Zero slope
- Undefined slope
Positive slope
A line with a positive slope goes up from left to right on the graph. Positive slopes are usually found when objects are moving upwards (like when someone rides an elevator).
Negative slope
A line with a negative slope goes down from left to right on the graph. Negative slopes are usually found when objects are going downhill (like when someone is walking downstairs).
Zero slope
A line with a zero slope is horizontal on the graph. Zero slopes are found when objects are at rest (like when someone stands next to something without moving it). When the y coordinate of the line is zero it will give a zero slope and the horizontal line is drawn.
Undefined slope
A line with an undefined slope is vertical on the graph. When the x coordinate of the line is zero it will give an undefined slope and the vertical line is drawn.
How to calculate slope?
There are two ways to calculate the slope of a line.
- By using two points
- By using a line equation
Let us take a few examples of slope to understand it.
Slope Using Two Points:
| Example 1: Calculate the steepness of the line if the x and y coordinate points are (12, 18) and (20, 28). Solution Step 1: First of all, write the given coordinate points of x and y. X1 = 12, Y1 = 18, X2 = 20, Y2 = 28 Step 2: Take the general formula of the slope. The slope of a line = m = 𝚫Y/𝚫X Step 3: Now calculate the change in values of x and y. For 𝚫X 𝚫X = X2 – X1 𝚫X = 20 – 12 𝚫X = 8 For 𝚫Y 𝚫Y = Y2 – Y1 𝚫Y = 28 – 18 𝚫Y = 10 Step 4: Now place the values of 𝚫X & 𝚫Y in the general expression of the slope. Slope of a line = m = 𝚫Y/𝚫X Slope of a line = m = 10/8 Slope of a line = m = 5/4 Slope of a line = m = 1.25 |
| Example 2: Calculate the steepness of the line if the x and y coordinate points are (-2, -7) and (12, 21). Solution Step 1: First of all, write the given coordinate points of x and y. X1 = -2, Y1 = -7, X2 = 12, Y2 = 21 Step 2: Take the general formula of a slope. Slope of a line = m = 𝚫Y/𝚫X Step 3: Now calculate the change in values of x and y. For 𝚫X 𝚫X = X2 – X1 𝚫X = 12 – (-2) 𝚫X = 12 + 2 𝚫X = 14 For 𝚫Y 𝚫Y = Y2 – Y1 𝚫Y = 21 – (-7) 𝚫Y = 21 + 7 𝚫Y = 28 Step 4: Now place the values of 𝚫X & 𝚫Y in the general expression of a slope. Slope of a line = m = 𝚫Y/𝚫X Slope of a line = m = 28/14 Slope of a line = m = 14/7 Slope of a line = m = 2 |
Slope using Line Equation:
| Example 1: Evaluate the slope of the line with the help of the given line equation. 6y – 12x + 20 = 0 Solution Step 1: First of all, arrange the given line equation according to the equation of the slope-intercept form. Equation of slope intercept form: y = mx + b So, 6y – 12x = -20 6y = 12x – 20 y = (12x – 20)/6 y = 12x/6 – 20/6 y = 2x – 10/3 Step 2: Now compare the above expression with the equation of the slope-intercept form. y = mx + b y = 2x – 10/3 Hence, m = 2 and b = -10/3 |
| Example 2: Evaluate the slope of the line with the help of the given line equation. 4y + 16x – 24 = 0 Solution Step 1: First of all, arrange the given line equation according to the equation of the slope-intercept form. Equation of slope intercept form: y = mx + b So, 4y + 16x = 24 4y = -16x + 24 y = (-16x + 24)/4 y = -16x/4 + 24/4 y = -4x + 6 Step 2: Now compare the above expression with the equation of the slope-intercept form. y = mx + b y = -4x + 6 Hence, m = -4 and b = 6 |
Conclusion:
Now that you know what slope is and how to calculate it from this post. We also discussed the formula, and types, and solved examples of the slope. Once you practice the problems discussed above by hand, you will master the slope of the line.
