Algebra: Slope Figuring out the slope of a line
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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.
- And let me write down the two points we're going to 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, negative 1, 3.
- So, let's see.
- Where do we graph that?
- So, this is 0, 0.
- We go negative 1, this is negative 1 here.
- And then we're going to go 3 up.
- 1, 2, 3.
- Because this is 3 right here.
- So, negative 1, 3 is going to be right over there.
- OK, so that's the first point.
- The second point, I'm going to do it in a different color.
- The second point is 2, 1.
- 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 going to be here.
- So we've graphed our two points.
- And now the line that connects them, it's going to look
- something thing like this.
- And I hope I can draw it well.
- 35 00:01:36,3 --> 00:01:39,078 Through that point.
- Like that.
- Then I'm going to do it.
- And then I'm just going to try to continue the line from here.
- That might be the best technique.
- Something like that.
- 42 00:01:57,68 --> 00:01:58,57 So, let's look at that line.
- So what we want to do in this presentation is, figure out
- I think will help you.
- So, there's a couple ways to view slope.
- I think, intuitively, you know that the slope is the
- inclination of this line.
- And 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 going to be a negative number, the slope.
- So you know that immediately.
- And we'll have -- what we're going to do is figure out how
- to figure out the slope.
- So the 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.
- The change in y over change in x.
- And that also is equal to rise over run.
- And I'm going to 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, negative 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 minus 2, so that's the rise.
- So the rise is equal to minus 2.
- Because we have to go down 2 to get to the same y
- as this yellow point.
- And then we have to run right there.
- We have to run plus 3.
- So rise divided by run is equal to minus 2 over 3.
- Well, how would we do that if we didn't have this nice graph
- here to actually draw on?
- Well, what we can do is, we can say let's take this
- as a starting point.
- Change in y, 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, you see that?
- We just took 3 minus 1.
- So that's the change in y over, and we take the first x point.
- Negative 1, minus the second x point, minus
- 2, so 3 minus 1 is 2.
- And negative 1 minus 2 is equal to minus 3.
- So, same thing.
- We got minus 2 over 3.
- Now we could have done it the other way.
- And I'm running out of space here.
- But we could've made this the first point.
- If we made that the first point, then the change in y
- would have been -- I want to make it really cluttered,
- so to confuse you.
- Change in y would be this y.
- 1 minus 3 over change in x, would be 2, minus minus 1.
- Well, 1 minus 3 is minus 2.
- And 2 minus negative 1 is 3.
- So, once again, we got minus 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.
- Actually, I'm going to do a couple just so you see the
- algebra without even graphing it first.
- 113 00:05:22,45 --> 00:05:24,56 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 over 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 minus 2.
- Equals minus 3/2.
- Let's do another one.
- This time I'm going to try to make it color-coded so it'll
- more self-explanatory.
- Say, it's 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.
- Well, in y.
- We take the first y.
- Let's 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 minus 2, is 1.
- And 4 minus 1 is 3.
- So the slope in this example is 1/3.
- And we could have actually switched it around.
- We could have also done it other way.
- We could have said, 2 minus 3 over 1 minus 4.
- In which case we would have gotten negative
- 1 over negative 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.
- But the important thing to realize is, if we use the 3
- first, if we use the 3 first for the y, we also have to
- use the 4 first for the x.
- That's a common mistake.
- And 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 could start the slope problems.
- The next module, I'll actually show you how to figure
- out the y intercept.
- Because, as we said, before the equation of any line is,
- y is equal to m x plus b.
- And I'm going to go into some 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 actually write down
- the equation of a line, and figure out other points
- that are on it.
- So I'm going to do that in future modules.
- I hope I haven't confused you too much.
- And try some of those the slope modules.
- You should be able to do them.
- And I hope you have fun.
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