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Slope review

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.
start text, S, l, o, p, e, end text, equals, start fraction, start text, r, i, s, e, end text, divided by, start text, r, u, n, end text, end fraction, equals, start fraction, delta, y, divided by, delta, x, end fraction
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 left parenthesis, 0, comma, 5, right parenthesis and left parenthesis, 4, comma, 2, right parenthesis.
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.
start text, S, l, o, p, e, end text, equals, start fraction, delta, y, divided by, delta, x, end fraction, equals, start fraction, 2, minus, 5, divided by, 4, minus, 0, end fraction, equals, start fraction, minus, 3, divided by, 4, end fraction
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, equals, 11, point, 4, space, space, space, y, equals, 11, point, 5
Solution: x, equals, 12, point, 7, space, space, space, y, equals, 15, point, 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 left parenthesis, 11, point, 4, comma, 11, point, 5, right parenthesis and left parenthesis, 12, point, 7, comma, 15, point, 4, right parenthesis.
Slope=ΔyΔx=15.411.512.711.4=3.91.3=3913=3\begin{aligned} \text{Slope}=\dfrac{\Delta y}{\Delta x}&=\dfrac{15.4-11.5}{12.7-11.4}\\\\ &=\dfrac{3.9}{1.3}\\\\ &=\dfrac{39}{13}\\\\ &=3\end{aligned}
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, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
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.

Want to join the conversation?

  • primosaur ultimate style avatar for user BenGrapentine
    I dont understand this slope thing at all
    can you help me
    (3 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      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!
      (58 votes)
  • starky seedling style avatar for user jaiden.warrens8lol
    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
      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!
      (15 votes)
  • aqualine ultimate style avatar for user 22cristiansantiagohernandez
    Are you supposed to simplify 4.5 by 1.5??
    (3 votes)
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  • starky seedling style avatar for user Akila Kodikara
    This is confusing me?
    Any helpp for the last question?
    (10 votes)
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  • blobby green style avatar for user Ryin
    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
    (11 votes)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (2 votes)
  • starky tree style avatar for user Lillianna White
    I just don't understand.... how do you tell the slope?
    (10 votes)
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  • aqualine ultimate style avatar for user Colin, Lambert
    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
    (5 votes)
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  • blobby green style avatar for user Duanli Liu
    Is the slope y divide by x?
    (7 votes)
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  • blobby green style avatar for user vimbaneli
    so all the x numbers go on the bottom and y goes on top
    (3 votes)
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  • blobby green style avatar for user ekagra.dwivedi
    I am confused on how to find the slope in an equation
    (3 votes)
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    • primosaur ultimate style avatar for user BronsonFebruary
      Since it is different than everything you have been doing so far, it is natural to be confused.
      The slope is part of the equation of a line.
      Y = mX + Y intercept
      Is the equation you will see a lot.
      The m represents the coefficient multiplied by X. Together, m and X are the slope.
      Take two points on the line. Any two points that the line passes through.
      (2,3) (0,1)
      It is important to note you are not multiplying the numbers.
      The 2 and 0 represent the x coordinate and the 3 and 1 represent the y coordinate.
      To finish finding the slope, you take the 2nd y coordinate, in this case 1, and minus the 1st y coordinate from the 2nd.
      This equals -2.
      You then take the 2nd x coordinate and subtract the 1st x coordinate from the 2nd. This equals -2.
      You divide the y value and by the x value. -2/-2 = 1
      The slope is 1.
      Y (subscript) 2 - Y (subscript) 1 / X (subscript) 2 - X (subscript) 1 = slope
      Hope this helps.
      (2 votes)