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Welcome to the presentation on figuring out the slope.
Let's get started.
So let's say I have two points,
and as we learned in previous presentations
that all you need to define a line is two points,
and I think if you think about that that makes sense.
Let's say we have two points.
Let me write down the two points we're gonna have.
Let's say one point is-- why isn't it writing?
Sometimes this thing acts a little finicky.
Oh, that's because I was trying to write in black.
Let's say that one point is (-1,3).
So let's see, where do we graph that?
So this is (0,0).
We go -1, this is -1 here.
Let me just write -1.
And then, we're gonna go 3 up,
1, 2, 3, because this is 3 right here.
So (-1,3) is gonna be right over there.
Okay, so that's the first point.
And the second point-- I'm gonna do it in a different color.
Let's say the second point is (2,1).
So let's see where we would put that.
We would count 1, 2; this is 2, 1 because this is 1.
So the point's gonna be here, right?
So we've graphed our two points, and now the line that connects them
is gonna look something like this.
I hope I can draw it well.
Looks like--no no no, that's not a good line.
Let me try to do it better.
I'm just gonna go straight
from this point through that point, like that.
And then I'm just gonna try to continue the line from here.
That might be the best technique.
Something like that.
Whoops, the microphone dropped.
As you can tell, this is not a completely professional operation.
Okay, so let's look at that line.
So what we want to do in this presentation is figure out
the slope of that line.
So let's write out a couple things that will help you.
So there's a couple of ways to view slope.
I think, intuitively, you know that the slope is
the inclination of this line.
We can already see that this is a downward sloping line
because it comes from the top left to the bottom right.
So it's gonna be a negative number, the slope.
So you know that immediately.
What we're gonna do is
actually figure out how to figure out the slope.
So slope-- let me write this down.
Slope--and oftentimes they'll use the variable m for slope;
I have no idea why,
because m clearly does not stand for slope.
That is equal to-- there's a couple of things
you might hear-- change in y over change in x.
That triangle, which is pronounced delta,
just a Greek letter, that means change.
So change in y over change in x,
and that also is equal to rise/run.
I'm gonna explain what all of this means in a second.
So let's start at one of these points.
Let's start at this green point, (-1,3).
So how much do we have to rise and how much do we have to run
to get to the second point, (2,1)?
So let's do the rise first.
Well, we have to go -2, so that's the rise.
So the rise is equal to -2, right?
Because we had to down 2 to get to the same y
as this yellow point.
And then we have to run, right there, we have to run +3.
So rise/run=-2/3.
Well, how would we do that if we didn't have this
nice graph here to draw on?
Well, what we can do is-- we can say, let's take this
as the starting point.
Change in y over change in x is equal to--
we take the first y point, which is 3,
and we subtract the second y point, which is 1.
Right? You see that?
We just took 3-1, so that's the change in y
over and we take the first x point,
-1 minus the second x point, -2.
So 3-1=2 and -1-2=-3.
So same thing. We got -2/3.
Now we could have done it the other way,
and I'm running out of space here,
but we could have made this the first point.
If we made that the first point,
then the change in y would have been--
let me see if I wanna make it cluttered so to confuse you.
Change in y would be this y, 1-3 over change in x
should be 2, minus -1.
Well, 1-3=-2, and 2-(-1)=3.
So once again we've got -2/3.
So it doesn't matter which point we start with
as long as if we use the y in this coordinate first,
then we have to use the x in that coordinate first.
Let's do some more problems.
And actually I'm gonna do a couple just so you see
the algebra without even graphing it first.
So let's say, I wanted to figure out the slope
between the points (5,2) and (3,5).
Well, let's take this as our starting point.
So change in y over change in x, or rise/run.
Well, change in y would be this 5.
5 minus this 2,
over this 3, minus this 5,
and that gets us 3, this is a 5,
over -2=-3/2.
Let's do another one.
This time I'm gonna try to make it color-coded
so it'll be more self-explanatory.
So (1,2), that's the first point.
And then the second point is (4,3).
So once again we say slope
is equal to change in y over change in x.
Change in y, we take the first y,
we'll start here and we'll call that y1.
So that's 3 minus the second y,
which is that 2.
And then all of that over, once again, the first x which is 4,
minus the second x, which is that 1,
and this equals 3-2=1
and 4-1=3.
So the slope in this example is 1/3.
And we could have actually switched it around.
We could have done it the other way.
We could have said
2-3/1-4,
in which case we would have gotten -1/-3,
well, that just equals 1/3 again
because the negatives cancel out.
So I'll let you think about why this and this
come out to the same thing.
The important thing to realize is is if used
the 3 first, if we used the 3 first for the y,
we also have to use the 4 first for the x.
That's a common mistake.
Also you always have to be very careful with
the negative signs when you do these type of problems.
But I think that will give you at least enough of a sense
that you can start with slope problems.
The next module I'll show you how to figure out
the y-intercept, because as we said before,
the equation of any line is y=mx+b,
and I'm gonna go into this more detail,
where m is the slope.
So if you know the slope of a line,
and you know the y-intercept of a line.
You know everything you need to know about the line,
and you can write down the equation of the line
and figure out other points that are on it.
So I'm gonna do that in the future modules.
I hope I haven't confused you too much.
Try some of the slope modules;
you should be able to do them, and I hope you have fun.