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Course: Geometry (SCPS pilot: textbook aligned) > Unit 3

Lesson 4: Slopes of lines
Districts
Geometry (SCPS pilot: textbook aligned)
Parallel and perpendicular lines
Slopes of lines
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Slope review

Google Classroom
The slope of a line is a measure of its steepness. Mathematically, slope is calculated as "rise over run" (change in y divided by change in x).

What is slope?

Slope is a measure of the steepness of a line.
Slope=riserun=ΔyΔx
Want an in-depth introduction to slope? Check out this video.

Example: Slope from graph

We're given the graph of a line and asked to find its slope.
A coordinate plane. The x- and y-axes each scale by one. A graph of a line intersects the points zero, five and four, two.
The line appears to go through the points (0,5) and (4,2).
A coordinate plane. The x- and y-axes each scale by one. A graph of a line intersects the points zero, five and four, two. Both of these points are plotted on the graph.
Slope=ΔyΔx=2−54−0=−34
In other words, for every three units we move vertically down the line, we move four units horizontally to the right.
A coordinate plane. The x- and y-axes each scale by one. A graph of a line intersects the points zero, five and four, two. Both of these points are plotted on the graph. There is a horizontal dotted segment from zero, five to four, five that is labeled four. There is a vertical dotted segment from four, five to four, two that is labeled negative three.
Want to learn more about finding slope from graphs? Check out this video.

Example: Slope from two points

We're told that a certain linear equation has the following two solutions:
Solution: x=11.4   y=11.5
Solution: x=12.7   y=15.4
And we're asked to find the slope of the graph of that equation.
The first thing to realize is that each solution is a point on the line. So, all we need to do is find the slope of the line through the points (11.4,11.5) and (12.7,15.4).
Slope=ΔyΔx=15.4−11.512.7−11.4=3.91.3=3913=3
The slope of the line is 3.
Want to learn more about finding slope from two points? Check out this video.

Practice

Problem 1
What is the slope of the line below?
Give an exact number.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
A coordinate plane. The x- and y-axes each scale by one. A graph of a line intersects the points one, two and four, four.

Want more practice? Check out this Slope from graphs exercise and this Slope from points exercise.

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  • primosaur ultimate style avatar for user BenGrapentine
    Posted 4 years ago. Direct link to BenGrapentine's post “I dont understand this sl...”
    I dont understand this slope thing at all
    can you help me
    (44 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      Posted 4 years ago. Direct link to Polina Vitić's post “You can always figure out...”
      Incredible Answer
      Great Answer
      Good Answer
      You can always figure out the slope of a line if you have 2 points. If you are not given 2 points, you can find 2 points on the graph and use them to find the slope.

      Here are some good things to know:
      - m = slope
      - (x₁, y₁) = point 1
      - (x₂, y₂) = point 2
      - rise = the difference in the y-values (y₂ - y₁)
      - run = the difference in the x-values (x₂ - x₁)

      In the slope formula, the slope (m) is equal to rise over run:

      m = rise / run
      = (y₂ - y₁) / (x₂ - x₁)

      Let's say we are given a line with points (4, 2) and (6, 1). If we say that point 1 is (4, 2) and point 2 is (6, 1), then:

      x₁ = 4 and x₂ = 6
      y₁ = 2 and y₂ = 1

      Now we just need to plug these values into the slope formula:

      m = rise / run
      = (y₂ - y₁) / (x₂ - x₁)
      = (1 - 2) / (6 - 4)
      = (-1) / 2
      = -1/2

      So the slope (m) is -1/2.

      The main thing to keep track of is which point is (x₁, y₁) and which point is (x₂, y₂). You don't want to mix these up.

      A few tips for graphs of slopes:
      - a perfectly horizontal line has no slope
      - a perfectly vertical line has a slope that is not defined
      - a line that goes upwards (from left to right) has a positive slope
      - a line that goes downwards (from left to right) has a negative slope

      Hope this helps!
      (143 votes)
  • aqualine ultimate style avatar for user 22cristiansantiagohernandez
    Posted 6 years ago. Direct link to 22cristiansantiagohernandez's post “Are you supposed to simpl...”
    Are you supposed to simplify 4.5 by 1.5??
    (45 votes)
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    • blobby green style avatar for user trinnydonhoff
      Posted 2 years ago. Direct link to trinnydonhoff's post “for the sake of the equat...”
      Great Answer
      Good Answer
      for the sake of the equation make 4.5 into 45 and make 1.5 into 15. From here, simplify 45/15. once you are done simplifying, 45/15 into 3/1, you have to reverse it to make it into the correct answer for slope equations (difference of y/difference of x).
      (32 votes)
  • starky seedling style avatar for user jaiden.warrens8lol
    Posted 5 years ago. Direct link to jaiden.warrens8lol's post “why would you have to do ...”
    why would you have to do the close one said it passes throw 0,5 and 4,2 why can't you just do it were it passes throw the x and y
    (16 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      Posted 4 years ago. Direct link to Polina Vitić's post “You can calculate the slo...”
      Great Answer
      Good Answer
      You can calculate the slope from the x- and y-intercepts. But in this case, we can't tell exactly what the x-intercept is just from looking at the graph - we can only see that it is somewhere between 6 and 7.

      In order to accurately calculate the slope, we need to use points where we know the exact value. We know the exact value of every point on the grid where the graph lines intersect. So when the line crosses one of those points, we know the exact coordinates for that point on the line.

      For example:
      (-4, 8)
      (0, 5)
      (4, 2)
      (8, -1)

      Any of these points may be used to calculate the slope, and you should get the same answer no matter which 2 points you use.

      Hope this helps!
      (26 votes)
  • duskpin tree style avatar for user MarsKitty322
    Posted 5 years ago. Direct link to MarsKitty322's post “Why the formula for slope...”
    Why the formula for slope is △ y/ △ x not △ x/ △ y?
    (13 votes)
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    • stelly blue style avatar for user Kim Seidel
      Posted 5 years ago. Direct link to Kim Seidel's post “Here are my thoughts... S...”
      Great Answer
      Good Answer
      Here are my thoughts...
      Slope is a measurement of how steep the line is. The steepness is determined by how fast the line rises/falls. Thus, the predominate measure of importance is the change in Y. If you had △ x/ △ y, then the measurement is more descriptive of how fast the line is moving left/right, which doesn't really describe steepness.
      (31 votes)
  • blobby green style avatar for user Ryin
    Posted 4 years ago. Direct link to Ryin's post “How can it be a fraction,...”
    How can it be a fraction, because when I put in a number that we used in class, it said that what I put in was wrong. If you guys could explain that concept to me that would be greatly appreciated
    (16 votes)
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    • stelly blue style avatar for user Kim Seidel
      Posted 4 years ago. Direct link to Kim Seidel's post “The slope of a line is de...”
      Good Answer
      The slope of a line is defined as a fraction: rise over run; or (y2-y1)/(x2-x1). So slope is always a fraction. Even if you get a number like 5 as a slope, you need to change it into 5/1 (fraction form) to understand what it means related to the lines movement.
      (16 votes)
  • starky sapling style avatar for user Space1
    Posted 4 years ago. Direct link to Space1's post “how do you graph a linear...”
    how do you graph a linear equation in slope intercept form such as:
    y= 1/2 x - 3
    (10 votes)
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    • stelly blue style avatar for user Kim Seidel
      Posted 4 years ago. Direct link to Kim Seidel's post “Slope-intercept form: y=m...”
      Good Answer
      Slope-intercept form: y=mx+b where...
      m = the slope
      b = the y-intercept at the point (0, b)

      So, in your equation, your y-intercept = (0, -3).
      Start by graphing that point.
      Then use the slope to find more points.
      Your slope = 1/2. This means from the point (0, -3), go up 1 unit and right 2 units to find the next point. You can repeat this as many times as you want to find points on the line.
      Don't forget to actually draw the line.

      FYI... Search for the leason on graphing from slope intercept form. There is a search bar at the top of all KA screens.
      (19 votes)
  • male robot hal style avatar for user Keith98A313
    Posted 9 months ago. Direct link to Keith98A313's post “This is genuinely so much...”
    This is genuinely so much easier to learn than the way my tutor taught it.
    (17 votes)
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  • aqualine ultimate style avatar for user Colin, Lambert
    Posted 5 years ago. Direct link to Colin, Lambert's post “I was wondering if you on...”
    I was wondering if you only have lets say, 2 points on a line, and I was wondering if you could solve for the slope, even if thats all you have
    (8 votes)
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  • marcimus purple style avatar for user AnnalynnG
    Posted 7 months ago. Direct link to AnnalynnG's post “I dont like this”
    I dont like this
    (10 votes)
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  • starky tree style avatar for user Lillianna White
    Posted 3 years ago. Direct link to Lillianna White's post “I just don't understand.....”
    I just don't understand.... how do you tell the slope?
    (11 votes)
    Default Khan Academy avatar avatar for user