Intuition for solving one-step equations

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.240
So once again, we
have three equal,

00:00:03.240 --> 00:00:05.502
or we say three
identical objects.

00:00:05.502 --> 00:00:06.960
They all have the
same mass, but we

00:00:06.960 --> 00:00:09.290
don't know what the
mass is of each of them.

00:00:09.290 --> 00:00:12.260
But what we do know is that
if you total up their mass,

00:00:12.260 --> 00:00:15.194
it's the same exact mass
as these nine objects

00:00:15.194 --> 00:00:15.860
right over here.

00:00:15.860 --> 00:00:18.440
And each of these nine objects
have a mass of 1 kilograms.

00:00:18.440 --> 00:00:20.647
So in total, you have 9
kilograms on this side.

00:00:20.647 --> 00:00:22.230
And over here, you
have three objects.

00:00:22.230 --> 00:00:23.490
They all have the same mass.

00:00:23.490 --> 00:00:24.698
And we don't know what it is.

00:00:24.698 --> 00:00:26.206
We're just calling that mass x.

00:00:26.206 --> 00:00:27.580
And what I want
to do here is try

00:00:27.580 --> 00:00:29.740
to tackle this a little
bit more symbolically.

00:00:29.740 --> 00:00:32.090
In the last video, we
said, hey, why don't we

00:00:32.090 --> 00:00:36.494
just multiply 1/3 of this
and multiply 1/3 of this?

00:00:36.494 --> 00:00:37.910
And then, essentially,
we're going

00:00:37.910 --> 00:00:39.600
to keep things
balanced, because we're

00:00:39.600 --> 00:00:42.030
taking 1/3 of the same mass.

00:00:42.030 --> 00:00:44.120
This total is the
same as this total.

00:00:44.120 --> 00:00:45.740
That's why the
scale is balanced.

00:00:45.740 --> 00:00:48.640
Now, let's think about
how we can represent this

00:00:48.640 --> 00:00:49.357
symbolically.

00:00:49.357 --> 00:00:51.190
So the first thing I
want you to think about

00:00:51.190 --> 00:00:56.830
is, can we set up
an equation that

00:00:56.830 --> 00:01:01.490
expresses that we have these
three things of mass x,

00:01:01.490 --> 00:01:05.071
and that in total, their mass
is equal to the total mass

00:01:05.071 --> 00:01:05.570
over here?

00:01:05.570 --> 00:01:07.500
Can we express that
as an equation?

00:01:07.500 --> 00:01:10.946
And I'll give you a
few seconds to do it.

00:01:10.946 --> 00:01:12.070
Well, let's think about it.

00:01:12.070 --> 00:01:14.280
Over here, we have three
things with mass x.

00:01:14.280 --> 00:01:16.330
So their total
mass, we could write

00:01:16.330 --> 00:01:22.010
as-- we could write their
total mass as x plus x plus x.

00:01:22.010 --> 00:01:25.940
And over here, we have nine
things with mass of 1 kilogram.

00:01:25.940 --> 00:01:29.970
I guess we could
write 1 plus 1 plus 1.

00:01:29.970 --> 00:01:30.530
That's 3.

00:01:30.530 --> 00:01:33.292
Plus 1 plus 1 plus 1 plus 1.

00:01:33.292 --> 00:01:34.000
How many is that?

00:01:34.000 --> 00:01:39.330
1, 2, 3, 4, 5, 6, 7, 8, 9.

00:01:39.330 --> 00:01:41.542
And actually, this is a
mathematical representation.

00:01:41.542 --> 00:01:43.295
If we set it up as
an equation, it's

00:01:43.295 --> 00:01:44.490
an algebraic representation.

00:01:44.490 --> 00:01:46.490
It's not the simplest
possible way we can do it,

00:01:46.490 --> 00:01:48.130
but it is a reasonable
way to do it.

00:01:48.130 --> 00:01:51.290
If we want, we can say, well, if
I have an x plus another x plus

00:01:51.290 --> 00:01:53.460
another x, I have three x's.

00:01:53.460 --> 00:01:57.280
So I could rewrite this as 3x.

00:01:57.280 --> 00:01:59.090
And 3x will be equal to?

00:01:59.090 --> 00:02:02.180
Well, if I sum up all of
these 1's right over here-- 1

00:02:02.180 --> 00:02:02.892
plus 1 plus 1.

00:02:02.892 --> 00:02:03.600
We're doing that.

00:02:03.600 --> 00:02:05.807
We have 9 of them, so
we get 3x is equal to 9.

00:02:05.807 --> 00:02:07.140
And let me make sure I did that.

00:02:07.140 --> 00:02:11.616
1, 2, 3, 4, 5, 6, 7, 8, 9.

00:02:11.616 --> 00:02:12.990
So that's how we
would set it up.

00:02:12.990 --> 00:02:15.480
And so the next question
is, what would we do?

00:02:15.480 --> 00:02:17.429
What can we do mathematically?

00:02:17.429 --> 00:02:19.220
Actually, to either
one of these equations,

00:02:19.220 --> 00:02:21.050
but we'll focus on
this one right now.

00:02:21.050 --> 00:02:23.630
What can we do
mathematically in order

00:02:23.630 --> 00:02:25.930
to essentially solve for the x?

00:02:25.930 --> 00:02:29.430
In order to figure out what
that mystery mass actually is?

00:02:29.430 --> 00:02:33.160
And I'll give you another
second or two to think about it.

00:02:33.160 --> 00:02:34.620
Well, when we did
it the last time

00:02:34.620 --> 00:02:36.244
with just the scales
we said, OK, we've

00:02:36.244 --> 00:02:38.490
got three of these x's here.

00:02:38.490 --> 00:02:40.320
We want to have just one x here.

00:02:40.320 --> 00:02:42.090
So we can say,
whatever this x is,

00:02:42.090 --> 00:02:43.787
if the scale stays
balanced, it's

00:02:43.787 --> 00:02:45.745
going to be the same as
whatever we have there.

00:02:45.745 --> 00:02:49.075
There might be a temptation
to subtract two of the x's

00:02:49.075 --> 00:02:50.950
maybe from this side,
but that won't help us.

00:02:50.950 --> 00:02:53.190
And we can even see it
mathematically over here.

00:02:53.190 --> 00:02:55.630
If we subtract two
x's from both sides,

00:02:55.630 --> 00:02:58.174
on the left-hand side you're
going to have 3x minus 2x.

00:02:58.174 --> 00:02:59.590
And on the right-hand
side, you're

00:02:59.590 --> 00:03:01.610
going to have 9 minus 2x.

00:03:01.610 --> 00:03:03.860
And you're just going to
be left with 3 of something

00:03:03.860 --> 00:03:05.790
minus 2 something is
just 1 of something.

00:03:05.790 --> 00:03:08.856
So you will just have an x there
if you get rid of two of them.

00:03:08.856 --> 00:03:11.230
But on the right-hand side,
you're going to get 9 minus 2

00:03:11.230 --> 00:03:11.370
x's.

00:03:11.370 --> 00:03:13.100
So the x's still
didn't help you out.

00:03:13.100 --> 00:03:16.020
You still have a mystery
mass on the right-hand side.

00:03:16.020 --> 00:03:17.430
So that doesn't help.

00:03:17.430 --> 00:03:20.065
So instead, what we say is--
and we did this the last time.

00:03:20.065 --> 00:03:22.190
We said, well, what if we
took 1/3 of these things?

00:03:22.190 --> 00:03:25.180
If we take 1/3 of these things
and take 1/3 of these things,

00:03:25.180 --> 00:03:27.680
we should still get the
same mass on both sides

00:03:27.680 --> 00:03:29.860
because the original
things had the same mass.

00:03:29.860 --> 00:03:31.818
And the equivalent of
doing that mathematically

00:03:31.818 --> 00:03:35.260
is to say, why don't we
multiply both sides by 1/3?

00:03:35.260 --> 00:03:37.860
Or another way to say it is we
could divide both sides by 3.

00:03:37.860 --> 00:03:41.670
Multiplying by 1/3 is the
same thing as dividing by 3.

00:03:41.670 --> 00:03:44.340
So we're going to multiply
both sides by 1/3.

00:03:44.340 --> 00:03:47.330
When you multiply both sides
by 1/3-- visually over here,

00:03:47.330 --> 00:03:49.520
if you had three x's,
you multiply it by 1/3,

00:03:49.520 --> 00:03:51.340
you're only going
to have one x left.

00:03:51.340 --> 00:03:53.910
If you have nine of
these one-kilogram boxes,

00:03:53.910 --> 00:03:59.130
you multiply it by 1/3, you're
only going to have three left.

00:03:59.130 --> 00:04:00.830
And over here, you
can even visually--

00:04:00.830 --> 00:04:03.690
if you divide by 3, which is
the same thing as multiplying

00:04:03.690 --> 00:04:07.200
by 1/3, you divide by 3.

00:04:07.200 --> 00:04:09.560
So you divide by 3.

00:04:09.560 --> 00:04:13.670
You have an x is equal
to a 1 plus 1 plus 1.

00:04:13.670 --> 00:04:15.360
An x is equal to 3.

00:04:15.360 --> 00:04:18.380
Or you see here,
an x is equal to 3.

00:04:18.380 --> 00:04:19.680
Over here you do the math.

00:04:19.680 --> 00:04:21.333
1/3 times 3 is 1.

00:04:21.333 --> 00:04:23.320
You're left with 1x.

00:04:23.320 --> 00:04:27.420
So you're left with x
is equal to 9 times 1/3.

00:04:27.420 --> 00:04:29.900
Or you could even view
it as 9 divided by 3,

00:04:29.900 --> 00:04:37.380
which is equal to 3.


Solving two-step equations

WEBVTT
Kind: captions
Language: en

00:00:00.754 --> 00:00:03.588
Now we mixed up things a little bit more:

00:00:03.588 --> 00:00:07.466
on the left side of the scale,
not only do we have

00:00:07.466 --> 00:00:11.487
these identical unknown masses with mass X,
these three blue things,

00:00:11.487 --> 00:00:14.087
we also have some of the 1kg masses
over here,

00:00:14.087 --> 00:00:16.498
actually, we have two of them.

00:00:16.498 --> 00:00:18.588
Now, we are going to figure out what X is.

00:00:18.588 --> 00:00:22.687
But before we even do that, I want you to think about a mathematical equation

00:00:22.687 --> 00:00:27.554
that can represent what is going on;
that equates what we have on the left hand,

00:00:27.554 --> 00:00:31.620
with what we have on the right side of the scale.

00:00:31.620 --> 00:00:33.354
I will give you a few seconds to think about it...

00:00:35.287 --> 00:00:37.620
So let's think about what we have on the left side:

00:00:37.620 --> 00:00:46.020
we have 3 masses with mass X, so you can say
we have 3x

00:00:46.020 --> 00:00:52.154
and then we have 2 masses of 1 kilogram,
so in total we have 2 kg. So + 2.

00:00:52.154 --> 00:00:57.354
So one way to think about the total mass on the left-hand side is 3x + 2.

00:00:57.354 --> 00:01:00.848
Three masses with mass X,
plus two kilograms.

00:01:00.848 --> 00:01:02.938
That is what we have on the left-hand side.

00:01:02.938 --> 00:01:05.585
Now, let us think about what we have on the right-hand side.

00:01:05.585 --> 00:01:07.154
We can simply count them:

00:01:07.154 --> 00:01:11.855
[counts to 14]

00:01:11.855 --> 00:01:19.220
Fourteen blocks, each has a mass of 1 kg,
so the total mass will be 14 kg.

00:01:19.220 --> 00:01:23.620
And we see that the scale is balanced, not tilting down or upwards.

00:01:23.620 --> 00:01:28.666
So this mass over here must be equal to this total mass.

00:01:28.666 --> 00:01:34.087
The scale is balanced,
so we can write an 'equal'-sign.

00:01:34.087 --> 00:01:37.220
(let me do that in a white coulour,
I do not like that brown)

00:01:37.220 --> 00:01:41.258
Now, what I want you to think about,

00:01:41.274 --> 00:01:44.890
and you can think about it either through the symbols or through the scales, is:

00:01:44.890 --> 00:01:48.104
how would you go about --
let us think about a few things:

00:01:48.104 --> 00:01:54.090
how would you first go about at least getting rid of these little 1kg blocks?

00:01:54.090 --> 00:01:56.557
I will give you a second to think about that...

00:01:59.023 --> 00:02:01.107
Well, the simplest thing is:

00:02:01.107 --> 00:02:03.940
you can take these 1kg blocks off of the left-hand side,

00:02:03.940 --> 00:02:07.098
but remember, if you just took these blocks off of the left-hand side,

00:02:07.098 --> 00:02:10.357
and it was balanced before,
now the left-hand side will be lighter

00:02:10.357 --> 00:02:13.757
and it will move up. But we want to keep it balanced so we can keep saying 'equal'.

00:02:13.757 --> 00:02:16.107
That this mass is equal to that mass.

00:02:16.107 --> 00:02:21.490
So, if we remove 2 block from the left-hand side,
we need to remove 2 from the right-hand side.

00:02:21.490 --> 00:02:26.490
So, we can remove two there,
and then we can remove two over there.

00:02:26.490 --> 00:02:31.340
Mathematically, what we are doing is:
we are subtracting 2 kilograms from each side.

00:02:31.340 --> 00:02:34.157
We are subtracting 2 from this side,

00:02:34.157 --> 00:02:38.909
So on the left-hand side we now have
3x + 2, minus 2

00:02:38.909 --> 00:02:42.624
we are left with just 3x,

00:02:42.624 --> 00:02:47.557
and on the right-hand side we had 14 and we took away 2 (let me write this:)

00:02:47.557 --> 00:02:53.023
we took away 2, so we are going to be left with 12 blocks.

00:02:53.023 --> 00:02:56.423
And you see that there, the ones that I have not crossed out, there are 12 left,

00:02:56.423 --> 00:02:58.957
and here you have 3 of those X-blocks.

00:02:58.957 --> 00:03:02.361
Since we removed the same amount from both sides,

00:03:02.361 --> 00:03:08.357
our scale is still balanced. And our equation:
3x is now equal to 12.

00:03:08.357 --> 00:03:12.690
Now, this turns into a problem very similar to what we saw in the last video,

00:03:12.690 --> 00:03:17.083
so now I ask you: what can we do to isolate one x,

00:03:17.083 --> 00:03:21.428
to only have one 'X' on the left-hand side of the scale,

00:03:21.428 --> 00:03:24.552
while keeping the scale balanced?

00:03:28.163 --> 00:03:30.629
The easiest way to think about it is:

00:03:30.629 --> 00:03:35.154
If I want one X on this left-hand side, that is a third of the total X's here.

00:03:35.154 --> 00:03:38.963
So what if I were to multiply the left-hand side by one-third --

00:03:38.963 --> 00:03:43.745
-- but if I want to keep the scale balanced,
I have to multiply the right-hand side by one-third.

00:03:43.745 --> 00:03:46.229
If we can do that mathematically,

00:03:46.229 --> 00:03:49.563
Over here I can multiply the left-hand side by 1/3,

00:03:49.563 --> 00:03:55.163
and if I want to keep my scale balanced I also have to multiply the right-hand side by 1/3.

00:03:55.163 --> 00:04:00.563
Multiplying it physically literally means: just keeping a third of what we had originally

00:04:00.563 --> 00:04:03.896
We would get rid of two of these.

00:04:03.896 --> 00:04:06.029
If we want to keep a third of what we had here originally,

00:04:06.029 --> 00:04:08.896
-- there are 12 blocks left over after removing those first two --

00:04:08.896 --> 00:04:15.429
so, 1/3 of 12: we are only going to have four of these little 1kg boxes left.

00:04:15.429 --> 00:04:21.363
Let me remove all but four.
(so, remove those, and those...)

00:04:21.363 --> 00:04:24.659
And I have left [counts them] 4 here.

00:04:24.659 --> 00:04:27.563
And so, what you are left with,
the only thing you have left,

00:04:27.563 --> 00:04:32.554
is this 'X' - I will shade it in to show this is the one we actually have left -

00:04:32.554 --> 00:04:37.058
and then we have these 1 kilogram boxes.

00:04:37.058 --> 00:04:40.866
You see it mathematically over here:
1/3 * 3x

00:04:40.866 --> 00:04:45.763
-- or you could have said 3x divided by 3 --
either way, that gives us

00:04:45.763 --> 00:04:49.736
-- these threes cancel out, so that would give you an 'X'

00:04:49.736 --> 00:04:56.633
and on the right-hand side: 12 * 1/3 - which is the same as 12/3, is equal to 4.

00:04:56.633 --> 00:05:02.600
And, since we did the same thing to both sides, the scale is still balanced.

00:05:02.600 --> 00:05:07.894
So you see that the mass of this thing must be the same as the mass of these 4 left-over blocks.

00:05:07.894 --> 00:05:10.829
x must be equal to 4 kilograms.


How to divide from both sides of an equation

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:02.220
So we have our scale again.

00:00:02.220 --> 00:00:04.892
And we've got some masses on the
left hand side and some masses

00:00:04.892 --> 00:00:05.850
on the right hand side.

00:00:05.850 --> 00:00:07.580
And we see that our
scale is balanced.

00:00:07.580 --> 00:00:10.540
We have the same total
mass on the left hand side

00:00:10.540 --> 00:00:12.450
that we have on the
right hand side.

00:00:12.450 --> 00:00:15.000
Instead of labeling the mystery
masses as question mark,

00:00:15.000 --> 00:00:16.309
I've labeled them all x.

00:00:16.309 --> 00:00:19.600
And since they all have an x on
it, we know that each of these

00:00:19.600 --> 00:00:20.770
have the same mass.

00:00:20.770 --> 00:00:23.220
But what I'm curious about
is, what is that mass?

00:00:23.220 --> 00:00:27.060
What is the mass of
each of these mystery

00:00:27.060 --> 00:00:28.660
masses, I guess we could say?

00:00:28.660 --> 00:00:30.970
And so I'll let think
about that for a second.

00:00:30.970 --> 00:00:37.140
How would you figure out what
this x value actually is?

00:00:37.140 --> 00:00:40.870
How many kilograms is the
mass of each of these things?

00:00:40.870 --> 00:00:44.760
What could you do to either one
or both sides of this scale?

00:00:44.760 --> 00:00:46.760
I'll give you a few seconds
to think about that.

00:00:49.540 --> 00:00:51.040
So you might be
tempted to say, well

00:00:51.040 --> 00:00:53.810
if I could end up with just one
mystery mass on the left hand

00:00:53.810 --> 00:00:57.900
side, and if I keep my scale
balanced, then that thing's

00:00:57.900 --> 00:01:00.400
going to be equal to whatever
I have on the right hand side.

00:01:00.400 --> 00:01:02.750
And that part would actually
be a true statement.

00:01:02.750 --> 00:01:05.640
But then to get only one
of these mystery masses

00:01:05.640 --> 00:01:07.890
on the left hand side, you
might say, well why don't I

00:01:07.890 --> 00:01:09.390
just remove two of them?

00:01:09.390 --> 00:01:13.594
You might just say, well
why don't I just remove--

00:01:13.594 --> 00:01:15.260
let me do it a good
color for removing--

00:01:15.260 --> 00:01:17.780
why don't I just remove
that one and that one?

00:01:17.780 --> 00:01:20.030
And then I'll just be left
with that right over there.

00:01:20.030 --> 00:01:22.470
But if you just
removed these two,

00:01:22.470 --> 00:01:24.740
then the left hand side
is going to become lighter

00:01:24.740 --> 00:01:27.580
or it's going to have a lower
mass than the right hand side.

00:01:27.580 --> 00:01:30.350
So it's going to move up
and the right hand side

00:01:30.350 --> 00:01:32.232
is going to move down.

00:01:32.232 --> 00:01:33.940
And then you might
say, OK, I understand.

00:01:33.940 --> 00:01:36.232
Whatever I have to do
to the left hand side,

00:01:36.232 --> 00:01:37.690
I have to do to
the right hand side

00:01:37.690 --> 00:01:39.390
in order to keep
my scale balanced.

00:01:39.390 --> 00:01:41.220
So you might say,
well why don't I

00:01:41.220 --> 00:01:43.440
remove two of these
mystery masses

00:01:43.440 --> 00:01:44.801
from the right hand side?

00:01:44.801 --> 00:01:46.300
But that's a problem
too because you

00:01:46.300 --> 00:01:47.810
don't know what this
mystery mass is.

00:01:47.810 --> 00:01:49.420
You could try to
remove two from this,

00:01:49.420 --> 00:01:51.669
but how many of these blocks
represent a mystery mass?

00:01:51.669 --> 00:01:53.540
We actually don't know.

00:01:53.540 --> 00:01:55.990
But you might then
say, well let's see,

00:01:55.990 --> 00:01:57.920
I've got three of
these things here.

00:01:57.920 --> 00:02:01.850
If I essentially multiply
what I have here by 1/3

00:02:01.850 --> 00:02:04.010
or if I only leave a
1/3 of the stuff here,

00:02:04.010 --> 00:02:06.090
and if I only leave a
1/3 of the stuff here,

00:02:06.090 --> 00:02:07.820
then the scale
should be balanced.

00:02:07.820 --> 00:02:12.390
If this has the
total mass as this,

00:02:12.390 --> 00:02:14.470
then 1/3 of this
total mass is going

00:02:14.470 --> 00:02:18.710
to be the same thing as
1/3 of that total mass.

00:02:18.710 --> 00:02:21.480
So let's just keep
only 1/3 of this here.

00:02:21.480 --> 00:02:24.792
So that's the equivalent
to multiplying by 1/3.

00:02:24.792 --> 00:02:26.500
So if we're only going
to keep 1/3 there,

00:02:26.500 --> 00:02:29.130
we're going to be left with
only one of the masses.

00:02:29.130 --> 00:02:31.150
And if we only keep
1/3 here, let's see,

00:02:31.150 --> 00:02:34.140
we have one, two, three,
four, five, six, seven, eight,

00:02:34.140 --> 00:02:36.110
nine masses.

00:02:36.110 --> 00:02:39.330
If we multiply this by 1/3,
or if we only keep 1/3 of it

00:02:39.330 --> 00:02:42.880
there, 1/3 times 9 is 3.

00:02:42.880 --> 00:02:44.750
So we're going to remove these .

00:02:44.750 --> 00:02:46.580
And so we have 1/3
of what we originally

00:02:46.580 --> 00:02:48.927
had on the right hand side
and 1/3 of what we originally

00:02:48.927 --> 00:02:50.010
had on the left hand side.

00:02:50.010 --> 00:02:52.170
And they will be
balanced because we

00:02:52.170 --> 00:02:55.470
took 1/3 of the
same total masses.

00:02:55.470 --> 00:02:57.610
And so what you're
left with is just one

00:02:57.610 --> 00:03:00.900
of these mystery masses,
this x thing right over here,

00:03:00.900 --> 00:03:02.370
whatever x might be.

00:03:02.370 --> 00:03:05.182
And you have three kilograms
on the right hand side.

00:03:05.182 --> 00:03:06.640
And so you can make
the conclusion,

00:03:06.640 --> 00:03:08.223
and the whole time
you kept this thing

00:03:08.223 --> 00:03:12.456
balanced, that x is equal to 3.


How to represent a relationship with a simple equation

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.100
I now want to refigure out
what this mystery mass is,

00:00:04.100 --> 00:00:07.850
but we're going to start using a
little bit more of mathematics.

00:00:07.850 --> 00:00:10.240
And mathematics really
are just a language,

00:00:10.240 --> 00:00:15.270
symbols for representing ideas,
for representing relationships

00:00:15.270 --> 00:00:16.379
between things.

00:00:16.379 --> 00:00:18.250
And so the first
thing I want you to do

00:00:18.250 --> 00:00:20.810
is think about if you can
express a relationship

00:00:20.810 --> 00:00:23.676
mathematically between
this side of the scale

00:00:23.676 --> 00:00:24.800
and that side of the scale.

00:00:24.800 --> 00:00:26.010
And I'll give you some hints.

00:00:26.010 --> 00:00:28.010
We know that they
have equal mass.

00:00:28.010 --> 00:00:29.530
So maybe you can
set up some type

00:00:29.530 --> 00:00:32.159
of relationship using an
equal sign, somehow showing

00:00:32.159 --> 00:00:35.299
that this right over
here is equal to that.

00:00:35.299 --> 00:00:37.090
And I'll give you a
few seconds to do that.

00:00:39.759 --> 00:00:41.300
So let's think about
it a little bit.

00:00:41.300 --> 00:00:43.140
What do we have on this side?

00:00:43.140 --> 00:00:45.010
Well, we have our mystery mass.

00:00:45.010 --> 00:00:47.510
And I'll represent
that mystery mass

00:00:47.510 --> 00:00:49.759
by the question mark
right over here.

00:00:49.759 --> 00:00:51.300
But that's not the
only thing that we

00:00:51.300 --> 00:00:52.520
have on the left-hand side.

00:00:52.520 --> 00:00:55.500
We also have these
other 3 kilograms.

00:00:55.500 --> 00:00:56.960
So let me write over here.

00:00:56.960 --> 00:00:59.020
We'll assume that we're
dealing with kilograms.

00:00:59.020 --> 00:01:01.700
So we have the mystery
mass in kilograms

00:01:01.700 --> 00:01:03.800
plus 3 more kilograms.

00:01:03.800 --> 00:01:06.469
That's what we have here
on the left-hand side.

00:01:06.469 --> 00:01:08.510
Now, what do we have here
on the right-hand side?

00:01:08.510 --> 00:01:15.280
Well, we just have 1, 2, 3, 4,
5, 6, 7, 8, 9, 10 kilograms.

00:01:15.280 --> 00:01:16.770
So we just have 10.

00:01:16.770 --> 00:01:19.770
We just have 10 on
the right-hand side.

00:01:19.770 --> 00:01:21.410
And what else do we know?

00:01:21.410 --> 00:01:24.500
Well, we know that this scale
is balanced, that the mass here

00:01:24.500 --> 00:01:26.850
is equal to the mass here.

00:01:26.850 --> 00:01:29.990
Because the scale is balanced
the way it's been drawn,

00:01:29.990 --> 00:01:34.730
we know that these
two things are equal.

00:01:34.730 --> 00:01:37.160
So we have just
set up an equation.

00:01:37.160 --> 00:01:39.520
We're using question
mark as our unknown.

00:01:39.520 --> 00:01:41.950
We don't know what
this mystery mass is.

00:01:41.950 --> 00:01:44.680
If we add 3 kilograms
to it, then we

00:01:44.680 --> 00:01:48.800
see that it has the exact
same mass as 10 kilograms.

00:01:48.800 --> 00:01:52.900
Now my question to you is,
what can we do to this equation

00:01:52.900 --> 00:01:55.560
so that we can essentially
solve for the unknown,

00:01:55.560 --> 00:01:58.875
so that we can figure
out what the unknown is?

00:02:01.860 --> 00:02:04.410
Well, we saw in the
last little problem

00:02:04.410 --> 00:02:07.390
that we had that if we wanted
to figure out this mystery mass,

00:02:07.390 --> 00:02:11.290
we had to remove 3
kilograms from both sides.

00:02:11.290 --> 00:02:13.677
If we just removed 3
kilograms from one side,

00:02:13.677 --> 00:02:15.510
then the scale wouldn't
be balanced anymore.

00:02:15.510 --> 00:02:17.400
And we really wouldn't be able
to say that the mystery mass is

00:02:17.400 --> 00:02:18.764
equal to the thing on the right.

00:02:18.764 --> 00:02:20.430
In order to say they're
equal, the stuff

00:02:20.430 --> 00:02:22.300
has to actually be balanced.

00:02:22.300 --> 00:02:27.580
So in the last video,
we removed 3 of these.

00:02:27.580 --> 00:02:30.260
We removed 3 kilograms
from both sides

00:02:30.260 --> 00:02:32.880
in order to keep
the scale balanced.

00:02:32.880 --> 00:02:36.020
So mathematically, we'll do
the exact same thing over here.

00:02:36.020 --> 00:02:38.730
We will remove 3,
not from one side.

00:02:38.730 --> 00:02:40.840
If we remove 3 from
one side, then it

00:02:40.840 --> 00:02:42.610
wouldn't be equal anymore.

00:02:42.610 --> 00:02:45.580
We need to remove
3 from both sides.

00:02:45.580 --> 00:02:47.560
So we need to remove 3.

00:02:47.560 --> 00:02:51.450
We need to subtract 3 from
both sides of this equation

00:02:51.450 --> 00:02:53.830
in order to keep
the scale balanced.

00:02:53.830 --> 00:02:55.940
So on the left-hand side,
what are we left with?

00:02:55.940 --> 00:02:58.460
Well, just like over here, we're
left with just the question

00:02:58.460 --> 00:02:58.960
mark.

00:02:58.960 --> 00:03:00.690
3 minus 3 is 0.

00:03:00.690 --> 00:03:03.580
So on the left-hand side, we're
left with just the question

00:03:03.580 --> 00:03:04.310
mark.

00:03:04.310 --> 00:03:08.930
And on the right-hand side,
we're left with 10 minus 3,

00:03:08.930 --> 00:03:09.850
which is 7.

00:03:12.480 --> 00:03:14.310
And we get the
exact same result.

00:03:14.310 --> 00:03:17.360
Question mark is equal to 7.

00:03:17.360 --> 00:03:18.860
And if we're dealing
with kilograms,

00:03:18.860 --> 00:03:22.320
then this is 7 kilograms.


Why we do the same thing to both sides of equations

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:04.790
We've got a scale here, and as
you see, the scale is balanced.

00:00:04.790 --> 00:00:07.550
And we have a
question to answer.

00:00:07.550 --> 00:00:09.720
We have this mystery
mass over here.

00:00:09.720 --> 00:00:13.560
It's a big question mark on
this blue mass right over here.

00:00:13.560 --> 00:00:16.170
And we also have
a bunch of 1-- I

00:00:16.170 --> 00:00:18.410
guess we could call
them 1-kilogram masses.

00:00:18.410 --> 00:00:21.680
So these are all each
a 1-kilogram mass.

00:00:21.680 --> 00:00:24.120
And my question to
you is, what could we

00:00:24.120 --> 00:00:27.450
do to either side of
this scale in order

00:00:27.450 --> 00:00:29.475
to figure out what
the mystery mass is?

00:00:29.475 --> 00:00:31.100
Or maybe we can't
figure it out at all.

00:00:31.100 --> 00:00:34.420
Is there something that we can
do, either removing or adding

00:00:34.420 --> 00:00:36.420
these things, so that
we can figure out

00:00:36.420 --> 00:00:38.190
what this mystery mass is?

00:00:38.190 --> 00:00:40.990
And I'll give you a couple of
seconds to think about that.

00:00:43.990 --> 00:00:47.210
Well, to figure out what
this mystery mass is,

00:00:47.210 --> 00:00:51.200
we essentially just want this
on one side of this scale.

00:00:51.200 --> 00:00:52.680
But that by itself isn't enough.

00:00:52.680 --> 00:00:55.190
We could just remove these
3, but that won't do the job,

00:00:55.190 --> 00:00:57.450
because if we just
remove these 3,

00:00:57.450 --> 00:00:59.260
then the left-hand
side of the scale

00:00:59.260 --> 00:01:02.650
is clearly going to have less
mass, and it's going to go up,

00:01:02.650 --> 00:01:04.390
and the right side
is going to go down.

00:01:04.390 --> 00:01:06.431
And that's not going to
give as much information.

00:01:06.431 --> 00:01:09.620
It's just going to tell us that
this blue thing has a lower

00:01:09.620 --> 00:01:11.760
mass than what's over here.

00:01:11.760 --> 00:01:13.870
So just removing this
won't help us much.

00:01:13.870 --> 00:01:16.719
It won't let us know that
this is equal to that.

00:01:16.719 --> 00:01:19.010
Well, what we've got to do
if we want to keep the scale

00:01:19.010 --> 00:01:22.180
balanced is we've got to
remove the same amount of mass

00:01:22.180 --> 00:01:24.530
from both sides of the scale.

00:01:24.530 --> 00:01:27.070
So if we want to remove
3 things here-- so let

00:01:27.070 --> 00:01:30.250
me try my best to
remove 3 things here.

00:01:30.250 --> 00:01:33.434
If we want to remove 3--
let me do it like this.

00:01:33.434 --> 00:01:34.350
I'll just color on it.

00:01:34.350 --> 00:01:35.540
I'll just erase it.

00:01:35.540 --> 00:01:37.850
So if we want to
remove 3 things there,

00:01:37.850 --> 00:01:41.780
if we did this by itself,
just removed these 3 things,

00:01:41.780 --> 00:01:44.340
then the two sides would not
have an equal mass anymore.

00:01:44.340 --> 00:01:46.270
This side over here
would have a lower mass.

00:01:46.270 --> 00:01:48.495
So we've got to remove
3 from both sides.

00:01:48.495 --> 00:01:49.870
So if we really
want to make sure

00:01:49.870 --> 00:01:51.244
that our scale is
balanced, we've

00:01:51.244 --> 00:01:54.160
got to remove 3 from both sides.

00:01:54.160 --> 00:01:57.220
And so if we started off
with the scales balanced

00:01:57.220 --> 00:01:59.480
and then we removed
3 from both sides,

00:01:59.480 --> 00:02:01.360
the scale will
still be balanced.

00:02:01.360 --> 00:02:03.360
And then when we do that,
we have a clearer idea

00:02:03.360 --> 00:02:05.900
of what the mass of
this object actually is.

00:02:05.900 --> 00:02:07.860
Now, when we remove
3 from both sides,

00:02:07.860 --> 00:02:10.500
the scale will
still be balanced.

00:02:10.500 --> 00:02:12.600
And we know that this mass
is equal to whatever's

00:02:12.600 --> 00:02:13.530
left over here.

00:02:13.530 --> 00:02:17.784
It's equal to 1,
2, 3, 4, 5, 6, 7.

00:02:17.784 --> 00:02:19.450
And if we're assuming
they're kilograms,

00:02:19.450 --> 00:02:21.410
we'll know that
the question mark

00:02:21.410 --> 00:02:26.520
mass is equal to 7 kilograms,
that this right over here

00:02:26.520 --> 00:02:30.920
is a 7-kilogram mass.


Why all the letters in algebra?

WEBVTT
Kind: captions
Language: en

00:00:00.145 --> 00:00:03.553
I'm here with Jesse Ro, whose a math teacher at Summit San Jose

00:00:03.553 --> 00:00:05.160
and a Khan Academy teaching fellow

00:00:05.160 --> 00:00:08.403
and you had some interesting ideas or questions.

00:00:08.403 --> 00:00:11.599
Yeah, one question that students ask a lot when they start Algebra

00:00:11.599 --> 00:00:16.106
is why do we need letters, why can't we just use numbers for everything?

00:00:16.106 --> 00:00:18.076
Why letters? So why do we have all these

00:00:18.076 --> 00:00:21.811
Xs and Ys and Zs and ABCs when we start dealing with Algebra?

00:00:21.811 --> 00:00:22.680
Yeah, exactly.

00:00:22.680 --> 00:00:28.286
That's interesting, well why don't we let people think about that for a second.

00:00:28.286 --> 00:00:30.232
So Sal, how would you answer this question?

00:00:30.232 --> 00:00:32.193
Why do we need letters in Algebra?

00:00:32.193 --> 00:00:35.824
So why letters. So there are a couple of ways I'd think about it.

00:00:35.824 --> 00:00:38.026
One is if you have an unknown.

00:00:38.026 --> 00:00:44.239
So if I were to write X plus three is equal to ten

00:00:44.239 --> 00:00:46.577
the reason why we're doing this is that we don't know what X is

00:00:46.577 --> 00:00:48.298
It's literally an unknown.

00:00:48.298 --> 00:00:50.197
And so we're going to solve for it in some way.

00:00:50.197 --> 00:00:51.710
But it did not have to be the letter X.

00:00:51.710 --> 00:00:55.703
We could have literally written blank plus three is equal to ten.

00:00:55.703 --> 00:00:59.725
Or we could have written Question Mark plus three is equal to ten.

00:00:59.725 --> 00:01:03.147
So it didn't have to be letters, but we needed some type of symbol.

00:01:03.147 --> 00:01:07.434
It literally could've been Smiley Face plus three is equal to ten.

00:01:07.434 --> 00:01:12.181
But until you know it, you need some type of a symbol to represent whatever that number is.

00:01:12.181 --> 00:01:15.700
Now we can go and solve this equation and then know what that symbol represents.

00:01:15.700 --> 00:01:17.916
But if we knew it ahead of time, it wouldn't be an unknown.

00:01:17.916 --> 00:01:20.387
It wouldn't be something that we didn't know.

00:01:20.387 --> 00:01:23.576
So that's one reason why I would use letters

00:01:23.576 --> 00:01:26.489
and where just numbers by itself wouldn't be helpful.

00:01:26.489 --> 00:01:28.942
The other is when you're describing relationships between numbers.

00:01:28.942 --> 00:01:32.286
So I could do something like - I could say - that

00:01:32.286 --> 00:01:38.021
whenever you give me a three, I'm going to give you a four.

00:01:38.021 --> 00:01:43.762
And I could say, if you give me a five, I'm going to give you a six.

00:01:43.762 --> 00:01:46.050
And i could keep going on and on forever.

00:01:46.050 --> 00:01:51.626
If you give me a 7.1, I'm going to give you an 8.1.

00:01:51.626 --> 00:01:54.431
And I could keep listing this on and on forever.

00:01:54.431 --> 00:01:57.477
Maybe you could give me any number, and I could tell you what I'm going to give you.

00:01:57.477 --> 00:02:00.879
But I would obviously run out of space and time if I were to list all of them.

00:02:00.879 --> 00:02:06.259
And we could do that much more elegantly if we used letters to describe the relationship.

00:02:06.259 --> 00:02:11.296
Maybe what you give me we call X, and what I give you we call Y.

00:02:11.296 --> 00:02:14.678
And so I say, look, whatever you give me, I'm going to add one to it.

00:02:14.678 --> 00:02:16.867
And that's what I'm going to give back to you.

00:02:16.867 --> 00:02:20.670
And so now, this very simple equation here

00:02:20.670 --> 00:02:24.717
can describe an infinite number of relationships between X

00:02:24.717 --> 00:02:28.219
or an infinite number of corresponding Ys and Xs.

00:02:28.219 --> 00:02:31.497
So now someone knows whatever X you give me

00:02:31.497 --> 00:02:34.612
you give me three, I add one to it, and I'm going to give you four.

00:02:34.627 --> 00:02:38.200
You give me 7.1, I'm going to add one to it and give you 8.1.

00:02:38.200 --> 00:02:41.166
So there is no more elegant way that you could've done it than by using symbols.

00:02:41.166 --> 00:02:43.959
With that said, I didn't have to use Xs and Ys.

00:02:43.959 --> 00:02:46.648
This is just a convention that kind of comes to use from history.

00:02:46.648 --> 00:02:49.725
I could've defined what you give me as Star

00:02:49.725 --> 00:02:54.827
and what I give you as Smiley Face

00:02:54.827 --> 00:02:57.733
and this also would've been a valid way to express this.

00:02:57.733 --> 00:03:01.733
So the letters are really just symbols. Nothing more.


Difference between equations and functions

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.697
SALMAN KHAN: I'm here with
Jesse Roe of Summit Prep.

00:00:03.697 --> 00:00:04.780
What classes do you teach?

00:00:04.780 --> 00:00:07.040
JESSE ROE: I teach algebra,
geometry, and algebra II.

00:00:07.040 --> 00:00:08.270
SALMAN KHAN: And now
you're with us, luckily,

00:00:08.270 --> 00:00:10.240
for the summer, doing
a whole bunch of stuff

00:00:10.240 --> 00:00:11.115
as a teaching fellow.

00:00:11.115 --> 00:00:12.656
JESSE ROE: Yeah, as
a teaching fellow

00:00:12.656 --> 00:00:14.760
I've been helping with
organizing and developing

00:00:14.760 --> 00:00:17.294
new content, mostly on the
exercise side of the site.

00:00:17.294 --> 00:00:19.710
SALMAN KHAN: And the reason
why we're doing this right now

00:00:19.710 --> 00:00:23.020
is you had some very
interesting ideas or questions.

00:00:23.020 --> 00:00:25.020
JESSE ROE: Yeah, so
as an algebra teacher,

00:00:25.020 --> 00:00:27.710
when I introduce that concept
of algebra to students,

00:00:27.710 --> 00:00:28.930
I get a lot of questions.

00:00:28.930 --> 00:00:30.470
One of those
questions is, what's

00:00:30.470 --> 00:00:33.280
the difference between an
equation and a function?

00:00:33.280 --> 00:00:36.720
SALMAN KHAN: The difference
between an equation verses

00:00:36.720 --> 00:00:39.680
a function, that's an
interesting question.

00:00:39.680 --> 00:00:42.142
Let's pause it and
let the viewers

00:00:42.142 --> 00:00:43.600
try to think about
it a little bit.

00:00:43.600 --> 00:00:45.670
And then maybe we'll
give a stab at it.

00:00:45.670 --> 00:00:46.670
JESSE ROE: Sounds great.

00:00:49.340 --> 00:00:51.320
So Sal, how would you
answer this question?

00:00:51.320 --> 00:00:53.720
What's the difference between
an equation and a function?

00:00:53.720 --> 00:00:55.950
SALMAN KHAN: Let me think
about it a little bit.

00:00:55.950 --> 00:00:56.660
So let me think.

00:00:56.660 --> 00:00:58.540
I think there's
probably equations

00:00:58.540 --> 00:01:00.780
that are not functions
and functions that

00:01:00.780 --> 00:01:01.530
are not equations.

00:01:01.530 --> 00:01:03.571
And then there are probably
things that are both.

00:01:03.571 --> 00:01:05.330
So let me think of it that way.

00:01:05.330 --> 00:01:08.170
So I'm going to draw-- if
this is the world of equations

00:01:08.170 --> 00:01:12.555
right over here, so
this is equations.

00:01:15.310 --> 00:01:19.460
And then over here is
the world of functions.

00:01:19.460 --> 00:01:21.306
That's the world of functions.

00:01:21.306 --> 00:01:22.680
I do think there
is some overlap.

00:01:22.680 --> 00:01:25.520
We'll think it through
where the overlap is,

00:01:25.520 --> 00:01:26.717
the world of functions.

00:01:26.717 --> 00:01:29.050
So an equation that is not a
function that's sitting out

00:01:29.050 --> 00:01:35.490
here, a simple one would
be something like x plus 3

00:01:35.490 --> 00:01:37.760
is equal to 10.

00:01:37.760 --> 00:01:40.354
I'm not explicitly talking
about inputs and outputs

00:01:40.354 --> 00:01:41.770
or relationship
between variables.

00:01:41.770 --> 00:01:44.282
I'm just stating an equivalence.

00:01:44.282 --> 00:01:48.230
The expression x plus
3 is equal to 10.

00:01:48.230 --> 00:01:50.870
So this, I think, traditionally
would just be an equation,

00:01:50.870 --> 00:01:52.480
would not be a function.

00:01:52.480 --> 00:01:55.450
Functions essentially
talk about relationships

00:01:55.450 --> 00:01:56.200
between variables.

00:01:56.200 --> 00:01:58.410
You get one or more
input variables,

00:01:58.410 --> 00:02:01.590
and we'll give you only
one output variable.

00:02:01.590 --> 00:02:04.900
I'll put value.

00:02:04.900 --> 00:02:06.232
And you can define a function.

00:02:06.232 --> 00:02:07.440
And I'll do that in a second.

00:02:07.440 --> 00:02:09.030
You could define a
function as an equation,

00:02:09.030 --> 00:02:10.449
but you can define a function
a whole bunch of ways.

00:02:10.449 --> 00:02:12.160
You can visually
define a function,

00:02:12.160 --> 00:02:16.790
maybe as a graph-- so
something like this.

00:02:16.790 --> 00:02:19.050
And maybe I actually
mark off the values.

00:02:19.050 --> 00:02:21.320
So that's 1, 2, 3.

00:02:21.320 --> 00:02:23.510
Those are the
potential x values.

00:02:23.510 --> 00:02:25.605
And then on the
vertical axis, I show

00:02:25.605 --> 00:02:27.480
what the value of my
function is going to be,

00:02:27.480 --> 00:02:29.000
literally my function of x.

00:02:29.000 --> 00:02:32.520
And maybe that is 1, 2, 3.

00:02:32.520 --> 00:02:34.200
And maybe this
function is defined

00:02:34.200 --> 00:02:36.670
for all non-negative values.

00:02:36.670 --> 00:02:38.690
So this is 0 of x.

00:02:38.690 --> 00:02:42.530
And so let me just draw--
so this right over here,

00:02:42.530 --> 00:02:45.010
at least for what I've drawn
so far, defines that function.

00:02:45.010 --> 00:02:46.676
I didn't even have
to use an equal sign.

00:02:46.676 --> 00:02:51.732
If x is 2, at least the way
I drew it, y is equal to 3.

00:02:51.732 --> 00:02:52.690
You give me that input.

00:02:52.690 --> 00:02:55.877
I gave you the value
of only one output.

00:02:55.877 --> 00:02:57.960
So that would be a legitimate
function definition.

00:02:57.960 --> 00:02:59.335
Another function
definition would

00:02:59.335 --> 00:03:02.050
be very similar to what you
do in a computer program,

00:03:02.050 --> 00:03:12.380
something like, let's say, that
you input the day of the week.

00:03:12.380 --> 00:03:21.660
And if day is equal to Monday,
maybe you output cereal.

00:03:21.660 --> 00:03:24.480
So that's what we're
going to eat that day.

00:03:24.480 --> 00:03:30.890
And otherwise, you
output meatloaf.

00:03:30.890 --> 00:03:34.590
So this would also
be a function.

00:03:34.590 --> 00:03:35.590
We only have one output.

00:03:35.590 --> 00:03:38.420
For any one day of
the week, we can only

00:03:38.420 --> 00:03:40.060
tell you cereal or meatloaf.

00:03:40.060 --> 00:03:42.510
There's no days where you
are eating both cereal

00:03:42.510 --> 00:03:46.590
and meatloaf, which
sounds repulsive.

00:03:46.590 --> 00:03:49.114
And then if I were to
think about something

00:03:49.114 --> 00:03:50.780
that could be an
equation or a function,

00:03:50.780 --> 00:03:53.300
I guess the way I think about
it is an equation is something

00:03:53.300 --> 00:03:55.210
that could be used
to define a function.

00:03:55.210 --> 00:04:03.200
So for example, we could say
that y is equal to 4x minus 10.

00:04:03.200 --> 00:04:06.840
This is a potential
definition for defining y

00:04:06.840 --> 00:04:08.000
as a function of x.

00:04:08.000 --> 00:04:10.400
You give me any value of x.

00:04:10.400 --> 00:04:13.180
Then I can find the
corresponding value of y.

00:04:13.180 --> 00:04:16.589
So this is at least how
I would think about it.


Finding average speed or rate

WEBVTT
Kind: captions
Language: en

00:00:00.259 --> 00:00:01.800
SALMAN KHAN: I have
some footage here

00:00:01.800 --> 00:00:04.584
of one of the most exciting
moments in sports history.

00:00:04.584 --> 00:00:06.000
And to make it
even more exciting,

00:00:06.000 --> 00:00:08.265
the commentator is
speaking in German.

00:00:08.265 --> 00:00:10.870
And I'm assuming that
this is OK under fair use,

00:00:10.870 --> 00:00:14.062
because I'm really using
it for a math problem.

00:00:14.062 --> 00:00:15.520
But I want you to
watch this video,

00:00:15.520 --> 00:00:20.100
and then I'll ask you
a question about it.

00:00:20.100 --> 00:00:22.060
[CHEERING]

00:00:22.060 --> 00:00:40.700
COMMENTATOR: [SPEAKING GERMAN]

00:00:40.700 --> 00:00:42.950
SALMAN KHAN: So you see,
it's exciting in any language

00:00:42.950 --> 00:00:44.220
that you might watch it.

00:00:44.220 --> 00:00:49.930
But my question to you is,
how fast was Usain Bolt going?

00:00:49.930 --> 00:00:52.440
What was his average
speed when he

00:00:52.440 --> 00:00:54.550
ran that 100 meters right there?

00:00:54.550 --> 00:00:57.630
And I encourage you to watch
the video as many times

00:00:57.630 --> 00:00:58.980
as you need to do it.

00:00:58.980 --> 00:01:02.150
And now I'll give you a little
bit of time to think about it,

00:01:02.150 --> 00:01:03.930
and then we will solve it.

00:01:07.720 --> 00:01:11.560
So we needed to figure out
how fast was Usain Bolt going

00:01:11.560 --> 00:01:12.659
over the 100 meters.

00:01:12.659 --> 00:01:14.450
So we're really thinking
about, in the case

00:01:14.450 --> 00:01:17.520
of this problem, average
speed or average rate.

00:01:17.520 --> 00:01:20.050
And you might already be
familiar with the notion

00:01:20.050 --> 00:01:26.460
that distance is equal
to rate or speed--

00:01:26.460 --> 00:01:28.750
I'll just write
rate-- times time.

00:01:28.750 --> 00:01:30.780
And I could write
times like that,

00:01:30.780 --> 00:01:33.840
but once we start doing algebra,
the traditional multiplication

00:01:33.840 --> 00:01:36.260
symbol can seem very confusing
because it looks just

00:01:36.260 --> 00:01:37.770
like the variable x.

00:01:37.770 --> 00:01:40.350
So instead, I will
write times like this.

00:01:40.350 --> 00:01:45.710
So distance is equal
to rate times time.

00:01:45.710 --> 00:01:48.190
And hopefully, this makes
some intuitive sense for you.

00:01:48.190 --> 00:01:51.320
If your rate or your
speed were 10 meters

00:01:51.320 --> 00:01:52.850
per second-- just as an example.

00:01:52.850 --> 00:01:55.020
That's not necessarily
how fast he went.

00:01:55.020 --> 00:01:56.740
But if you went 10
meters per second,

00:01:56.740 --> 00:02:04.050
and if you were to do
that for two seconds,

00:02:04.050 --> 00:02:06.040
then it should hopefully
make intuitive sense

00:02:06.040 --> 00:02:07.140
that you went 20 meters.

00:02:07.140 --> 00:02:10.710
You went 10 meters per
second for two seconds.

00:02:10.710 --> 00:02:12.490
And it also works
out mathematically.

00:02:12.490 --> 00:02:14.980
10 times 2 is equal to 20.

00:02:14.980 --> 00:02:17.500
And then you have seconds in
the denominator and seconds

00:02:17.500 --> 00:02:18.670
up here in the numerator.

00:02:18.670 --> 00:02:20.300
I just wrote seconds
here with an s.

00:02:20.300 --> 00:02:21.390
I wrote it out there.

00:02:21.390 --> 00:02:23.250
But they also cancel
out, and you're just

00:02:23.250 --> 00:02:25.660
left with the units of meters.

00:02:25.660 --> 00:02:28.620
So you're just left
with 20 meters.

00:02:28.620 --> 00:02:30.940
So hopefully this
makes intuitive sense.

00:02:30.940 --> 00:02:32.760
With that out of the
way, let's actually

00:02:32.760 --> 00:02:37.260
think about the problem at hand.

00:02:37.260 --> 00:02:39.990
What information do
we actually have?

00:02:39.990 --> 00:02:42.780
So do we have the distance?

00:02:42.780 --> 00:02:48.240
So what is the distance
in the video we just did?

00:02:48.240 --> 00:02:53.610
And I'll give you a second
or two to think about it.

00:02:53.610 --> 00:02:56.070
Well, this race
was the 100 meters.

00:02:56.070 --> 00:03:01.020
So the distance was 100 meters.

00:03:01.020 --> 00:03:02.180
Now, what else do we know?

00:03:02.180 --> 00:03:04.280
Do we know-- well, we're
trying to figure out the rate.

00:03:04.280 --> 00:03:05.863
That's what we're
going to figure out.

00:03:05.863 --> 00:03:09.780
What else do we know out of
this equation right over here?

00:03:09.780 --> 00:03:11.840
Well, do we know the time?

00:03:11.840 --> 00:03:13.180
Do we know the time?

00:03:13.180 --> 00:03:18.240
What was the time that it
took Usain Bolt to run the 100

00:03:18.240 --> 00:03:18.840
meters?

00:03:18.840 --> 00:03:23.150
And I'll give you another few
seconds to think about that.

00:03:23.150 --> 00:03:25.900
Well luckily, they were
timing the whole thing.

00:03:25.900 --> 00:03:28.846
And they also showed
that it's a world record.

00:03:28.846 --> 00:03:30.470
But this right over
here is in seconds.

00:03:30.470 --> 00:03:33.370
It's how long it took Usain
Bolt to run the 100 meters.

00:03:33.370 --> 00:03:38.640
It was 9.58 seconds.

00:03:38.640 --> 00:03:40.670
And I'll just write
s for seconds.

00:03:40.670 --> 00:03:44.260
So given this information here,
what you need to attempt to do

00:03:44.260 --> 00:03:48.139
is now give us our rate in
terms of meters per second.

00:03:48.139 --> 00:03:49.930
I want you to think if
you could figure out

00:03:49.930 --> 00:03:51.830
the rate in terms of
meters per second.

00:03:51.830 --> 00:03:54.275
We know the distance,
and we know the time.

00:03:57.060 --> 00:03:59.300
Well, let's substitute
these values

00:03:59.300 --> 00:04:00.890
into this equation
right over here.

00:04:00.890 --> 00:04:02.490
We know the distance
is 100 meters.

00:04:06.850 --> 00:04:08.880
And it's equal to-- we
don't know the rate,

00:04:08.880 --> 00:04:11.210
so I'll just write
rate right over here.

00:04:11.210 --> 00:04:13.580
And let me write it
in that same color.

00:04:13.580 --> 00:04:18.250
It's equal to rate times--
and what's the time?

00:04:18.250 --> 00:04:19.269
We do know the time.

00:04:19.269 --> 00:04:25.780
It's 9.58 seconds.

00:04:25.780 --> 00:04:27.750
And we care about rate.

00:04:27.750 --> 00:04:30.050
We care about solving for rate.

00:04:30.050 --> 00:04:31.900
So how can we do that?

00:04:31.900 --> 00:04:34.590
Well, if you look at this right
hand side of the equation,

00:04:34.590 --> 00:04:37.520
I have 9.58 seconds times rate.

00:04:37.520 --> 00:04:39.290
If I were able to
divide this right hand

00:04:39.290 --> 00:04:41.810
side by 9.58 seconds,
I'll just have

00:04:41.810 --> 00:04:43.250
rate on the right hand side.

00:04:43.250 --> 00:04:45.140
And that's what I
want to solve for.

00:04:45.140 --> 00:04:48.740
So you say, well, why don't I
just divide the right hand side

00:04:48.740 --> 00:04:53.330
by 9.58 seconds?

00:04:53.330 --> 00:04:56.360
Because if I did that,
the units cancel out,

00:04:56.360 --> 00:04:57.920
if we're doing
dimensional analysis.

00:04:57.920 --> 00:05:00.420
Don't worry too much if that
word doesn't make sense to you.

00:05:00.420 --> 00:05:04.340
But the units cancel out,
and the 9.58 cancels out.

00:05:04.340 --> 00:05:08.110
But I can't just divide one
side of an equation by a number.

00:05:08.110 --> 00:05:11.670
When we started off, this
is equal to this up here.

00:05:11.670 --> 00:05:14.000
If I divide the
right side by 9.58,

00:05:14.000 --> 00:05:16.820
in order for the equality
to still be true,

00:05:16.820 --> 00:05:19.590
I need to divide the left
side by the same thing.

00:05:19.590 --> 00:05:21.445
So I can't just
divide the right side.

00:05:21.445 --> 00:05:23.070
I have to divide the
left side in order

00:05:23.070 --> 00:05:25.410
for the equality
to still be true.

00:05:25.410 --> 00:05:27.500
If I said one thing is
equal to another thing,

00:05:27.500 --> 00:05:30.000
and I divide the other thing
by something, in order for them

00:05:30.000 --> 00:05:32.124
to still be equal, I have
to divide the first thing

00:05:32.124 --> 00:05:33.140
by that same amount.

00:05:33.140 --> 00:05:35.810
So I divide by 9.58 seconds.

00:05:35.810 --> 00:05:37.310
So on our right
hand side-- and this

00:05:37.310 --> 00:05:41.030
was the whole point--
these two cancel out.

00:05:41.030 --> 00:05:42.580
And then on the
left hand side, I'm

00:05:42.580 --> 00:05:44.970
left with 100 divided by 9.58.

00:05:44.970 --> 00:05:47.460
And my units are
meters per second,

00:05:47.460 --> 00:05:51.480
which are the exact units that
I want for rate, or for speed.

00:05:51.480 --> 00:05:57.450
And so let's get the calculator
out to divide 100 by 9.58.

00:05:57.450 --> 00:06:04.920
So I've got 100 meters
divided by 9.58 seconds

00:06:04.920 --> 00:06:07.585
gives me 10 point--
this says we've

00:06:07.585 --> 00:06:09.440
got about three
significant digits here--

00:06:09.440 --> 00:06:11.850
so let's say 10.4.

00:06:11.850 --> 00:06:14.140
So this gives us 10.4.

00:06:14.140 --> 00:06:17.170
And I'll write it
in the rate color.

00:06:17.170 --> 00:06:23.290
10.4-- and the units are
meters per second-- meters

00:06:23.290 --> 00:06:29.912
per second is equal to my rate.

00:06:29.912 --> 00:06:30.870
Now, the next question.

00:06:30.870 --> 00:06:32.370
So we got this in
meters per second.

00:06:32.370 --> 00:06:34.440
But unfortunately,
meters per second,

00:06:34.440 --> 00:06:36.390
they're not the--
when we drive a car,

00:06:36.390 --> 00:06:39.130
we don't see the speedometer
in meters per second.

00:06:39.130 --> 00:06:42.700
We see either kilometers
per hour or miles per hour.

00:06:42.700 --> 00:06:47.080
So the next task I have for
you is to express this speed,

00:06:47.080 --> 00:06:49.140
or this rate-- and this
is his average speed,

00:06:49.140 --> 00:06:51.040
or his average rate,
over the 100 meters.

00:06:51.040 --> 00:06:56.600
But to think about this in
terms of kilometers per hour.

00:06:56.600 --> 00:06:59.200
So try to figure out
if you can rewrite

00:06:59.200 --> 00:07:01.215
this in kilometers per hour.

00:07:04.350 --> 00:07:06.310
Well, let's just take
this step by step.

00:07:06.310 --> 00:07:07.970
So I'm going to
write-- so let me just

00:07:07.970 --> 00:07:09.770
go down here, start over.

00:07:09.770 --> 00:07:12.800
So I started off with 10.4.

00:07:12.800 --> 00:07:20.260
And I'll write meters in
blue, and seconds in magenta.

00:07:20.260 --> 00:07:22.934
Now, we want to get to
kilometers per hour.

00:07:22.934 --> 00:07:24.350
Right now we're
meters per second.

00:07:24.350 --> 00:07:25.560
So let's take baby steps.

00:07:25.560 --> 00:07:30.562
Let's first think about it in
terms of kilometers per second.

00:07:30.562 --> 00:07:32.020
And I'll give you
a second to think

00:07:32.020 --> 00:07:33.686
about what we would
do this to turn this

00:07:33.686 --> 00:07:37.660
into kilometers per second.

00:07:37.660 --> 00:07:41.080
Well, the intuition here, if I'm
going 10.4 meters per second,

00:07:41.080 --> 00:07:45.190
how many kilometers
is 10.4 meters?

00:07:45.190 --> 00:07:48.100
Well, kilometers is a much
larger unit of measurement.

00:07:48.100 --> 00:07:49.590
It's 1,000 times larger.

00:07:49.590 --> 00:07:54.210
So 10.4 meters will be a much
smaller number of kilometers.

00:07:54.210 --> 00:07:56.919
And in particular, I'm
going to divide by 1,000.

00:07:56.919 --> 00:07:58.460
Another way to think
about it, if you

00:07:58.460 --> 00:08:00.290
want to focus on
the units, we want

00:08:00.290 --> 00:08:03.150
to get rid of this in meters,
and we want a kilometers.

00:08:03.150 --> 00:08:05.440
So we want a
kilometers, and we want

00:08:05.440 --> 00:08:07.540
to get rid of these meters.

00:08:07.540 --> 00:08:09.240
So if we had meters
in the numerator,

00:08:09.240 --> 00:08:10.790
we could divide by meters here.

00:08:10.790 --> 00:08:12.100
They would cancel out.

00:08:12.100 --> 00:08:14.100
But the intuitive way to
think about it is we're

00:08:14.100 --> 00:08:16.030
going from a smaller
unit, meters,

00:08:16.030 --> 00:08:18.000
to a larger unit, kilometers.

00:08:18.000 --> 00:08:20.580
So 10.4 meters are going
to be a much smaller number

00:08:20.580 --> 00:08:21.580
of kilometers.

00:08:21.580 --> 00:08:24.690
But if we look at it
this way, how many meters

00:08:24.690 --> 00:08:25.550
are in 1 kilometer?

00:08:25.550 --> 00:08:29.360
1 kilometer is equal
to 1,000 meters.

00:08:29.360 --> 00:08:31.700
This right over here, 1
kilometer over 1,000 meters,

00:08:31.700 --> 00:08:33.750
this is 1 over 1.

00:08:33.750 --> 00:08:35.500
We're not changing
the fundamental value.

00:08:35.500 --> 00:08:37.929
We're essentially just
multiplying it by one.

00:08:37.929 --> 00:08:40.860
But when we do this,
what do we get?

00:08:40.860 --> 00:08:42.500
Well, the meters cancel out.

00:08:42.500 --> 00:08:44.510
We're left with
kilometers and seconds.

00:08:44.510 --> 00:08:48.666
And the numbers, you get
10.4 divided by 1,000.

00:08:48.666 --> 00:08:52.880
10.4 divided by 1,000
is going to give you--

00:08:52.880 --> 00:08:55.700
so if you divide by 10,
you're going to get 1.04.

00:08:55.700 --> 00:08:58.760
You divide by 100,
you get 0.104.

00:08:58.760 --> 00:09:05.090
You divide by 1,000,
you get 0.0104.

00:09:05.090 --> 00:09:07.590
So that's just 10.4
divided by 1,000.

00:09:07.590 --> 00:09:13.230
And then our units are
kilometers per second.

00:09:13.230 --> 00:09:15.040
So that's the
kilometers, and then I

00:09:15.040 --> 00:09:19.230
have my seconds right over here.

00:09:19.230 --> 00:09:20.990
So let me write the equal sign.

00:09:20.990 --> 00:09:24.687
Now, let's try to convert
this to kilometers per hour.

00:09:24.687 --> 00:09:26.270
And I'll give you a
little bit of time

00:09:26.270 --> 00:09:29.630
to think about that one.

00:09:29.630 --> 00:09:33.580
Well, hours, there's
3,600 seconds in an hour.

00:09:33.580 --> 00:09:35.890
So however many kilometers
I do in a second,

00:09:35.890 --> 00:09:39.360
I'm going to do 3,600
times that in an hour.

00:09:39.360 --> 00:09:43.890
And the units will
also work out.

00:09:43.890 --> 00:09:46.090
If I do this many
in a second, so it's

00:09:46.090 --> 00:09:53.815
going to be times 3,600, there
are 3,600 seconds in an hour.

00:09:53.815 --> 00:09:55.440
And another way to
think about it is we

00:09:55.440 --> 00:09:56.850
want hours in the denominator.

00:09:56.850 --> 00:09:58.130
We had seconds.

00:09:58.130 --> 00:10:00.250
So if we multiply
by seconds per hour,

00:10:00.250 --> 00:10:02.240
there are 3,600
seconds per hour,

00:10:02.240 --> 00:10:03.740
the seconds are
going to cancel out,

00:10:03.740 --> 00:10:06.310
and we're going to be left
with hours in the denominator.

00:10:06.310 --> 00:10:09.770
So seconds cancel out, and we're
left with kilometers per hour.

00:10:09.770 --> 00:10:12.650
But now we have to multiply
this number times 3,600.

00:10:12.650 --> 00:10:14.350
I'll get the calculator
out for that.

00:10:17.100 --> 00:10:31.800
So we have 0.0104 times 3,600
gives us, I'll just say 37.4.

00:10:31.800 --> 00:10:44.245
So this is equal to 37.4
kilometers per hour.

00:10:48.340 --> 00:10:51.950
So that's his average speed
in kilometers per hour.

00:10:51.950 --> 00:10:55.290
And now the last thing I want to
do, for those of us in America,

00:10:55.290 --> 00:10:57.160
we'll convert into
imperial units,

00:10:57.160 --> 00:10:59.780
or sometimes called English
units, which are ironically

00:10:59.780 --> 00:11:01.150
not necessarily used in the UK.

00:11:01.150 --> 00:11:03.320
They tend to be used in America.

00:11:03.320 --> 00:11:05.552
So let's convert this
into miles per hour.

00:11:05.552 --> 00:11:07.510
And the one thing I will
tell you, just in case

00:11:07.510 --> 00:11:15.249
you don't know, is that 1.61
kilometers is equal to 1 mile.

00:11:15.249 --> 00:11:16.790
So I'll give you a
little bit of time

00:11:16.790 --> 00:11:20.980
to convert this
into miles per hour.

00:11:20.980 --> 00:11:23.040
Well, as you see
from this, a mile

00:11:23.040 --> 00:11:25.320
is a slightly larger
or reasonably larger

00:11:25.320 --> 00:11:27.260
unit than a kilometer.

00:11:27.260 --> 00:11:30.156
So if you're going 37.4
kilometers in a certain amount

00:11:30.156 --> 00:11:32.280
of time, you're going to
go slightly smaller amount

00:11:32.280 --> 00:11:33.821
of miles in a certain
amount of time.

00:11:33.821 --> 00:11:36.950
Or in particular, you're
going to divide by 1.61.

00:11:36.950 --> 00:11:38.400
So let me rewrite it.

00:11:38.400 --> 00:11:46.550
If I have 37.4
kilometers per hour,

00:11:46.550 --> 00:11:47.810
we're going to a larger unit.

00:11:47.810 --> 00:11:49.090
We're going to miles.

00:11:49.090 --> 00:11:51.690
So we're going to divide by
something larger than one.

00:11:51.690 --> 00:11:56.090
So we have one-- let
me write it in blue-- 1

00:11:56.090 --> 00:12:04.010
mile is equal to
1.61 kilometers.

00:12:04.010 --> 00:12:07.960
Or you could say there's one
1.61th mile per kilometer.

00:12:07.960 --> 00:12:09.920
It also, once again,
works out with units.

00:12:09.920 --> 00:12:12.536
We want to get rid of the
kilometers in the numerator.

00:12:12.536 --> 00:12:14.160
So we would want it
in the denominator.

00:12:14.160 --> 00:12:15.493
We want a mile in the numerator.

00:12:15.493 --> 00:12:17.650
So that's why we have a
mile in the numerator here.

00:12:17.650 --> 00:12:20.060
So let's once again
multiply, or I

00:12:20.060 --> 00:12:22.360
guess in this case
we're dividing by 1.61.

00:12:22.360 --> 00:12:29.450
And we get-- let's just divide
our previous value by 1.61.

00:12:29.450 --> 00:12:34.340
And we get 23 point--
I'll just round up-- 23.3.

00:12:34.340 --> 00:12:41.750
This is equal to 23.3.

00:12:41.750 --> 00:12:48.275
And then we have miles per hour.

00:12:51.760 --> 00:12:53.570
Which is obviously very fast.

00:12:53.570 --> 00:12:55.300
He's the fastest human.

00:12:55.300 --> 00:12:57.690
But it's not maybe as fast
as you might have imagined.

00:12:57.690 --> 00:13:01.060
In a car, 23.3 miles per
hour doesn't seem so fast.

00:13:01.060 --> 00:13:03.300
And especially relative
to the animal world,

00:13:03.300 --> 00:13:05.590
it's not particularly
noteworthy.

00:13:05.590 --> 00:13:08.890
This is actually slightly
slower than a charging elephant.

00:13:08.890 --> 00:13:13.930
Charging elephants have been
clocked at 25 miles per hour.


Combining like terms, but more complicated

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:04.750
Now we have a very, very,
very hairy expression.

00:00:04.750 --> 00:00:09.375
And once again, I'm going to
see if you can simplify this.

00:00:09.375 --> 00:00:11.000
And I'll give you
little time to do it.

00:00:13.680 --> 00:00:17.480
So this one is even crazier than
the last few we've looked at.

00:00:17.480 --> 00:00:20.960
We've got y's and
xy's, and x squared

00:00:20.960 --> 00:00:25.730
and x's, well more just xy's and
y squared and on and on and on.

00:00:25.730 --> 00:00:28.370
And there will be a temptation,
because you see a y here

00:00:28.370 --> 00:00:29.940
and a y here to
say, oh, maybe I can

00:00:29.940 --> 00:00:32.549
add this negative
3y plus this 4xy

00:00:32.549 --> 00:00:34.830
somehow since I see a y and a y.

00:00:34.830 --> 00:00:37.360
But the important
thing to realize here

00:00:37.360 --> 00:00:40.380
is that a y is
different than an xy.

00:00:40.380 --> 00:00:41.810
Think about it
they were numbers.

00:00:41.810 --> 00:00:46.672
If y was 3 and an x was a
2, then a y would be a 3

00:00:46.672 --> 00:00:49.270
while an xy would have been a 6.

00:00:49.270 --> 00:00:52.010
And a y is very different
than a y squared.

00:00:52.010 --> 00:00:54.710
Once again, if the why it
took on the value 3, then

00:00:54.710 --> 00:00:57.270
the y squared would
be the value 9.

00:00:57.270 --> 00:00:59.580
So even though you see
the same letter here,

00:00:59.580 --> 00:01:03.640
they aren't the same-- I
guess you cannot add these two

00:01:03.640 --> 00:01:05.480
or subtract these two terms.

00:01:05.480 --> 00:01:07.610
A y is different
than a y squared,

00:01:07.610 --> 00:01:09.860
is different than an xy.

00:01:09.860 --> 00:01:12.640
Now with that said, let's
see if there is anything

00:01:12.640 --> 00:01:14.480
that we can simplify.

00:01:14.480 --> 00:01:16.700
So first, let's think
about the y terms.

00:01:16.700 --> 00:01:18.060
So you have a negative 3y there.

00:01:18.060 --> 00:01:20.140
Do we have any more y term?

00:01:20.140 --> 00:01:20.710
Yes, we do.

00:01:20.710 --> 00:01:22.889
We have this 2y
right over there.

00:01:22.889 --> 00:01:24.930
So I'll just write it
out-- I'll just reorder it.

00:01:24.930 --> 00:01:28.860
So we have negative 3y plus 2y.

00:01:31.740 --> 00:01:33.770
Now, let's think
about-- and I'm just

00:01:33.770 --> 00:01:36.380
going in an arbitrary order,
but since our next term is an xy

00:01:36.380 --> 00:01:38.970
term-- let's think about
all of the xy terms.

00:01:38.970 --> 00:01:41.945
So we have plus 4xy
right over here.

00:01:41.945 --> 00:01:43.570
So let me just write
it down-- I'm just

00:01:43.570 --> 00:01:47.200
reordering the whole
expression-- plus 4xy.

00:01:47.200 --> 00:01:50.990
And then I have minus
4xy right over here.

00:01:54.710 --> 00:01:56.550
Then let's go to
the x squared terms.

00:01:56.550 --> 00:01:59.810
I have negative 2 times x
squared, or minus 2x squared.

00:01:59.810 --> 00:02:01.830
So let's look at this.

00:02:01.830 --> 00:02:04.590
So I have minus 2x squared.

00:02:04.590 --> 00:02:06.360
Do I have any other x squared?

00:02:06.360 --> 00:02:07.080
Yes, I do.

00:02:07.080 --> 00:02:09.240
I have this 3x squared
right over there.

00:02:09.240 --> 00:02:12.720
So plus 3x squared.

00:02:12.720 --> 00:02:17.990
And then let's see, I have
an x term right over here,

00:02:17.990 --> 00:02:20.220
and that actually looks
like the only x term.

00:02:20.220 --> 00:02:22.390
So that's plus 2x.

00:02:22.390 --> 00:02:24.190
And then I only
have one y squared

00:02:24.190 --> 00:02:29.320
term-- I'll circle that in
orange-- so plus y squared.

00:02:29.320 --> 00:02:31.880
So all I have done is I've
reordered the statement

00:02:31.880 --> 00:02:34.820
and I've color coded it based
on the type of term we have.

00:02:34.820 --> 00:02:37.160
And now it should be
a little bit simpler.

00:02:37.160 --> 00:02:38.740
So let's try it out.

00:02:38.740 --> 00:02:41.580
If I have negative
3 of something

00:02:41.580 --> 00:02:43.524
plus 2 of that something,
what do I have?

00:02:43.524 --> 00:02:45.690
Or another way to say it,
if I have two of something

00:02:45.690 --> 00:02:48.730
and I subtract 3 of that,
what am I left with?

00:02:48.730 --> 00:02:50.940
Well, I'm left with negative
1 of that something.

00:02:50.940 --> 00:02:55.872
So I could write negative 1y, or
I could just write negative y.

00:02:55.872 --> 00:02:57.580
And another way you
could think about it,

00:02:57.580 --> 00:02:59.496
but I like to think about
it intuitively more,

00:02:59.496 --> 00:03:00.960
is what's the coefficient here?

00:03:00.960 --> 00:03:02.254
It is negative 3.

00:03:02.254 --> 00:03:03.420
What's the coefficient here?

00:03:03.420 --> 00:03:04.160
It's 2.

00:03:04.160 --> 00:03:05.576
Where obviously
both are dealing--

00:03:05.576 --> 00:03:07.440
they're both y
terms, not xy terms,

00:03:07.440 --> 00:03:09.880
not y squared terms, just y.

00:03:09.880 --> 00:03:12.560
And so negative 3
plus 2 is negative 1,

00:03:12.560 --> 00:03:16.000
or negative 1y is the
same thing as negative y.

00:03:16.000 --> 00:03:19.300
So those simplify to
this right over here.

00:03:19.300 --> 00:03:21.180
Now let's look at the xy terms.

00:03:21.180 --> 00:03:24.560
If I have 4 of
this, 4 xy's and I

00:03:24.560 --> 00:03:29.060
were to take away 4 xy's,
how many xy's am I left with?

00:03:29.060 --> 00:03:30.890
Well, I'm left with no xy's.

00:03:30.890 --> 00:03:33.170
Or you could say add
the coefficients,

00:03:33.170 --> 00:03:36.740
4 plus negative 4,
gives you 0 xy's.

00:03:36.740 --> 00:03:39.450
Either way, these
two cancel out.

00:03:39.450 --> 00:03:41.120
If I have 4 of something
and I take away

00:03:41.120 --> 00:03:44.350
those 4 of that something,
I'm left with none of them.

00:03:44.350 --> 00:03:46.940
And so I'm left with no xy's.

00:03:46.940 --> 00:03:48.830
And then I have
right over here--

00:03:48.830 --> 00:03:51.910
I could have written
0xy, but that

00:03:51.910 --> 00:03:54.420
seems unnecessary--
then right over here

00:03:54.420 --> 00:03:56.160
I have my x squared terms.

00:03:56.160 --> 00:03:58.700
Negative 2 plus 3 is 1.

00:03:58.700 --> 00:04:01.240
Or another way of saying
it, if I have 3x squared

00:04:01.240 --> 00:04:02.830
and I were to take
away 2 of those x

00:04:02.830 --> 00:04:05.540
squared, so I'm left
with the 1x squared.

00:04:05.540 --> 00:04:09.140
So this right over here
simplifies to 1x squared.

00:04:09.140 --> 00:04:10.950
Or I could literally
just write x squared.

00:04:10.950 --> 00:04:13.450
1x squared is the same
thing as x squared.

00:04:13.450 --> 00:04:16.779
So plus x squared,
and then these there's

00:04:16.779 --> 00:04:18.260
nothing really left to simplify.

00:04:18.260 --> 00:04:23.675
So plus 2x plus y squared.

00:04:27.365 --> 00:04:27.990
And we're done.

00:04:27.990 --> 00:04:29.000
And obviously you
might have gotten

00:04:29.000 --> 00:04:31.250
an answer in some other
order, but the order

00:04:31.250 --> 00:04:33.059
in which I write these
terms don't matter.

00:04:33.059 --> 00:04:35.100
It just matters that you
were able to simplify it

00:04:35.100 --> 00:04:37.535
to these four terms.


How to simplify a big expression by combining like terms

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:03.100
We have a hairy-looking
expression here.

00:00:03.100 --> 00:00:07.600
And your goal is to try to
simplify it as much as you can.

00:00:07.600 --> 00:00:09.600
And I'll give you a little
bit of time to do it.

00:00:12.420 --> 00:00:14.330
Let's just think about
it, step by step.

00:00:14.330 --> 00:00:16.610
And it might help if we
were to actually reorder

00:00:16.610 --> 00:00:18.670
the terms in this expression.

00:00:18.670 --> 00:00:20.810
So let me put all
the x terms first.

00:00:20.810 --> 00:00:24.240
So I have 5x-- that's
that term-- minus 2x.

00:00:26.750 --> 00:00:32.060
Then I have plus 7y plus 3y.

00:00:35.020 --> 00:00:43.447
Then i have plus 8z,
and then I have minus z.

00:00:43.447 --> 00:00:45.530
And then the last term
that I haven't included yet

00:00:45.530 --> 00:00:46.630
is that plus 5.

00:00:49.750 --> 00:00:51.440
Now we'll just think it through.

00:00:51.440 --> 00:00:55.885
If I have 5 x's and
I were to take away 2

00:00:55.885 --> 00:00:58.687
x's, is how many x's am
I going to be left with?

00:00:58.687 --> 00:01:00.270
Well, I'm going to
be left with 3 x's.

00:01:00.270 --> 00:01:01.269
That's true of anything.

00:01:01.269 --> 00:01:03.920
There's not some fancy
algebraic magic going on here.

00:01:03.920 --> 00:01:06.615
5 of anything minus
2 of that same thing,

00:01:06.615 --> 00:01:08.490
you're going to be left
with 3 of that thing.

00:01:08.490 --> 00:01:11.210
In this case, that
thing are x's.

00:01:11.210 --> 00:01:17.126
So this is going
to simplify to 3x.

00:01:17.126 --> 00:01:18.500
Now, in a lot of
algebra classes,

00:01:18.500 --> 00:01:20.291
you'll hear people say,
oh, well, you know,

00:01:20.291 --> 00:01:23.930
the coefficient on 5x is 5.

00:01:23.930 --> 00:01:27.060
And the coefficient on
this subtracting the 2x,

00:01:27.060 --> 00:01:29.640
the coefficient
here is negative 2,

00:01:29.640 --> 00:01:32.170
and we had to add
the coefficients.

00:01:32.170 --> 00:01:36.880
Let me write that word
down-- coefficient.

00:01:36.880 --> 00:01:38.840
These right over here
are the coefficients.

00:01:38.840 --> 00:01:42.810
They're the number that you're
multiplying the variable by.

00:01:42.810 --> 00:01:45.600
So you're the 5 or the
negative 2 in this case.

00:01:45.600 --> 00:01:47.100
And so you could
just say, oh, I had

00:01:47.100 --> 00:01:48.705
to just add the coefficients.

00:01:48.705 --> 00:01:50.830
And that's OK, and there's
nothing wrong with that.

00:01:50.830 --> 00:01:52.621
But I really want to
emphasize that there's

00:01:52.621 --> 00:01:54.690
a very common sense
intuition here.

00:01:54.690 --> 00:01:57.930
If you have 5 of something, you
take away 2 of that something,

00:01:57.930 --> 00:02:00.570
you are left with 3
of that something.

00:02:00.570 --> 00:02:01.937
And you have to be very careful.

00:02:01.937 --> 00:02:04.270
You have to make sure that
you're adding and subtracting

00:02:04.270 --> 00:02:05.090
the same things.

00:02:05.090 --> 00:02:07.440
Here, we're dealing with x's.

00:02:07.440 --> 00:02:10.430
So we can take 5 x's
and take away 2 x's.

00:02:10.430 --> 00:02:13.910
We can't think about merging
the x's and the y's, at least

00:02:13.910 --> 00:02:16.870
not in any simple way right
now, because that, frankly,

00:02:16.870 --> 00:02:19.070
wouldn't make any
intuitive sense.

00:02:19.070 --> 00:02:20.340
Now let's think about the y's.

00:02:20.340 --> 00:02:22.824
If I have 7 of something,
and I were to add 3 more

00:02:22.824 --> 00:02:24.240
of that something,
well, then, I'm

00:02:24.240 --> 00:02:25.780
going to have 10
of that something.

00:02:25.780 --> 00:02:31.420
So this part right over here
is going to simplify to 10y.

00:02:31.420 --> 00:02:35.030
Once again, you could say
the coefficient on 7y is 7.

00:02:35.030 --> 00:02:37.260
The coefficient on 3y is 3.

00:02:37.260 --> 00:02:40.900
We added the coefficients--
7 plus 3-- to get 10y.

00:02:40.900 --> 00:02:43.147
But I really want to
emphasize the intuition here.

00:02:43.147 --> 00:02:45.730
It's much more if you've got 7
of something, you add another 3

00:02:45.730 --> 00:02:47.910
to that something, you've
got 10 of that something.

00:02:47.910 --> 00:02:49.470
Now let's look at the z's.

00:02:49.470 --> 00:02:53.197
If I've got 8 of something
and I take away 1 of them,

00:02:53.197 --> 00:02:54.780
I'm going to have 7
of that something.

00:02:54.780 --> 00:02:56.222
So that is 7z.

00:02:56.222 --> 00:02:57.430
And you might say, hey, wait.

00:02:57.430 --> 00:02:59.721
What was the coefficient
right here on this negative z?

00:02:59.721 --> 00:03:02.040
I don't see any number
out front of the z.

00:03:02.040 --> 00:03:04.087
Well, implicitly, I
could have put a 1 here,

00:03:04.087 --> 00:03:05.420
and it's exactly the same thing.

00:03:05.420 --> 00:03:08.360
Subtracting a z is the exact
same thing as subtracting 1z.

00:03:08.360 --> 00:03:11.072
The word "onesie" strikes
a part of my brain

00:03:11.072 --> 00:03:12.530
because I have very
young children,

00:03:12.530 --> 00:03:14.556
but that's a different
type of onesie.

00:03:14.556 --> 00:03:15.930
And then you could
see, oh, yeah,

00:03:15.930 --> 00:03:19.090
you definitely did add the
two coefficients, the 8

00:03:19.090 --> 00:03:20.415
and the negative 1.

00:03:20.415 --> 00:03:22.320
But once again,
common sense tells you

00:03:22.320 --> 00:03:24.840
if you have 8 of something,
and you take away 1 of them,

00:03:24.840 --> 00:03:26.720
you have 7 of that something.

00:03:26.720 --> 00:03:31.860
And then finally,
you have a plus 5.

00:03:31.860 --> 00:03:32.740
So we're done.

00:03:32.740 --> 00:03:41.980
This simplified to 3x
plus 10y plus 7z plus 5.


How to combine like terms

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.690
Let's say that you
started off with 3 apples.

00:00:03.690 --> 00:00:09.680
And then I were to give you
another 7, another 7 apples.

00:00:09.680 --> 00:00:13.030
So my question to you-- and
this might be very obvious--

00:00:13.030 --> 00:00:17.930
is how many apples
do you now have?

00:00:17.930 --> 00:00:21.910
And I'll give you a second
to think about that.

00:00:21.910 --> 00:00:23.420
Well, this is fairly basic.

00:00:23.420 --> 00:00:24.344
You had 3 apples.

00:00:24.344 --> 00:00:25.760
Now, I'm going to
give you 7 more.

00:00:25.760 --> 00:00:27.370
You now have 3 plus 7.

00:00:27.370 --> 00:00:30.030
You now have 10 apples.

00:00:30.030 --> 00:00:32.770
But let's say I want to do
the same type of thinking,

00:00:32.770 --> 00:00:35.352
but I'm too lazy to
write the word "apples."

00:00:35.352 --> 00:00:37.310
Let's say instead of
writing the word "apples,"

00:00:37.310 --> 00:00:39.840
I just use the letter a.

00:00:39.840 --> 00:00:42.330
And let's say this is,
say, a different scenario.

00:00:42.330 --> 00:00:46.450
You start off with 4 apples.

00:00:46.450 --> 00:00:51.220
And to that, I add
another 2 apples.

00:00:51.220 --> 00:00:53.570
How many apples do you now have?

00:00:53.570 --> 00:00:56.650
Instead of writing apples, I'm
just going to write a's here.

00:00:56.650 --> 00:00:59.260
So how many of these
a's do you now have?

00:00:59.260 --> 00:01:01.320
And once again, I'll
give you a few seconds

00:01:01.320 --> 00:01:04.819
to think about that.

00:01:04.819 --> 00:01:07.200
This also might be a little
bit of common sense for you.

00:01:07.200 --> 00:01:10.960
If you had 4 of these apples or
whatever these a's represented,

00:01:10.960 --> 00:01:13.220
if you had 4 of them and
then you add 2 more of them,

00:01:13.220 --> 00:01:16.410
you're now going to
have 6 of these apples.

00:01:16.410 --> 00:01:18.660
But once again, we
started off assuming

00:01:18.660 --> 00:01:20.180
that a's represent apples.

00:01:20.180 --> 00:01:22.140
But they could have
represented anything.

00:01:22.140 --> 00:01:25.030
If you have 4 of whatever
a represents, and then you

00:01:25.030 --> 00:01:27.670
have another 2 of
whatever a represents,

00:01:27.670 --> 00:01:29.980
you'll now have 6 of
whatever a represents.

00:01:29.980 --> 00:01:32.896
Or if you just think of it if
I have 4 a's, and then I add

00:01:32.896 --> 00:01:34.520
another 2 a's, I'm
going to have 6 a's.

00:01:34.520 --> 00:01:40.490
You can literally think of 4
a's as a plus a plus a plus a.

00:01:40.490 --> 00:01:45.320
And if to that, I add another
2 a's-- so plus a plus

00:01:45.320 --> 00:01:47.310
a, that's 2 a's
right over there--

00:01:47.310 --> 00:01:49.030
how many a's do I now have?

00:01:49.030 --> 00:01:52.400
Well, that's 1, 2, 3, 4, 5, 6.

00:01:52.400 --> 00:01:55.717
I now have 6 a's.

00:01:55.717 --> 00:01:57.800
So thinking of it that
way, let's get a little bit

00:01:57.800 --> 00:02:00.210
more abstract.

00:02:00.210 --> 00:02:04.840
Let's say that I have 5
x's, whatever x represents.

00:02:04.840 --> 00:02:06.510
x could be some number.

00:02:06.510 --> 00:02:08.940
So I have 5 of whatever
that number is.

00:02:08.940 --> 00:02:14.590
And from that, I subtract 2
of whatever that number is.

00:02:14.590 --> 00:02:16.310
What would this evaluate to?

00:02:16.310 --> 00:02:22.110
How many of these
x's would I now have?

00:02:22.110 --> 00:02:27.760
So it's essentially 5x minus
2x is going to be what times x?

00:02:27.760 --> 00:02:30.280
Once again, I'll give you a
few seconds to think about it.

00:02:33.050 --> 00:02:36.350
Well, if I have 5 of something
and I subtract 2 of those away,

00:02:36.350 --> 00:02:39.050
I'm going to have 3 of
that something left.

00:02:39.050 --> 00:02:41.720
So this is going
to be equal to 3x.

00:02:41.720 --> 00:02:44.984
5x minus 2x is equal to 3x.

00:02:44.984 --> 00:02:46.900
And if you really think
about what that means,

00:02:46.900 --> 00:02:52.490
five x's are just x plus
x plus x plus x plus x.

00:02:52.490 --> 00:02:56.900
And then we're going to
take away two of those x's.

00:02:56.900 --> 00:03:00.490
So take away one x,
take away two x's.

00:03:00.490 --> 00:03:05.328
You are going to be
left with three x's.


Newton's first law of motion concepts

WEBVTT
Kind: captions
Language: en

00:00:00.369 --> 00:00:02.660
Now that we know a little
bit about Newton's First Law,

00:00:02.660 --> 00:00:04.260
let's give ourselves
a little quiz.

00:00:04.260 --> 00:00:06.470
And what I want you
to do is figure out

00:00:06.470 --> 00:00:10.370
which of these statements
are actually true.

00:00:10.370 --> 00:00:13.990
And our first statement is,
"If the net force on a body

00:00:13.990 --> 00:00:18.530
is zero, its velocity
will not change."

00:00:18.530 --> 00:00:20.030
Interesting.

00:00:20.030 --> 00:00:24.100
Statement number two, "An
unbalanced force on a body

00:00:24.100 --> 00:00:27.680
will always impact
the object's speed."

00:00:27.680 --> 00:00:29.650
Also an interesting statement.

00:00:29.650 --> 00:00:31.880
Statement number
three, "The reason

00:00:31.880 --> 00:00:34.240
why initially
moving objects tend

00:00:34.240 --> 00:00:36.870
to come to rest in
our everyday life

00:00:36.870 --> 00:00:41.560
is because they are being
acted on by unbalanced forces."

00:00:41.560 --> 00:00:45.760
And statement four, "An
unbalanced force on an object

00:00:45.760 --> 00:00:50.180
will always change the
object's direction."

00:00:50.180 --> 00:00:51.675
So I'll let you
think about that.

00:00:56.440 --> 00:00:59.426
So let's think about these
statement by statement.

00:00:59.426 --> 00:01:01.050
So our first statement
right over here,

00:01:01.050 --> 00:01:03.160
"If the net force
on a body is zero,

00:01:03.160 --> 00:01:05.750
its velocity will not change."

00:01:05.750 --> 00:01:07.600
This is absolutely true.

00:01:07.600 --> 00:01:09.080
This is actually
even another way

00:01:09.080 --> 00:01:13.200
of rephrasing
Newton's First Law.

00:01:13.200 --> 00:01:16.520
If I have some type
of object that's

00:01:16.520 --> 00:01:20.020
just traveling through
space with some velocity--

00:01:20.020 --> 00:01:22.640
so it has some speed
going in some direction,

00:01:22.640 --> 00:01:23.752
and maybe it's deep space.

00:01:23.752 --> 00:01:25.210
And we can just,
for purity, assume

00:01:25.210 --> 00:01:27.125
that there's no
gravitational interactions.

00:01:27.125 --> 00:01:29.070
There will always be
some minuscule ones,

00:01:29.070 --> 00:01:31.170
but we'll assume no
gravitational interactions.

00:01:31.170 --> 00:01:32.700
Absolutely no
particles that it's

00:01:32.700 --> 00:01:35.870
bumping into, absolute
vacuum of space.

00:01:35.870 --> 00:01:38.050
This thing will
travel on forever.

00:01:38.050 --> 00:01:39.760
Its velocity will not change.

00:01:39.760 --> 00:01:43.450
Neither its speed nor its
direction will change.

00:01:43.450 --> 00:01:46.040
So this one is absolutely true.

00:01:46.040 --> 00:01:49.640
Statement number two, "An
unbalanced force on a body

00:01:49.640 --> 00:01:53.970
will always impact
the object's speed."

00:01:53.970 --> 00:01:57.170
And the key word right
over here is "speed."

00:01:57.170 --> 00:02:00.090
If I had written "impact
the object's velocity," then

00:02:00.090 --> 00:02:01.430
this would be a true statement.

00:02:01.430 --> 00:02:04.760
An unbalanced force
on a body will always

00:02:04.760 --> 00:02:06.860
impact the object's velocity.

00:02:06.860 --> 00:02:07.770
That would be true.

00:02:07.770 --> 00:02:09.130
But we wrote "speed" here.

00:02:09.130 --> 00:02:10.910
Speed is the
magnitude of velocity.

00:02:10.910 --> 00:02:13.630
It does not take into
account the direction.

00:02:13.630 --> 00:02:16.160
And to see why this
second statement is false,

00:02:16.160 --> 00:02:18.070
you could think about
a couple of things.

00:02:18.070 --> 00:02:19.930
And we'll do more
videos on the intuition

00:02:19.930 --> 00:02:21.950
of centripetal acceleration
and centripetal

00:02:21.950 --> 00:02:24.190
forces, inward forces,
if this does not

00:02:24.190 --> 00:02:28.060
make complete intuitive sense
to you just at this moment.

00:02:28.060 --> 00:02:30.940
But imagine we're looking at
an ice skating rink from above.

00:02:30.940 --> 00:02:32.400
And you have an ice skater.

00:02:32.400 --> 00:02:34.300
This is the ice skater's head.

00:02:34.300 --> 00:02:37.010
And they are traveling
in that direction.

00:02:37.010 --> 00:02:39.530
Now imagine right
at that moment,

00:02:39.530 --> 00:02:42.181
they grab a rope that is
nailed to a stake in the ice

00:02:42.181 --> 00:02:43.430
skating rink right over there.

00:02:43.430 --> 00:02:46.013
We're viewing all of this from
above, and this right over here

00:02:46.013 --> 00:02:46.990
is the rope.

00:02:46.990 --> 00:02:49.050
Now what is going to happen?

00:02:49.050 --> 00:02:52.570
Well, the skater
is going to travel.

00:02:52.570 --> 00:02:55.782
Their direction is
actually going to change.

00:02:55.782 --> 00:02:57.240
And they could hold
on to the rope,

00:02:57.240 --> 00:02:58.610
and as long as they
hold on to the rope,

00:02:58.610 --> 00:02:59.860
they'll keep going in circles.

00:02:59.860 --> 00:03:01.340
And when they let
go of the rope,

00:03:01.340 --> 00:03:05.150
they'll start going
in whatever direction

00:03:05.150 --> 00:03:07.280
they were traveling
in when they let go.

00:03:07.280 --> 00:03:09.580
They'll keep going
on in that direction.

00:03:09.580 --> 00:03:12.800
And if we assume very,
very, very small frictions

00:03:12.800 --> 00:03:14.230
from the ice skating
rink, they'll

00:03:14.230 --> 00:03:16.580
actually have the same speed.

00:03:16.580 --> 00:03:19.660
So the force, the inward
force, the tension

00:03:19.660 --> 00:03:22.710
from the rope pulling on the
skater in this situation,

00:03:22.710 --> 00:03:25.720
would have only changed
the skater's direction.

00:03:25.720 --> 00:03:28.360
So and unbalanced force
doesn't necessarily

00:03:28.360 --> 00:03:29.910
have to impact the
object's speed.

00:03:29.910 --> 00:03:31.090
It often does.

00:03:31.090 --> 00:03:33.480
But in that situation, it
would have only impacted

00:03:33.480 --> 00:03:35.240
the skater's direction.

00:03:35.240 --> 00:03:37.540
Another situation like
this-- and once again,

00:03:37.540 --> 00:03:41.390
this involves centripetal
acceleration, inward forces,

00:03:41.390 --> 00:03:44.310
inward acceleration--
is a satellite in orbit,

00:03:44.310 --> 00:03:46.880
or any type of thing in orbit.

00:03:46.880 --> 00:03:49.890
So if that is some
type of planet,

00:03:49.890 --> 00:03:52.820
and this is one of the
planet's moons right over here,

00:03:52.820 --> 00:03:56.810
the reason why it stays in orbit
is because the pull of gravity

00:03:56.810 --> 00:04:00.630
keeps making the object
change its direction, but not

00:04:00.630 --> 00:04:01.340
its speed.

00:04:01.340 --> 00:04:04.236
Its speed is the
exact right speed.

00:04:04.236 --> 00:04:05.610
So this was its
speed right here.

00:04:05.610 --> 00:04:07.318
If the planet wasn't
there, it would just

00:04:07.318 --> 00:04:09.580
keep going on in that
direction forever and forever.

00:04:09.580 --> 00:04:11.460
But the planet right
over here, there's

00:04:11.460 --> 00:04:12.970
an inward force of gravity.

00:04:12.970 --> 00:04:16.320
And we'll talk more about the
force of gravity in the future.

00:04:16.320 --> 00:04:17.860
But this inward
force of gravity is

00:04:17.860 --> 00:04:21.810
going to accelerate this object
inwards while it travels.

00:04:21.810 --> 00:04:23.860
And so after some
period of time,

00:04:23.860 --> 00:04:25.960
this object's velocity
vector-- if you

00:04:25.960 --> 00:04:28.430
add the previous velocity
with how much it's

00:04:28.430 --> 00:04:31.640
changed its new velocity vector.

00:04:31.640 --> 00:04:34.430
Now this is after its traveled
a little bit-- its new velocity

00:04:34.430 --> 00:04:38.090
vector might look
something like this.

00:04:38.090 --> 00:04:40.650
And it's traveling at
the exact right speed

00:04:40.650 --> 00:04:45.910
so that the force
of gravity is always

00:04:45.910 --> 00:04:50.370
at a right angle to
its actual trajectory.

00:04:50.370 --> 00:04:54.372
It's the exact right speed so it
doesn't go off into deep space

00:04:54.372 --> 00:04:56.080
and so it doesn't
plummet into the earth.

00:04:56.080 --> 00:04:58.700
And we'll cover that
in much more detail.

00:04:58.700 --> 00:05:01.160
But the simple answer is,
unbalanced force on a body

00:05:01.160 --> 00:05:02.930
will always impact its velocity.

00:05:02.930 --> 00:05:06.260
It could be its speed,
its direction, or both,

00:05:06.260 --> 00:05:07.910
but it doesn't have to be both.

00:05:07.910 --> 00:05:11.170
It could be just the speed
or just the direction.

00:05:11.170 --> 00:05:14.510
So this is an
incorrect statement.

00:05:14.510 --> 00:05:16.220
Now the third
statement, "The reason

00:05:16.220 --> 00:05:17.860
why initially
moving objects tend

00:05:17.860 --> 00:05:19.880
to come to rest in
our everyday life

00:05:19.880 --> 00:05:23.670
is because they are being
acted on by unbalanced forces."

00:05:23.670 --> 00:05:24.870
This is absolutely true.

00:05:24.870 --> 00:05:27.230
And this is the example we gave.

00:05:27.230 --> 00:05:29.780
If I take an object,
if I take my book

00:05:29.780 --> 00:05:31.660
and I try to slide
it across the desk,

00:05:31.660 --> 00:05:33.600
the reason why it
eventually comes to stop

00:05:33.600 --> 00:05:37.160
is because we have the
unbalanced force of friction--

00:05:37.160 --> 00:05:39.160
the grinding of the
surface of the book

00:05:39.160 --> 00:05:41.720
with the grinding of the table.

00:05:41.720 --> 00:05:43.470
If I'm inside of a
pool or even if there's

00:05:43.470 --> 00:05:45.410
absolutely no
current in the pool,

00:05:45.410 --> 00:05:48.670
and if I were to try to
push some type of object

00:05:48.670 --> 00:05:50.350
inside the water,
it eventually comes

00:05:50.350 --> 00:05:55.800
to stop because of all of the
resistance of the water itself.

00:05:55.800 --> 00:05:58.560
It's providing an unbalanced
force in a direction

00:05:58.560 --> 00:05:59.970
opposite it's motion.

00:05:59.970 --> 00:06:01.740
That is what's slowing it down.

00:06:01.740 --> 00:06:03.210
So in our everyday
life, the reason

00:06:03.210 --> 00:06:04.970
why we don't see these
things go on and on

00:06:04.970 --> 00:06:07.250
forever is that we have
these frictions, these air

00:06:07.250 --> 00:06:12.690
resistants, or the friction
with actual surfaces.

00:06:12.690 --> 00:06:15.980
And then the last statement, "An
unbalanced force on an object

00:06:15.980 --> 00:06:19.870
will always change the
object's direction."

00:06:19.870 --> 00:06:22.620
Well, this one actually is
maybe the most intuitive.

00:06:22.620 --> 00:06:25.130
We always have this situation.

00:06:25.130 --> 00:06:27.460
Let's say I have a
block right over here,

00:06:27.460 --> 00:06:29.910
and it's traveling with some
velocity in that direction--

00:06:29.910 --> 00:06:32.480
five meters per second.

00:06:32.480 --> 00:06:37.550
If I apply an unbalanced
force in that same direction--

00:06:37.550 --> 00:06:39.050
so that's my force
right over there.

00:06:39.050 --> 00:06:40.940
If I apply it in
that same direction,

00:06:40.940 --> 00:06:43.360
I'm just going to accelerate
it in that same direction.

00:06:43.360 --> 00:06:45.030
So I won't
necessarily change it.

00:06:45.030 --> 00:06:47.510
Even if I were to act against
it, I might decelerate it,

00:06:47.510 --> 00:06:50.184
but I won't necessarily
change its direction.

00:06:50.184 --> 00:06:52.600
I could change its direction
by doing something like this,

00:06:52.600 --> 00:06:53.650
but I don't necessarily.

00:06:53.650 --> 00:06:55.790
I'm not always
necessarily changing

00:06:55.790 --> 00:06:57.330
the object's direction.

00:06:57.330 --> 00:07:00.530
So this is not true.

00:07:00.530 --> 00:07:02.990
An unbalanced force on
an object will not always

00:07:02.990 --> 00:07:04.310
change the object's direction.

00:07:04.310 --> 00:07:07.810
It can, like these
circumstances, but not always.

00:07:07.810 --> 00:07:13.290
So "always" is what makes
this very, very, very wrong.


Newton's first law of motion

WEBVTT
Kind: captions
Language: en

00:00:00.385 --> 00:00:02.909
Human beings have always observed that if you have

00:00:02.909 --> 00:00:05.862
an object that is moving,

00:00:05.862 --> 00:00:08.434
so this is a moving object,

00:00:08.434 --> 00:00:10.830
traveling to the right here,

00:00:10.830 --> 00:00:12.800
that it seems to stop on its own.

00:00:12.800 --> 00:00:15.211
That if you do nothing to this moving object,

00:00:15.211 --> 00:00:19.500
on its own, this object is going to come to a stop.

00:00:19.500 --> 00:00:22.784
It is going to come to rest.

00:00:23.215 --> 00:00:24.537
And on the other side of things,

00:00:24.537 --> 00:00:26.213
if you want to keep an object moving,

00:00:26.213 --> 00:00:28.477
you have to keep applying a force to it.

00:00:28.477 --> 00:00:30.334
We've never in our everyday experience

00:00:30.334 --> 00:00:31.628
seen an object that just keeps moving

00:00:31.643 --> 00:00:34.841
on and on forever without anyone acting on it.

00:00:34.841 --> 00:00:37.063
It seems like something will always stop.

00:00:37.063 --> 00:00:39.756
And this is why, for most of human history,

00:00:39.756 --> 00:00:42.245
probably pre-history, but we definitely know the ancient Greeks

00:00:42.245 --> 00:00:45.643
all the way until the early 1600s,

00:00:45.935 --> 00:00:49.641
so for at least 2000 years,

00:00:49.641 --> 00:00:54.553
the assumption was "objects have a natural tendency to stop."

00:00:54.553 --> 00:01:02.248
Objects ... have ... tendency ... to come to rest or to stop.

00:01:03.355 --> 00:01:07.003
And if you want to keep them moving,

00:01:13.322 --> 00:01:17.326
you have to apply some type of a net force to it.

00:01:19.542 --> 00:01:21.784
And once again, this is completly consistent with

00:01:21.784 --> 00:01:23.230
everyday human experience,

00:01:23.230 --> 00:01:25.925
this is what we've all experienced our entire lives.

00:01:26.294 --> 00:01:30.364
But then these gentlemen show up in the 1600s,

00:01:30.364 --> 00:01:34.188
and you might be surprised to see three gentlemen here,

00:01:34.188 --> 00:01:36.663
because this is about Newton's first law of motion.

00:01:36.663 --> 00:01:38.775
And, indeed, one of these gentlemen is

00:01:38.775 --> 00:01:40.434
Sir Isaac Newton.

00:01:40.434 --> 00:01:41.899
That's Newton right over there [middle].

00:01:41.899 --> 00:01:44.514
But these other two guys get at least as much credit for it

00:01:44.514 --> 00:01:46.746
because they actually described really what

00:01:46.746 --> 00:01:48.114
Newton's first Law describes,

00:01:48.114 --> 00:01:49.745
and they did it before Newton.

00:01:49.745 --> 00:01:51.436
This is Galileo.

00:01:53.085 --> 00:01:56.033
And this is Rene Descartes.

00:01:56.705 --> 00:01:58.649
And they describe it in different ways,

00:01:58.649 --> 00:02:00.548
and Newton frankly gets the credit for it

00:02:00.548 --> 00:02:02.766
because he really encapsulates into a broader framework

00:02:02.766 --> 00:02:05.016
with his other Laws, and the Laws of Gravitation,

00:02:05.016 --> 00:02:07.968
which was really the basics of classical mechanics,

00:02:07.968 --> 00:02:11.114
and seem to describe, at least until the 20th Century,

00:02:11.114 --> 00:02:14.413
most of how reality actually worked.

00:02:14.413 --> 00:02:17.381
And their big insight, and it was very unintuitive at the time,

00:02:17.381 --> 00:02:19.965
{so now we come to the 1600s}

00:02:19.965 --> 00:02:21.215
Is that these three gentlemen said,

00:02:21.215 --> 00:02:23.097
maybe it works the other way.

00:02:23.097 --> 00:02:36.828
Maybe objects have a tendency to maintain their velocity, their speed and their direction.

00:02:39.328 --> 00:02:41.706
And if their speed is zero, they'll maintain that restfulness.

00:02:41.706 --> 00:02:46.093
Unless they're acted upon by an unbalanced force.

00:02:58.433 --> 00:03:00.582
So the completly opposite way of thinking.

00:03:00.582 --> 00:03:03.831
For over 2000 years, objects tend to stop on their own,

00:03:03.831 --> 00:03:05.914
if you want to keep the movement, apply a force.

00:03:05.914 --> 00:03:07.332
These guys say,

00:03:07.332 --> 00:03:10.087
Objects have a tendency to maintain their motion forever

00:03:10.087 --> 00:03:13.093
and the only way that you're going to stop them is

00:03:13.093 --> 00:03:16.115
if you act on it, or accelerate them, or change their velocity,

00:03:16.115 --> 00:03:17.949
so either their speed or direction some way,

00:03:17.949 --> 00:03:21.617
is to act on them with an unbalanced force.

00:03:21.617 --> 00:03:22.560
But you might be saying,

00:03:22.560 --> 00:03:24.051
Hey, come on Sal, what's going on?

00:03:24.051 --> 00:03:25.206
You just went through this,

00:03:25.206 --> 00:03:26.532
you said for most of most of human history,

00:03:26.532 --> 00:03:28.164
including my own personal history,

00:03:28.164 --> 00:03:29.826
this is what I observed [top right].

00:03:29.826 --> 00:03:32.863
How can these guys say that this thing has a tendency to

00:03:32.863 --> 00:03:34.333
go on forever?

00:03:34.333 --> 00:03:36.635
This seems to break down.

00:03:36.635 --> 00:03:37.783
And their big insight was,

00:03:37.783 --> 00:03:40.498
well, maybe these things don't have, by themselves,

00:03:40.498 --> 00:03:42.968
a tendency to stop, but because of interactions

00:03:42.968 --> 00:03:45.516
with their environment, forces are being generated

00:03:45.516 --> 00:03:47.865
that are acting against their motion.

00:03:47.865 --> 00:03:50.600
So when you think you're leaving this thing alone,

00:03:50.600 --> 00:03:54.234
there is actualy a net force that is trying to stop it.

00:03:54.234 --> 00:03:57.750
And in this particular example over here,

00:03:57.750 --> 00:04:01.134
the net force is the force of friction.

00:04:01.134 --> 00:04:04.386
It's the interaction between the block and the ground.

00:04:04.386 --> 00:04:07.135
So, when you think you're leaving this thing alone,

00:04:07.135 --> 00:04:10.851
you actually have a net force going against its motion,

00:04:10.851 --> 00:04:13.100
which is the force of friction.

00:04:13.100 --> 00:04:14.359
And these guys realize that,

00:04:14.359 --> 00:04:15.251
because they said,

00:04:15.251 --> 00:04:17.388
look, if it was an innate property of the block,

00:04:17.388 --> 00:04:18.719
regardless of the environment,

00:04:18.719 --> 00:04:20.684
it should kind of always come to a stop

00:04:20.684 --> 00:04:22.437
in maybe a similar way.

00:04:22.437 --> 00:04:25.251
But they saw, if you made this surface a little bit smoother

00:04:26.559 --> 00:04:28.486
this thing would travel further and further.

00:04:28.486 --> 00:04:31.060
Maybe if you eliminated this friction,

00:04:31.060 --> 00:04:33.352
if you made this surface completely friction-less,

00:04:33.352 --> 00:04:36.312
completely smooth, this thing indeed would travel forever.

00:04:36.312 --> 00:04:38.902
And they didn't have the luxury of launching satellites,

00:04:38.902 --> 00:04:40.787
and doing things in deep space,

00:04:40.787 --> 00:04:44.769
so it was a very, very unintuitive thought experiment.

00:04:44.769 --> 00:04:46.584
And you might say, what about this other thing,

00:04:46.584 --> 00:04:48.011
what happens when I am applying the force?

00:04:48.011 --> 00:04:49.492
Becuase in my everyday life,

00:04:49.492 --> 00:04:52.008
If I want to drag my TV set across the room

00:04:52.008 --> 00:04:53.791
I apply a force to it.

00:04:53.791 --> 00:04:55.442
And what these guys would tell you

00:04:55.442 --> 00:04:56.298
is all you were doing,

00:04:56.298 --> 00:04:58.644
if you were keeping the velocity of that TV constant,

00:04:58.644 --> 00:05:03.425
all you were doing was counteracting this net negative force.

00:05:03.425 --> 00:05:06.410
So if this was a TV dragging across your carpet,

00:05:06.410 --> 00:05:09.575
this is the force of friction acting against the motion

00:05:09.575 --> 00:05:10.811
of the object,

00:05:10.811 --> 00:05:14.541
and so you are essentially just balancing it when you push it.

00:05:14.541 --> 00:05:15.791
If you balance it perfectly,

00:05:15.791 --> 00:05:17.791
you will be able to maintain it's velocity.

00:05:17.791 --> 00:05:19.175
If you want to accelerate it,

00:05:19.175 --> 00:05:24.599
you will have to apply even more force in the direction you are actually pushing it.

00:05:24.599 --> 00:05:25.199
Many thanks to Sal! :)