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Review: The 4 Types of Slope of a Line (⤢⤡⟷↕)

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Writing Equations of Lines in Slope Intercept Form

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Rise Over Run (Slope) - Official Music Video

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Slope is a number that shows how steep a line is. Master finding slope on graphs and with points in this interactive, free lesson. Start learning now!

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Straight-Line Equations: Slope-Intercept Form | Purplemath

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Covers the 'slope-intercept', or 'y = mx + b', form of straight-line equations.

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Using Slope and y-Intercept to Graph Lines| Purplemath

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Demonstrates, step-by-step and with illustrations, how to use slope and the y-intercept to graph straight lines.

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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.

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,
P ₁ = (x ₁, y ₁)
P ₂ = (x ₂, y ₂)

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 = y ₂ - y ₁
Δx = x ₂ - x ₁

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

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,

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 c 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,

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 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

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 l ₁ and l ₂ in the coordinate plane, inclined at angle θ 1 and θ ₂ respectively with the x-axis, such that the given angles follow the external angle theorem as, θ ₂= θ ₁ + 90º.

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 l ₁ and l ₂ in the coordinate plane, inclined at angle θ 1 and θ ₂ respectively with the x-axis, such that the, θ 2 = θ ₁.

Therefore, their slopes can be given as,
⇒ m ₁ = m ₂

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.
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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 = (y ₂ - y ₁)/(x ₂ - x ₁)

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 (y ₂ - y ₁)/(x ₂ - x ₁), where θ is the angle which the line makes with the positive x-axis and (x ₁, y ₁) and (x ₂, y ₂) 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 = (y ₂ - y ₁)/(x ₂ - x ₁), where (x ₁, y ₁) and (x ₂, y ₂) are the coordinates of the two points lying on the line.

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Slope - Definition, Types, Examples | Slope of Line Formula

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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.

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IXL | Find the slope from a graph | 8th grade math

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Improve your math knowledge with free questions in “Find the slope from a graph” and thousands of other math skills.

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IXL | Find the slope from a graph | Level J math

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Improve your math knowledge with free questions in "Find the slope from a graph" and thousands of other math skills.