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Learn to calculate the slope of a line with these simple methods
Co-authored byGrace Imson, MAand Johnathan Fuentes
Last Updated: November 2, 2023Fact Checked
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- Finding the Slope |
- Types of Slopes |
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If you’re taking algebra, finding the slope of a line is an important concept to understand. But there are multiple ways to find the slope, and your teacher may expect you to learn them all. Feeling a bit overwhelmed? Don’t fret. This guide explains how to find the slope of a line using (x, y) points from graphs. We’ll also explain how the slope formula works, and how to recognize positive, negative, zero, and undefined slopes. Keep reading to learn how to calculate the slope of a line and ace your next quiz, exam, or homework assignment.
Practice Problems
Find the Slope of a Line Practice Problems
Find the Slope of a Line Practice Problems ANSWER KEY
Things You Should Know
- Slope = Rise divided by Run and is represented by the variable m. The slope m is found in the slope-intercept formula, y = mx + b
- When given two (x, y) points on a line, Run = x2 - x1 and Rise = y2 - y1. Therefore, m = (y2 - y1)/(x2 - x1).
- Find the slope by finding two (x, y) points on a line, labeling one (x1, y1) and the other (x2, y2). The slope m = (y2 - y1)/(x2 - x1).
Method 1
Method 1 of 2:
Finding the Slope
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1
This is the slope formula, which states Slope = Rise over Run. When plotting a line on a graph, the “Rise” refers to the change in y that corresponds to a specific change in x. This change in x is called the “Run.” For instance, if y increases by 4 when x increases by 2, then Rise = 4 and Run = 2. To find the slope, divide 4/2 to get 2.[1]
- The slope of a line is represented by the variable m. In this example, m = 2.
- The slope m is part of the formula y = mx + b. This is called the “slope-intercept formula.”
- You can use y = mx + b to calculate a value of y that corresponds to a particular value of x. Each pair of corresponding x and y values is called a “point”, written as (x, y).[2]
- If you find multiple (x, y) points for the same equation, you can plot those points on a graph and draw a straight line through them.
2
Find two different points that the line passes through. Each point has an x value and a y value, written together as (x, y). You can pick any two points you like, as long as the line passes directly through them.[3]
- In the example above, we picked the points (2, 1) and (5, 3).
- The first point has an x value of 2 and a y value of 1, so it’s written as (2, 1).
- The second point has an x value of 5 and a y value of 3. It’s written as (5, 3).
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3
Choose one point to be (x1, y1) and the other to be (x2, y2). In other words, the first (x, y) pair can be (x1, y1), and the second paid can be (x2, y2). In the example above, (x1, y1) is (2, 1), while (x2, y2) is (5, 3).[4]
- For simplicity, you can always make the point farthest on the left (x1, y1). This point will have the lower value for x.
- In reality, you can make either point (x1, y1) or (x2, y2), as long as you remember which is which.
4
Calculate m by dividing the Rise by the Run. Rise = y2 - y1, while Run = x2 - x1. To illustrate this, let’s calculate m for our previous example using the coordinates (2, 1) and (5, 3):[5]
- (x1, y1) = (2, 1), and (x2, y2) is (5, 3).
- Rise = y2 - y1, or 3 - 1. Run = x2 - x1, or 5 - 2.
- Rise/Run = (3 - 1)/(5 - 2), or (2/3).
- Therefore, m = ⅔, or 0.67.
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Method 2
Method 2 of 2:
Types of Slopes
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Positive slope: A positive slope is “uphill” with a positive m value. “Uphill” means that y increases as x increases. In the above image, the lines all have positive slopes. Note that each line goes uphill from left to right, and that the y value for each point gets larger as the x value gets larger.[6]
- A positive slope can be a positive integer like 3, 5, or 16. It can also be a fraction or decimal, like ½, 0.75, or 1.86.
- Look closely at the y = mx + b formulas listed with each line. Notice that m is always a positive number.
2
Negative slope: A negative slope is “downhill” with a negative m value. “Downhill” means that y decreases as x increases. In the above image, the lines all have negative slopes. Note that each line goes downhill from left to right, and that the y value for each point gets smaller as the x value gets larger.[7]
- A negative slope can be a negative integer like -2, -6, or -19. It can also be a negative fraction or decimal, like -⅓, -0.9, or -2.21.
- Note the y = mx + b formulas for each line. You’ll see that m is always a negative number.
3
Zero slope: A horizontal line has a slope of m = 0. The y value always stays the same, even as x increases. In other words, the line doesn’t go “uphill” or “downhill”. In the graph above, you’ll see that all the lines are completely flat.[8]
- A horizontal line always has a slope of m = 0. This means the equation y = mx + b can be written as y = 0x + b. Since 0x = 0, the equation gets simplified to y = b, with b being the y corresponding value for all values of x.
- The formulas for the line above all follow the same format: y = some number.
4
Undefined slope: A vertical line has an “undefined” slope and one value for x. In other words, y can have any value, but x always has the same value. Vertical lines can’t be written with the formula y = mx + b. Instead, they are written as x = some number, and the vertical line passes through the x-axis at that exact number.[9]
- The image above shows lines with undefined slopes.
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Community Q&A
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Question
How do I find slope of 2x - 4y = 20?
Donagan
Top Answerer
Re-work the equation until y is isolated on one side. Then note the coefficient of the x term. That's the slope. In this example, we re-work the equation until we isolate y: y = x/2 - 5. The coefficient of x is ½, so the slope of the line is ½.
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How do I find the slope of a line y=9?
Community Answer
For all lines where y equals a constant and there is no x, the slope is 0.
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How do I use a protractor and trigonometry to find slope of a line?
Donagan
Top Answerer
The slope of a line is a non-angular representation of the angle between the line and a horizontal line such as the x-axis. Use a protractor to measure that angle, and then convert the angle to a decimal or a fraction using a trig table. For example, if a protractor tells you that there is a 45° angle between the line and a horizontal line, a trig table will tell you that the tangent of 45° is 1, which is the line's slope. Most angles do not have such a simple tangent. For instance, a 30° angle has a tangent of 0.577. You could use that as the slope, or you could convert the decimal to a fraction, but in this case it would be a rather unwieldy fraction (577/1000 or 72/125).
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Tips
You can calculate the slope from a list of (x, y) points that correspond to a line. Choose one point from the list to be (x1, y1) and another to be (x2, y2).
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As long as you have at least two (x, y) points, you don’t need to plot the line on a graph to calculate the slope (though your assignments may still require you to do so).
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Expert Interview
Thanks for reading our article! If you’d like to learn more about mathematics, check out our in-depth interview with Grace Imson, MA.
References
- ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut27_graphline.htm
- ↑ https://www.alamo.edu/contentassets/ab5b70d70f264cec9097745e8f30ca0a/graphing/math0303-equations-of-a-line.pdf
- ↑ https://web.ics.purdue.edu/~braile/eas100/Slope.pdf
- ↑ https://web.ics.purdue.edu/~braile/eas100/Slope.pdf
- ↑ https://web.ics.purdue.edu/~braile/eas100/Slope.pdf
- ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut27_graphline.htm
- ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut27_graphline.htm
- ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut27_graphline.htm
- ↑ https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut27_graphline.htm
About This Article
Co-authored by:
Grace Imson, MA
Math Teacher
This article was co-authored by Grace Imson, MA and by wikiHow staff writer, Johnathan Fuentes. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 569,296 times.
45 votes - 60%
Co-authors: 45
Updated: November 2, 2023
Views:569,296
Categories: Coordinate Geometry
Article SummaryX
In geometry, the slope of a line describes how steep the line is, as well as the direction it’s going—that is, whether the line is going up or down. To find the slope of a line, all you have to do is divide the rise of the line by its run. To get the rise and run, pick any two coordinates along the line. For instance, your first coordinate might be at 2 on the x axis and 4 on the y axis, while your second coordinate might be at 5 on the x axis and 7 on the y axis. Next, write a fraction with the difference between your two y coordinates on top—this is the rise—and the difference between the x coordinates on the bottom—that’s the run. In our example, the rise would be 7-4, while the run would be 5-2. This means the slope of the line would be 3/3, or 1. To figure out the direction of the line, check whether your slope is positive or negative. Lines that go up from left to right always have a positive slope, while lines that go down from left to right always have a negative slope. To figure out how steep the line is, look at the magnitude of the number. Whether it’s positive or negative, the greater the magnitude, the steeper the slope. For instance, a line with a slope of -7 is steeper than a line with a slope of -2. Similarly, a line with a slope of 15 is steeper than a line with a slope of 3. If you want to learn how to reduce the numbers in your slope, keep reading the article!
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