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# Slope of a line: negative slope

Slope is like a hill's steepness. We find it by dividing the vertical change (rise) by the horizontal change (run). If we move right on a graph and go up, the slope is positive. If we go down, it's negative. We can find the slope between any two points on a line, and it's always the same. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• at Sal says delta. Where does this come from?
(8 votes)
• Delta is the fourth letter of the Greek alphabet which is usually used for denoting change between two values. For e.g. Δy means the change or difference between the two values of the y co-ordinates.
(22 votes)
• how do I find the slope of a triangle
(8 votes)
• Only lines have slope, not shapes. If you want to fund the slope of the sides of triangles, then you'll need to know two coordinate points on the sides and then use the slope formula (change in y)/(change in x).
Example:
There's is a Triangle ABC where A is (0, 0), B is (2, 3), and C is (5, 6). What is the slope of each side of the triangle.
The slope of AB is:
(change in y)_(3-0)
-------------=-----=3/2
(change in x)_
(2-0)
The slope of BC is:
(change in y)_(6-3)
-------------=------=1
(change in x)_
(5-2)
The slope of AC is:
(change in y)_(6-0)
-------------=-----=6/5
(change in x)_
(5-0)

I hope this helps!
(19 votes)
• Hey, How do I know if I am suppose to count the slope over the line or under the line between the two points?
(7 votes)
• You can do either way. One thing, the signs might differ a little when you're doing the sum, but ultimately, after you simplify the answer, you'll see that whichever direction you went, the answer will be the same.
NOTE THIS: If you're going left, it'll be negative.
if you're going right, it'll be positive.
if you're going downwards, it'll be negative.
if you're going upwards, it'll be positive.
if you don't understand this negative - positive thing, you can check out the topic - Quadrants.
(14 votes)
• I'm in high school I've got no clue still how to plot a point could anyone give me any evidence on how to understand it better
(3 votes)
• Okay. First let's visualize the coordinate plane (the graph). Going across, we have the x axis. Going up and down we have the y axis. On these axes are numbers. The numbers are spaced evenly. Using these numbers, you can find any point on either of the axes. For example, If I asked you to find 5 on the x axis on the following graph:
``y axis|6|4|2|_ 2 _ 4 _ 6 _ 8 _   <----- x axis``

you would tell me that it is right in between the "4" and the "6" on the x axis. They're just two number lines. Now, to find a point on the plane (that is, a point in the space) you need two numbers: one for it's x location, and one for it's y location. Think of it this way: the x value tells you how far across the point is. If a point has an x value of 4, for example, you know that it is on the number 4 on the x axis, or it is directly above or below the number 4. The same goes for the y value, only this time it tells you how far up the point is.

The only thing we haven't covered is how we notate all of this, and this is quite trivial. If I say point x has coordinates [4,5], that means that it has an x value (or x coordinate) is 4, and the y value (or y coordinate) is 5. On a graph, that would look like this:
``|6|4|       .  <== our point is "4 across" (directly above2                                    number 4) and it is |_  2 _ 4 _ 6 _ 8 _      "five up" (directly across from                                        where the number 5                                        would be).``
(20 votes)
• At What does the triangle on x+3 stand for?
(5 votes)
• It's not a triangle, it's "delta". In this case it means that you have to find the absolute value of x1 (which is -3) minus x2 (which is 0). The result is 3.
(14 votes)
• if x = 19y - 3, is 19 the slope? and how do you solve that?
(7 votes)
• It depends on how you look at the problem. Normally mathematicians consider the xy-plane, it is the coordinate system you are probably most familiar with. Another plane is the yx-plane, then your y-axis is pointing to the left/right and your x-axis up/down. In the xy-plane (the coordinate system) we write a line as y=kx+m. The k in the equation y=kx+m is the slope of the line in the xy-plane (the coordinate system). The m in the equation y=kx+m is where the line hits the y-axis (that's like saying "where x=0"). So, if we put in x=0 we get y=k*0+m=m, so m is the value where the line hits the y-axis, just as I told you!

To solve your problem, we do like this.

We look att your equation x=19y-3, we now that the slope here is 19 and where the line hits the x-axis is where y=0, x=19*0-3=-3, so x=-3 is where the line hits the y-axis. But now we are talking about the yx-plane! Lets think about the xy-plane (the usual coordinate system).

So we have x=19y-3, and we want to solve for y.

x=19y-3
add 3 to both sides
x+3=19y
divide by 19
(x+3)/19=y
simplify
x/19+3/19=y
change the order of the equation
y=x/19+3/19

Now our equation is on the form y=kx+m.

Here k=1/19, because x*(1/19)=x/19

and m=3/19
(6 votes)
• I have a question about getting slope from a graph, is there a specific location in the slope that i have to pick or is it any point that intersect with the horizontal and vertical grids?
Also does the points have to be in one quadrant?

Because i was doing the problems and it seems they don't allow for more than 2 exact points to be the right answer.
(4 votes)
• As long as the line you are measuring the slope of is straight, it will have the same slope everywhere, so it doesn't matter which two points you pick. They don't have to be in the same quadrant, although it's usually easier to pick two points in the first quadrant if you can, though, because then you all positive coordinates and you're not having to worry about subtracting negatives and so on. It can also be easier to pick the points where the line intersects the axes, because there one of the coordinates will be zero, which again makes the subtraction easier. An example: if the two points on the line are (3,0) and (-1,8) then the slope is (8 - 0) / (-1 - 3) = 8 / (-4) = -2.
(6 votes)
• i did not understood you
(5 votes)
• facts
(2 votes)
• i have heard of other methods to find slope' what are they?
(4 votes)
• There one way to find slope but there are many shortcuts like y=mx+b when mx= slope and +b= y intercept
(2 votes)
• I am really attentive and focusing in class my math teacher told me to come on here for better understanding but i still dont understand can someone please help?
(3 votes)
• Visually, a line has negative slope if it goes down and right (or up and left). Mathematically, this means that as x increases, y decreases.
(4 votes)

## Video transcript

Find the slope of the line pictured on the graph. So the slope of a line is defined to be rise over run. Or you could also view it as change in y over change in x. And let me show you what that means. So let's start at some arbitrary point on this line, and they highlight some of these points. So let's start at one of these points right over here. So if we wanted to start one of these points-- and let's say we want to change our x in the positive direction. So we want to go to the right. So let's say we want to go from this point to this point over here. How much do we have to move in x? So if we want to move in x, we have to go from this point to this point. We're going from negative 3 to 0. So our change in x-- and this triangle, that's delta. That means "change in." Our change in x is equal to 3. So what was our change in y when our change in x is equal to 3? Well, when we moved from this point to this point, our x-value changed by 3, but what happened to our y-value? Well, our y-value went down. It went from positive 3 to positive 2. Our y-value went down by 1. So our change in y is equal to negative 1. So we rose negative 1. We actually went down. So our rise is negative 1 when our run-- when our change in x-- is 3. So change in y over change in x is negative 1 over 3, or we could say that our slope is negative 1/3. Let me scroll over a little bit. It is negative 1/3. And I want to show you that we can do this with any two points on the line. We could even go further than 3 in the x-direction. So let's go the other way. Let's start at this point right over here and then move backwards to this point over here, just to show you that we'll still get the same result. So to go from this point to that point, what is our change in x? So our change in x is this right over here. Our change in x is that distance right over there. We started at 3, and we went to negative 3. We went back 6. Over here, our change in x is equal to negative 6. We're starting at this point now. So over here our change in x is negative 6. And then when our change in x is negative 6, when we start at this point and we move 6 back, what is our change of y to get to that point? Well, our y-value went from 1. That was our y-value at this point. And then when we go back to this point, our y-value is 3. So what did we do? We moved up by 2. Our change in y is equal to 2. Slope is change in y over change in x, or rise over run. Change in y is just rise. Change in x is just run, how much you're moving in the horizontal direction. So rise over run in this example right over here is going to be 2 over negative 6, which is the same thing as negative 1/3. And you could verify it for yourself. Take any of these two points, start at one of these two points, and figure out what is the run to get to the next point, and then what is the rise to get the next point. And for any line, the slope won't change. Let me do it again. Over here, we had to move in the positive 3 direction, so that is our run. So this right here is positive 3. That's our run. But what's our rise? Well, we actually went down, so we have a negative rise. Our rise is negative 1. So we have negative 1 as our rise. We went down. And our run was positive 3. So our slope here is negative 1/3.