The Ultimate Guide to Finding the Slope of a Line

Performing common slope calculations so you can solve the toughest math problems, quickly and correctly

Co-authored by Jake Adams

Last Updated: January 31, 2024 Fact Checked

The slope of a line is a measure of how fast it is changing. This can be for a straight line -- where the slope tells you exactly how far up (positive slope) or down (negative slope) a line goes while it goes how far across. Slope can also be used for a line tangent to a curve. Or, it can be for a curved line when doing Calculus, where slope is also known as the "derivative" of a function. Either way, think of slope simply as the "rate of change" of a graph: if you make the variable "x" bigger, at what rate does "y" change? That is a way to see slope as a cause and an effect event.

Method 1

Method 1 of 3:

Finding the Slope of a Linear Equation

Download Article

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/b\/b6\/Find-the-Slope-of-an-Equation-Step-1-Version-2.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-1-Version-2.jpg","bigUrl":"\/images\/thumb\/b\/b6\/Find-the-Slope-of-an-Equation-Step-1-Version-2.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-1-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

1
Use slope to determine how steep, and in what direction (upward or downward), a line goes. Finding the slope of a line is easy, as long as you have or can setup a linear equation. This method works if and only if: ^{[1]
X
Research source}
- There are no exponents on the variables
- There are only two variables, neither of which are fractions (for example, you would not have ${\frac {1}{x}}$
- The equation can be simplified to the form $y=mx+b$ , where m and b are constants (numbers like 3, 10, -12, ${\frac {4}{3}},{\frac {3}{5}}$ ). ^{[2]
  X
  Research source}
2
Find the number in front of the x, usually written as "m," to determine slope. If your equation is already in the right form, $y=mx+b$ , then simply pick the number in the "m" position (but if there is no number written in front of x then the slope is 1). That is your slope! Note that this number, m , is always multiplied by the variable, in this case an "x." Check the following examples:
- $y=2x+6$
  - Slope = 2
- $y=2-x$
  - Slope = -1
- $y={\frac {3}{8}}x-10$
  - Slope = ${\frac {3}{8}}$ ^{[3]
    X
    Research source}
Advertisement
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/5c\/Find-the-Slope-of-an-Equation-Step-3-Version-2.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-3-Version-2.jpg","bigUrl":"\/images\/thumb\/5\/5c\/Find-the-Slope-of-an-Equation-Step-3-Version-2.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-3-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

3
Reorganize the equation so one variable is isolated if the slope isn't apparent. You can add, subtract, multiply, and more to isolate a variable, usually the "y." Just remember that, whatever you do to one side of the equal sign (like add 3) you must do to the other side as well. Your final goal is an equation similar to $y=mx+b$ . For example:
- Find the slope of $2y-3=8x+7$
- Set to the form $y=mx+b$ :
  - $2y-3+3=8x+7+3$
  - $2y=8x+10$
  - ${\frac {2y}{2}}={\frac {8x+10}{2}}$
  - $y=4x+5$
- Find the slope:
  - Slope = M = 4 ^{[4]
    X
    Research source}

Method 2

Method 2 of 3:

Finding the Slope with Two Points

Download Article

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/6\/69\/Find-the-Slope-of-an-Equation-Step-4-Version-2.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-4-Version-2.jpg","bigUrl":"\/images\/thumb\/6\/69\/Find-the-Slope-of-an-Equation-Step-4-Version-2.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-4-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

1
Use a graph and two points to find slope without the equation handy. If you've got a graph and a line, but no equation, you can still find the slope with ease. All you need are two points on the line, which you plug into the equation ${\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$ . While finding the slope, keep in mind the following information to help you check if you're on the right track: ^{[5]
X
Research source}
- Positive slopes go higher the further right you go.
- Negative slopes go lower the further right you go.
- Bigger slopes are steeper lines. Small slopes are always more gradual.
- Perfectly horizontal lines have a slope of zero.
- Perfectly vertical lines do not have a slope at all. Their slope is "undefined." ^{[6]
  X
  Research source}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/8\/87\/Find-the-Slope-of-an-Equation-Step-5-Version-2.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-5-Version-2.jpg","bigUrl":"\/images\/thumb\/8\/87\/Find-the-Slope-of-an-Equation-Step-5-Version-2.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-5-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

2
Find two points, putting them in simple (x,y) form. Use the graph (or the test question) to find the x and y coordinates of two points on the graph. They can be any two points that the line crosses through. For an example, assume that the line in this method goes through (2,4) and (6,6). ^{[7]
X
Research source}
- In each pair, the x coordinate is the first number, the y coordinate comes after the comma.
- Each x coordinate on a line has an associated y coordinate.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/4\/45\/Find-the-Slope-of-an-Equation-Step-6-Version-2.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-6-Version-2.jpg","bigUrl":"\/images\/thumb\/4\/45\/Find-the-Slope-of-an-Equation-Step-6-Version-2.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-6-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

3
Label your points x ₁ , y ₁ , x ₂ , y ₂ , keeping each point with its pair. Continuing our first example, with the points (2,4) and (6,6), label the x and y coordinates of each point. You should end up with: ^{[8]
X
Research source}
- x ₁ : 2
- y ₁ : 4
- x ₂ : 6
- y ₂ : 6 ^{[9]
  X
  Research source}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/97\/Find-the-Slope-of-an-Equation-Step-7-Version-2.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-7-Version-2.jpg","bigUrl":"\/images\/thumb\/9\/97\/Find-the-Slope-of-an-Equation-Step-7-Version-2.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-7-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

4
Plug your points into the "Point-Slope Formula" to get your slope. The following formula is used to find slope using any two points on a straight line: ${\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$ . Simply plug in your four points and simplify: ^{[10]
X
Research source}
- Original Points: (2,4) and (6,6).
- Plug into Point Slope:
  - ${\frac {6-4}{6-2}}$
- Simplify for Final Answer:
  - ${\frac {2}{4}}={\frac {1}{2}}$ = Slope
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/c\/c2\/Find-the-Slope-of-an-Equation-Step-8-Version-2.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-8-Version-2.jpg","bigUrl":"\/images\/thumb\/c\/c2\/Find-the-Slope-of-an-Equation-Step-8-Version-2.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-8-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

The slope of a line is “Rise over Run:” how much the line goes up divided by how much the line "runs" to the right. The “rise” of the line is the difference between the y-values (remember, the Y-axis goes up and down), and the “run” of the line is the difference between the x-values (and the X-axis goes left and right). ^{[11]
X
Research source}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/b\/b9\/Find-the-Slope-of-an-Equation-Step-9-Version-2.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-9-Version-2.jpg","bigUrl":"\/images\/thumb\/b\/b9\/Find-the-Slope-of-an-Equation-Step-9-Version-2.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-9-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

The equation of the slope is ${\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$ . This may also be shown using the Greek letter “Δ”, called “delta”, meaning “difference of”. Slope can also be shown as Δy/Δx, meaning "difference of y / difference of x:" this is the same exact question as "find the slope between

Method 3

Method 3 of 3:

Using Differential Calculus to Find the Slope of a Curve

Download Article

{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/a\/aa\/Find-the-Slope-of-an-Equation-Step-10-Version-2.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-10-Version-2.jpg","bigUrl":"\/images\/thumb\/a\/aa\/Find-the-Slope-of-an-Equation-Step-10-Version-2.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-10-Version-2.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

1
Review how to take a variety of derivatives from common functions. Derivatives give you the rate of change (or slope) at a single point on a line. The line can be curved or straight -- it doesn't matter. Think of it as how much the line is changing at any time, instead of the slope of the entire line. How you take derivatives changes depending on the type of function you have, so review how to take common derivatives before moving on.
- Review taking derivatives here
- The most simple derivatives, those for basic polynomial equations, are easy to find using a simple shortcut. This will be used for the rest of the method.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/5\/5c\/Find-the-Slope-of-an-Equation-Step-11.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-11.jpg","bigUrl":"\/images\/thumb\/5\/5c\/Find-the-Slope-of-an-Equation-Step-11.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-11.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

2
Understand what questions are asking for a slope using derivatives. You will not always be asked to explicitly find the derivative or slope of a curve. You might also be asked for the "rate of change at point (x,y). You could be asked for an equation for the slope of the graph, which simply means you need to take the derivative. Finally, you may be asked for "the slope of the tangent line at (x,y)." This, once again, just wants the slope of the curve at a specific point, (x,y).
- To calculate the slope of the tangent line, you need to take the first derivative, and add in the x value into your derivative to get your slope.
- For this method, consider the question: "What is the slope of the line $f(x)=2x^{2}+6x$ at the point (4,2)?" ^{[12]
  X
  Research source}
- The derivative is often written as $f'(x),y',$ or ${\frac {dy}{dx}}$
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/3\/37\/Find-the-Slope-of-an-Equation-Step-12.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-12.jpg","bigUrl":"\/images\/thumb\/3\/37\/Find-the-Slope-of-an-Equation-Step-12.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-12.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

3
Take the derivative of your function. You don't even really need you graph, just the function or equation for your graph. For this example, use the function from earlier, $f(x)=2x^{2}+6x$ . Following the methods outlined here , take the derivative of this simple function. ^{[13]
X
Research source}
- Derivative: $f'(x)=4x+6$
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/92\/Find-the-Slope-of-an-Equation-Step-13.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-13.jpg","bigUrl":"\/images\/thumb\/9\/92\/Find-the-Slope-of-an-Equation-Step-13.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-13.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

4
Plug in your point to the derivative equation to get your slope. The differential of a function will tell you the slope of the function at a given point. In other words, f’(x) is slope of the function at any point (x,f(x)) So, for the practice problem:
- What is the slope of the line $f(x)=2x^{2}+6x$ at the point (4,2)?
- Derivative of Equation:
  - $f'(x)=4x+6$
- Plug in Point for x:
  - $f'(x)=4(4)+6$
- Find the Slope:
- The slope of the $f(x)=2x^{2}+6x$ at (4,2) is 22.
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/7\/7a\/Find-the-Slope-of-an-Equation-Step-14.jpg\/v4-460px-Find-the-Slope-of-an-Equation-Step-14.jpg","bigUrl":"\/images\/thumb\/7\/7a\/Find-the-Slope-of-an-Equation-Step-14.jpg\/v4-728px-Find-the-Slope-of-an-Equation-Step-14.jpg","smallWidth":460,"smallHeight":345,"bigWidth":728,"bigHeight":546,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/3.0\/\">Creative Commons<\/a><br>\n<\/p><p><br \/>\n<\/p><\/div>"}

5
Check your point against a graph whenever possible. Know that not all points in calculus will have a slope. Calculus gets into complex equations and difficult graphs, and not all points will have a slope, or even exist on every graph. Whenever possible, use a graphing calculator to check the slope of your graph. If you can't, draw the tangent line using your point and the slope (remember -- "rise over run") and note if it looks like it could be correct. ^{[14]
X
Research source}
- Tangent lines are just lines with the exact same slope as your point on the curve. To draw one, go up (positive) or down (negative) your slope (in the case of the example, 22 points up). Then move over one and draw a point. Connect the dots, (4,2) and (26,3) for your line.

Practice Problems and Answers

Practice Problems and Answers to Find the Slope of an Equation

Community Q&A

Search

Add New Question

Question

What is the slope for the equation y=1?

Donagan

Top Answerer

The graph of y=1 is a straight, horizontal line, meaning that it does not rise or fall as it moves left or right. Its slope is therefore zero.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 11 Helpful 20
Question

What if the equation is like x+y=0 or x-y=0?

Community Answer

That's no problem. When x+y=0, y=-x. In this case the slope is -1. On the other hand, when x-y=0, y=x. Here the slope is +1.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 6 Helpful 18
Question

What's the difference between a slope = 0 and slope = undefined?

Donagan

Top Answerer

A zero slope is a horizontal line (parallel to the x-axis), and an undefined slope is a vertical line (parallel to the y-axis).

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 12 Helpful 12

See more answers

Ask a Question

200 characters left

Include your email address to get a message when this question is answered.

Submit

Video

Tips

Submit a Tip

All tip submissions are carefully reviewed before being published

Submit

Thanks for submitting a tip for review!

You Might Also Like

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

Log in

The Ultimate Guide to Finding the Slope of a Line

Steps

Finding the Slope of a Linear Equation

Finding the Slope with Two Points

Using Differential Calculus to Find the Slope of a Curve

Practice Problems and Answers

Community Q&A

Video

Tips

You Might Also Like

References

About This Article

Reader Success Stories

Did this article help you?

Quizzes

You Might Also Like

Featured Articles

Trending Articles

Featured Articles

Featured Articles

Watch Articles

Trending Articles

Explore Categories