Slope

The slope of a line is the measure of the steepness and the direction of the line. Finding the slope of lines in a coordinate plane can help in predicting whether the lines are parallel, perpendicular, or none without actually using a compass.

The slope of any line can be calculated using any two distinct points lying on the line. The slope of a line formula calculates the ratio of the "vertical change" to the "horizontal change" between two distinct points on a line. In this article, we will understand the method to find the slope and its applications.

1.	What is Slope?
2.	Slope of a Line
3.	Slope of a Line Formula
4.	How to Find Slope?
5.	Types of Slope
6.	Slope of Perpendicular Lines
7.	Slope of Parallel Lines
8.	FAQs on Slope

What is Slope?

The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line. The net change in y coordinate is Δy, while the net change in the x coordinate is Δx. So the change in y coordinate with respect to the change in x coordinate can be written as,

m = Δy/Δx
where,m is the slope

Note that tan θ = Δy/Δx

We also refer this tan θ to be the slope of the line.

Slope of a Line

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane. Calculating the slope of a line is similar to finding the slope between two different points. In general, to find the slope of a line, we need to have the values of any two different coordinates on the line.

Slope Between Two Points

The slope of a line can be calculated using two points lying on the straight line. Given the coordinates of the two points, we can apply the slope of line formula. Let coordinates of those two points be,
P₁ = (x₁, y₁)
P₂ = (x₂, y₂)

As we discussed in the previous sections, the slope is the "change in y coordinate with respect to the change in x coordinate of that line". So, putting the values of Δy and Δx in the equation of slope, we know that:
Δy = y₂ - y₁
Δx = x₂ - x₁

Hence, using these values in a ratio, we get :

Slope = m = tan θ = (y₂ - y₁)/(x₂ - x₁)

where, m is the slope, and θ is the angle made by the line with the positive x-axis.

Slope of a Line Formula

The slope of a line can be calculated from the equation of the line. The general slope of a line formula is given as,

y = mx + b

where,

m is the slope, such that m = tan θ = Δy/Δx
θ is the angle made by the line with the positive x-axis
Δy is the net change in y-axis
Δx is the net change in x-axis

Slope of a Line Example

Let us recall the definition of slope of a line and try solving the example given below.

Example: What is the equation of a line whose slope is 1, and that passes through the point (-1, -5)?

Solution:

We know that if the slope is given as 1, then the value of m will be 1 in the general equation y = mx + b. So, we substitute the value of m as 1, and we get,

y = x + b

Now, we already have the value of one point on the line. So, we put the value of the point (-1, -5) in the equation y = x + b, and we get,

b = -4

Hence, substituting the values of m and b in the general equation, we get our final equation as y = x - 4.

Equation is: y = x - 4

How to Find Slope?

We can find the slope of the line using different methods. The first method to find the value of the slope is by using the equation is given as,

m = (y₂ - y₁)/(x₂ - x₁)
where, m is the slope of the line.

Also, the change in x is run and the change in y is rise or fall. Thus, we can also define a slope as, m = rise/run

Finding Slope from a Graph

While finding the slope of a line from the graph, one method is to directly apply the formula given the coordinates of two points lying on the line. Let's say the values of the coordinates of the two points are not given. So, we have another method as well to find the slope of the line. In this method, we try to find the tangent of the angle made by the line with the x-axis. Hence, we find the slope as given below.

The slope of a line has only one value. So, the slopes found by Methods 1 and 2 will be equal. In addition to that, let's say we are given the equation of a straight line. The general equation of a line can be given as,

y = mx + b

The value of the slope is given as m; hence the value of m gives the slope of any straight line.

The below-given steps can be followed to find the slope of a line such that the coordinates of two points lying on the line are: (2, 4), (1, 2)

Step 1: Note the coordinates of the two points lying on the line, (x₂, y₂), (x₁, y₁). Here the coordinates are given as (2, 4), (1, 2).
Step 2: Apply the slope of line formula, m = (y₂ - y₁)/(x₂ - x₁) = (4 - 2)/(2 - 1) = 2.
Step 3: Therefore, the slope of the given line = 2.

Types of Slope

We can classify the slope into different types depending upon the relationship between the two variables x and y and thus the value of the gradient or slope of the line obtained. There are 4 different types of slopes, given as,

Positive slope
Negative slope
Zero slope
Undefined Slope

Positive Slope

Graphically, a positive slope indicates that while moving from left to right in the coordinate plane, the line rises, which also signifies that when x increases, so do y.

Negative Slope

Graphically, a negative slope indicates that while moving from left to right in the coordinate plane, the line falls, which also signifies that when x increases, y decreases.

Zero Slope

For a line with zero slope, the rise is zero, and thus applying the rise over run formula we get the slope of the line as zero.

Undefined Slope

For a line with an undefined slope, the value of the run is zero. The slope of a vertical line is undefined.

Slope of Horizontal Line

We know that, a horizontal line is a straight line that is parallel to the x-axis or is drawn from left to right or right to left in a coordinate plane. Therefore, the net change in the y-coordinates of the horizontal line is zero. The slope of a horizontal line can be given as,

Slope of a horizontal line, m = Δy/Δx = zero

Slope of Vertical Line

We know that, a vertical line is a straight line that is parallel to the y-axis or is drawn from top to bottom or bottom to top in a coordinate plane. Therefore, the net change in the x-coordinates of the vertical line is zero. The slope of a vertical line can be given as,

Slope of a vertical line, m = Δy/Δx = undefined

Slope of Perpendicular Lines

A set of perpendicular lines always has 90º angle between them. Let us suppose we have two perpendicular lines l₁ and l₂ in the coordinate plane, inclined at angle θ₁and θ₂ respectively with the x-axis, such that the given angles follow the external angle theorem as, θ₂= θ₁ + 90º.

Therefore, their slopes can be given as,
m₁ = tan θ₁
m₂ = tan (θ₁ + 90º) = - cot θ₁
⇒ m₁ × m₂ = -1

Thus, the product of slopes of two perpendicular lines is equal to -1.

Slope of Parallel Lines

A set of parallel lines always have an equal angle of inclination. Let us suppose we have two parallel lines l₁ and l₂ in the coordinate plane, inclined at angle θ₁and θ₂ respectively with the x-axis, such that the, θ₂= θ₁.

Therefore, their slopes can be given as,
⇒ m₁ = m₂

Thus, the slopes of the two parallel lines are equal.

Important Notes on Slope:

The slope of a line is the measure of the tangent of the angle made by the line with the x-axis.
The slope is constant throughout a straight line.
The slope-intercept form of a straight line can be given by y = mx + b
The slope is represented by the letter m, and is given by, m = tan θ = (y₂ - y₁)/(x₂ - x₁)

Challenging Question:

A line has the equation y = 2x - 7. Find the equation of a line that is perpendicular to the given line, and is passing through the origin.

☛ Related Topics:

Solved Examples on Slope

Example 1: Given a line with the equation, 2y = 8x + 9, find its slope.

Solution:

We know that the general formula of the slope is given as, y = mx + b

Hence, we try to bring the equation to this form. We make the coefficient of y = 1, and hence we get,

y = 4x + 4.5

Clearly, the coefficient of x is found to be 4. Hence, our slope will be same as the coefficient of x.

The slope is 4.
Example 2: The equation of a line is given as x = 5. Find the slope of the given line.

Solution:

The equation is given as x = 5. We can thus see that y is missing from our equation. Hence, we can assume the coefficient of y to be 0 for now. Thus we now get,

(0)y = x - 5

Now, we try to make the coefficient of y as 1. Let us try dividing both sides by Zero. We know that mathematically, if any real is divided by Zero, then the value can not be determined.

In this case, the coefficient of x divided by Zero will give us our slope. But we know that the answer will not be defined in such a case. So we can safely say that our slope is not defined in such cases.

Slope is not defined.
Example 3: If the rise is 10 units, while the run is just 5 units, find the slope of the line.

Solution:

We know that the slope of a line will be

m = Rise/Run

Now, substituting the values, we will get

m = Rise/Run = 10/5 = 2

Slope is 2.
Example 4: Find the slope of a line that is parallel to the x-axis and intersects the y - axis at y = 4.

Solution:

We know that the slope of any line is the tangent of its angle made with the x-axis. So, if the line is given to be parallel to the x-axis itself, then the angle made will be 0º. Hence, tan 0º will be 0. So the value of the slope is found to be,

m = tan 0 = 0

Hence, the value of the slope will be Zero.

Slope is 0.

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Practice Questions on Slope

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FAQs on Slope

What is the Slope of a Line?

The slope of a line, also known as the gradient is defined as the value of the steepness or the direction of a line in a coordinate plane. Slope can be calculated using different methods, given the equation of a line or the coordinates of points lying on the straight line.

What is the Formula to Find Slope of a Line?

We can calculate the slope of a line directly using the slope of a line formula given the coordinates of the two points lying on the line. The formula is given as,
Slope = m = tan θ = (y₂ - y₁)/(x₂ - x₁)

How to Calculate the Slope?

The slope is found by measuring the tangent of the angle made by the line with the x-axis. There are different methods to find the slope of a line. The expression that can be used to find the slope is given as tan θ or (y₂ - y₁)/(x₂ - x₁), where θ is the angle which the line makes with the positive x-axis and (x₁, y₁) and (x₂, y₂) are the coordinates of the two points lying on the line.

What are the 4 Different Types of Slopes?

The 4 different types of slopes are positive slope, negative slope, zero slope, and undefined slope.

What is an Undefined Slope?

Any slope that has an angle of 90º with the x-axis, will have an undefined value of the tangent of 90º. Hence, such lines will have an undefined value of the slope.

What Does Slope Look Like?

The slope is nothing but the measure of the tangent of the angle made with the x-axis. Hence, it is just the measure of an angle.

What are 3 Ways to Find Slope?

The ways to find slope are point slope form, slope-intercept form, and the standard form. We can apply any of the forms of the equation of a straight line given the required information to find the slope.

How do you Show that Three Points are Collinear by Slope?

To prove the collinearity of three points, say A, B, and C, we can apply the slope formula. The slope of lines AB and BC should be equal for the three given points to be collinear points.

How to Find Slope With Two Points?

The slope can be calculated using the coordinates of two points using the formula, m = (y₂ - y₁)/(x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points lying on the line.

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