Order of operations, more examples

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.570
Let's do some order of
operations problems, and for

00:00:03.570 --> 00:00:05.435
the sake of time I'll do
every other problem.

00:00:05.435 --> 00:00:08.730
So let's start with 1b.

00:00:08.730 --> 00:00:10.900
1b right there.

00:00:10.900 --> 00:00:21.290
They have 2 plus 7 times 11
minus 12 divided by 3.

00:00:21.290 --> 00:00:24.890
So just remember, the top
priority is always going to be

00:00:24.890 --> 00:00:26.320
your parentheses.

00:00:26.320 --> 00:00:28.900
So you have your parentheses--
Let me write it this way.

00:00:28.900 --> 00:00:31.730
Your top priority's going to
be your parentheses, after

00:00:31.730 --> 00:00:36.410
that you're going to have your
exponents, after that you have

00:00:36.410 --> 00:00:40.990
multiplying and dividing, and
after that you have addition

00:00:40.990 --> 00:00:42.190
and subtraction.

00:00:42.190 --> 00:00:44.320
So let's remember that and
tackle these order of

00:00:44.320 --> 00:00:45.930
operations problems.

00:00:45.930 --> 00:00:48.940
So priority, there's no
parentheses here, there's no

00:00:48.940 --> 00:00:51.840
exponents, so the priority's
going to go to multiplication

00:00:51.840 --> 00:00:53.060
and division.

00:00:53.060 --> 00:00:56.990
So you could view this as being
equivalent to-- So we're

00:00:56.990 --> 00:00:59.640
going to do our multiplication
before we do any addition or

00:00:59.640 --> 00:01:02.590
subtraction, and we're going
to do our division before

00:01:02.590 --> 00:01:05.040
doing any addition
or subtraction.

00:01:05.040 --> 00:01:08.100
Problem 1b is exactly equivalent
to this, the

00:01:08.100 --> 00:01:10.680
parentheses are just-- I'm
reinforcing the notion that

00:01:10.680 --> 00:01:14.730
I'm going to do my
multiplication and division

00:01:14.730 --> 00:01:17.210
before I do the addition
and the subtraction.

00:01:17.210 --> 00:01:27.230
So 7 times 11 is 77, and then
12 divided by 3 is 4.

00:01:27.230 --> 00:01:31.380
And the rest of the problem was
2 plus this thing, which

00:01:31.380 --> 00:01:34.420
is 77, minus this thing.

00:01:34.420 --> 00:01:36.770
And here, since everything is
in addition or subtraction,

00:01:36.770 --> 00:01:40.560
let's just go left to right.

00:01:40.560 --> 00:01:49.210
2 plus 77 is 79 minus 4,
which is equal to 75.

00:01:49.210 --> 00:01:52.380
So 1b is equal to 75.

00:01:52.380 --> 00:01:54.160
Let's do 1d.

00:01:54.160 --> 00:01:57.140
This is a nice hairy problem
right there.

00:01:57.140 --> 00:01:59.400
So 1d.

00:01:59.400 --> 00:02:05.900
It says 2 times 3
plus 2 minus 1.

00:02:05.900 --> 00:02:15.110
Closing two parentheses, all of
that over 4 minus 6 plus 2

00:02:15.110 --> 00:02:18.890
minus 3 minus 5.

00:02:18.890 --> 00:02:21.540
Let's see if we can simplify
this a little bit.

00:02:21.540 --> 00:02:24.060
As we said, parentheses
take our priority.

00:02:24.060 --> 00:02:28.050
So let's do the parentheses
first. 2 minus 1.

00:02:28.050 --> 00:02:30.120
2 minus 1 is just 1.

00:02:30.120 --> 00:02:32.000
3 minus 5.

00:02:32.000 --> 00:02:36.560
That is minus 2, or negative
2, I should say.

00:02:36.560 --> 00:02:38.895
6 plus 2 is 8.

00:02:41.500 --> 00:02:43.860
Now let's keep looking at the
parentheses to see where we

00:02:43.860 --> 00:02:44.730
can simplify things.

00:02:44.730 --> 00:02:46.190
We have this parentheses
right here.

00:02:46.190 --> 00:02:50.420
So 3 plus this 1 is now going
to be equal to 4.

00:02:50.420 --> 00:02:51.328
Actually, let me rewrite it.

00:02:51.328 --> 00:02:55.250
So we're going to have 2 times
this whole expression, 3 plus

00:02:55.250 --> 00:02:58.000
1, so it's 2 times 4.

00:02:58.000 --> 00:03:00.400
That right there is 4.

00:03:00.400 --> 00:03:05.820
All of that over 4 minus
8, that's negative 4.

00:03:05.820 --> 00:03:08.080
This right here is negative 4.

00:03:08.080 --> 00:03:11.150
And then minus this
negative 2.

00:03:11.150 --> 00:03:13.860
So minus negative 2.

00:03:13.860 --> 00:03:19.770
2 times 4 is 8, so this whole
thing simplifies to-- A minus

00:03:19.770 --> 00:03:21.980
of a negative, that's just
the plus of the plus, the

00:03:21.980 --> 00:03:23.470
negatives cancel out.

00:03:23.470 --> 00:03:27.280
So this whole thing simplifies
to 8 divided by negative 4 is

00:03:27.280 --> 00:03:29.410
negative 2 plus 2.

00:03:31.950 --> 00:03:33.620
So it equals 0.

00:03:33.620 --> 00:03:37.890
So this big, hairy thing
simplified to 0.

00:03:37.890 --> 00:03:39.140
Now let's do 2b.

00:03:42.130 --> 00:03:46.400
Let me clear some space here.

00:03:46.400 --> 00:03:51.820
I'll leave the order of
operations stuff there.

00:03:51.820 --> 00:03:57.880
Let me clear that and
let me clear this.

00:03:57.880 --> 00:03:59.800
All right, 2b.

00:04:02.700 --> 00:04:03.800
Evaluate the following

00:04:03.800 --> 00:04:05.280
expressions involving variables.

00:04:05.280 --> 00:04:05.810
Fair enough.

00:04:05.810 --> 00:04:10.600
So they wrote 2y squared, and
they're saying that x is equal

00:04:10.600 --> 00:04:14.270
to 1, which is irrelevant
because there is no x here,

00:04:14.270 --> 00:04:16.310
and y is equal to 5.

00:04:16.310 --> 00:04:19.690
So if y is equal to 5, this
thing becomes the same thing

00:04:19.690 --> 00:04:25.840
as 2 times 5 squared.

00:04:25.840 --> 00:04:27.890
And notice, I put parentheses
there.

00:04:27.890 --> 00:04:31.070
I could have written this as,
this is the same thing as 2

00:04:31.070 --> 00:04:33.590
times 5 squared.

00:04:33.590 --> 00:04:36.550
And if you look at the order of
operations, exponents take

00:04:36.550 --> 00:04:38.120
priority over multiplication.

00:04:38.120 --> 00:04:41.240
That's why in my head I just
automatically put those

00:04:41.240 --> 00:04:41.950
parentheses.

00:04:41.950 --> 00:04:44.170
We're going to do the
exponent first.

00:04:44.170 --> 00:04:53.130
So this is 25, so you get 2
times 25 is equal to 50.

00:04:53.130 --> 00:04:58.030
So that is 2b, this is equal
to-- use a darker color --that

00:04:58.030 --> 00:05:00.090
is equal to 50.

00:05:00.090 --> 00:05:01.340
Let's do 2d.

00:05:05.470 --> 00:05:11.506
They're giving us y squared
minus x, whole thing squared.

00:05:11.506 --> 00:05:16.330
x is equal to 2 and
y is equal to 1.

00:05:16.330 --> 00:05:17.920
Well, we just substitute.

00:05:17.920 --> 00:05:19.580
Where we see a y we put a 1.

00:05:19.580 --> 00:05:25.370
So this is going to be 1 squared
minus x squared--

00:05:25.370 --> 00:05:27.600
Sorry, minus x, not x squared.

00:05:27.600 --> 00:05:28.755
So we just put a regular
x there.

00:05:28.755 --> 00:05:30.410
That's where we put a 2.

00:05:30.410 --> 00:05:32.880
And then all of that squared.

00:05:32.880 --> 00:05:37.310
Well 1 squared is just
1, so that is just 1.

00:05:37.310 --> 00:05:43.420
1 minus 2 is negative 1.

00:05:43.420 --> 00:05:47.600
And then we're going to want to
square our negative 1, so

00:05:47.600 --> 00:05:50.460
that will be equal
to positive 1.

00:05:50.460 --> 00:05:52.180
So that is equal to 1.

00:05:52.180 --> 00:05:54.810
Negative times a negative
is a positive.

00:05:54.810 --> 00:05:56.110
All right, let's do 3b.

00:06:00.760 --> 00:06:02.210
Doing every other problem.

00:06:02.210 --> 00:06:04.050
I'll do it in yellow.

00:06:04.050 --> 00:06:05.160
Evaluate the following

00:06:05.160 --> 00:06:06.850
expressions involving variables.

00:06:06.850 --> 00:06:07.300
All right.

00:06:07.300 --> 00:06:08.010
Same idea.

00:06:08.010 --> 00:06:15.250
So they're giving us
4x over 9x squared.

00:06:15.250 --> 00:06:19.130
Oh, actually I said I'd
do 3b, I was doing 3a.

00:06:19.130 --> 00:06:19.890
So here we go.

00:06:19.890 --> 00:06:29.060
We have z squared over x
plus y plus x squared

00:06:29.060 --> 00:06:32.530
over x minus y.

00:06:32.530 --> 00:06:36.220
And they're telling us that x
is equal to 1, y is equal to

00:06:36.220 --> 00:06:39.310
negative 2, and z
is equal to 4.

00:06:39.310 --> 00:06:41.660
So let's just do our
substitutions first.

00:06:41.660 --> 00:06:44.130
So z squared, that's the same
thing as-- I'll do it in a

00:06:44.130 --> 00:06:53.480
different color --4 squared over
x, 1, plus y, negative 2,

00:06:53.480 --> 00:06:58.410
plus x squared, that's
1 squared, over x,

00:06:58.410 --> 00:07:01.770
which is 1, minus y.

00:07:01.770 --> 00:07:05.330
y is negative 2.

00:07:05.330 --> 00:07:10.700
So this is going to be equal to
4 squared is 16 over 1 plus

00:07:10.700 --> 00:07:15.830
negative 2, that's 1 minus 2--
it's just a negative 1 --plus

00:07:15.830 --> 00:07:21.310
1 squared, which is 1, over
1 minus negative 2.

00:07:21.310 --> 00:07:23.840
That's the same thing
as 1 plus 2.

00:07:23.840 --> 00:07:26.180
So it's 1/3.

00:07:26.180 --> 00:07:29.410
And so this will be 16 divided
by negative 1.

00:07:29.410 --> 00:07:37.100
We could write that as that's
equal to negative 16 plus 1/3.

00:07:37.100 --> 00:07:40.480
Now if we want to actually add
these as fractions we could

00:07:40.480 --> 00:07:42.460
have a common denominator.

00:07:42.460 --> 00:07:48.640
Negative 16 is the same thing
as minus 48 over 3, or

00:07:48.640 --> 00:07:51.210
negative 48 over 3.

00:07:51.210 --> 00:07:54.480
If you take 48 divided by 3
you'll get 16, and I'm just

00:07:54.480 --> 00:07:55.970
keeping the negative sign.

00:07:55.970 --> 00:07:59.800
And then you add
that plus 1/3.

00:07:59.800 --> 00:08:02.890
We have a common denominator
now, 3.

00:08:02.890 --> 00:08:10.740
Negative 48 plus 1
is negative 47.

00:08:10.740 --> 00:08:16.760
So our answer is negative
47 over 3.

00:08:16.760 --> 00:08:18.010
Problem 3d.

00:08:23.910 --> 00:08:25.970
Same type of situation.

00:08:25.970 --> 00:08:35.240
x squared minus z squared over
xz minus 2x times z minus x.

00:08:35.240 --> 00:08:41.510
x is equal to negative
1, z is equal to 3.

00:08:41.510 --> 00:08:43.299
Let's do our substitutions.

00:08:43.299 --> 00:08:45.560
So this is x squared.

00:08:45.560 --> 00:08:48.220
That's minus 1 squared.

00:08:48.220 --> 00:08:53.670
Minus z squared, so
minus 3 squared.

00:08:53.670 --> 00:08:56.816
All of that over x times z.

00:08:56.816 --> 00:09:06.200
x times z is minus 1 times 3,
minus 2 times x, x is negative

00:09:06.200 --> 00:09:12.280
1, times z minus x,
times 3 minus x.

00:09:12.280 --> 00:09:15.940
x is negative 1 minus x.

00:09:15.940 --> 00:09:18.370
Wherever we saw an x
we put a minus 1.

00:09:18.370 --> 00:09:20.440
So this is going to be equal
to-- Remember, you do your

00:09:20.440 --> 00:09:22.280
exponents first. Well,

00:09:22.280 --> 00:09:24.690
parentheses first, then exponents.

00:09:24.690 --> 00:09:29.330
So we have negative 1 squared,
that's just a positive 1.

00:09:29.330 --> 00:09:32.090
3 squared, that's just
a positive 9.

00:09:32.090 --> 00:09:36.420
So our numerator becomes 1 minus
9, that's minus 8 or

00:09:36.420 --> 00:09:38.170
negative 8.

00:09:38.170 --> 00:09:40.160
And then our denominator.

00:09:40.160 --> 00:09:45.040
Negative 1 times 3
is negative 3.

00:09:45.040 --> 00:09:47.300
And then let's go to our
parentheses here.

00:09:47.300 --> 00:09:50.540
We have 3 minus negative 1,
that's the same thing as 3

00:09:50.540 --> 00:09:52.390
plus plus 1.

00:09:52.390 --> 00:09:56.080
So that right there becomes 4.

00:09:56.080 --> 00:10:01.650
So our denominator becomes
negative 3 minus 2 times

00:10:01.650 --> 00:10:05.010
negative 1 times 4, so
that's negative 8.

00:10:05.010 --> 00:10:07.880
Minus negative 8.

00:10:07.880 --> 00:10:11.230
Minus of a negative is the
same thing as a plus.

00:10:11.230 --> 00:10:21.320
So this whole thing becomes
negative 8 over negative 3

00:10:21.320 --> 00:10:23.160
plus 8 is 5.

00:10:23.160 --> 00:10:28.350
So it's negative 8/5,
minus 8 over 5.

00:10:28.350 --> 00:10:31.170
All right, let me clear up
some space just so we can

00:10:31.170 --> 00:10:33.500
reference this problem
properly.

00:10:33.500 --> 00:10:34.930
Let me clear all of this
out of the way.

00:10:38.380 --> 00:10:39.350
These are interesting now.

00:10:39.350 --> 00:10:41.770
Problem 4: insert parentheses in
each expression to make it

00:10:41.770 --> 00:10:42.790
a true equation.

00:10:42.790 --> 00:10:44.040
Fascinating.

00:10:45.910 --> 00:10:48.110
So 4b.

00:10:48.110 --> 00:10:56.810
You have 12 divided by 4 plus
10 minus 3 times 3 plus 7 is

00:10:56.810 --> 00:10:59.100
equal to 11.

00:10:59.100 --> 00:11:01.270
So let's see what happens if we
just do traditional order

00:11:01.270 --> 00:11:03.500
of operations, and I'll do a
little bit in my head because

00:11:03.500 --> 00:11:06.570
this is going to take some
experimentation.

00:11:06.570 --> 00:11:08.420
Oh yeah, this is 4b,
12 divided by 4--

00:11:08.420 --> 00:11:09.740
Yep, that's the problem.

00:11:09.740 --> 00:11:14.220
So if did 12 divided by 4 first,
and we would get 3.

00:11:14.220 --> 00:11:15.570
So let me just do
this in yellow.

00:11:15.570 --> 00:11:20.670
So if we did regular order of
operations this would be a 3.

00:11:20.670 --> 00:11:25.010
This right here would
it be a 9.

00:11:25.010 --> 00:11:32.940
So you would have 3 plus 10,
which is 13, minus 9, 13 minus

00:11:32.940 --> 00:11:36.210
9 is 4 plus 7.

00:11:36.210 --> 00:11:36.990
Actually, that seems right.

00:11:36.990 --> 00:11:37.980
Let me make sure I
did that right.

00:11:37.980 --> 00:11:41.410
3 plus 10-- Right,
that looks right.

00:11:41.410 --> 00:11:44.060
So we really just have to do
regular order of operations.

00:11:44.060 --> 00:11:46.760
So it already looks like
a true equation.

00:11:46.760 --> 00:11:52.030
So if you do 12 divided by 4
plus 10 minus 3 times 3 plus

00:11:52.030 --> 00:11:53.730
7, I think it turns out right.

00:11:53.730 --> 00:11:55.030
Let me confirm.

00:11:55.030 --> 00:11:56.740
Make sure I'm not making
a mistake.

00:11:56.740 --> 00:12:04.900
12 divided by 4 is 3 plus 10
minus 3 times 3 is 9 plus 7.

00:12:04.900 --> 00:12:11.650
This is equal to 13 minus 9,
which is equal to-- So all of

00:12:11.650 --> 00:12:18.910
this is equal to 13 minus 9 is
equal to 4 plus 7 is, indeed,

00:12:18.910 --> 00:12:19.940
equal to 11.

00:12:19.940 --> 00:12:21.140
So that one wasn't too bad.

00:12:21.140 --> 00:12:23.400
You actually wouldn't have to
put any parentheses to make

00:12:23.400 --> 00:12:24.220
this a true expression.

00:12:24.220 --> 00:12:26.330
You would just have to follow
the order of operations.

00:12:26.330 --> 00:12:28.450
But throwing those parentheses
there makes it a little bit

00:12:28.450 --> 00:12:30.170
easier to read.

00:12:30.170 --> 00:12:32.210
Let's try 4d.

00:12:35.050 --> 00:12:43.390
12 minus 8 minus 4 times
5 is equal to minus 8.

00:12:43.390 --> 00:12:45.260
So first let's just see what
happens if we did traditional

00:12:45.260 --> 00:12:46.260
order of operations.

00:12:46.260 --> 00:12:49.170
If we did traditional order of
operations we would do this 4

00:12:49.170 --> 00:12:53.050
times 5 first, which would
give us 20 over there.

00:12:53.050 --> 00:13:00.380
And then we would have
12 minus 8 is 4.

00:13:00.380 --> 00:13:04.010
And then we would do 4 minus
20-- No, that's not right.

00:13:04.010 --> 00:13:06.410
That would give us
negative 16.

00:13:06.410 --> 00:13:07.465
So that's not going
to be right.

00:13:07.465 --> 00:13:09.100
So we can't just do traditional
order of

00:13:09.100 --> 00:13:10.050
operations.

00:13:10.050 --> 00:13:14.040
Sorry, this is a minus
8 right there.

00:13:14.040 --> 00:13:17.390
So let's see how we can
experiment with this.

00:13:17.390 --> 00:13:18.780
Let's try out a couple
of situations.

00:13:18.780 --> 00:13:25.650
What if we did 12 minus 8 minus
4 and then multiplied

00:13:25.650 --> 00:13:26.650
that times 5.

00:13:26.650 --> 00:13:27.740
Let's see what this give us.

00:13:27.740 --> 00:13:30.230
I'm just experimenting
with parentheses.

00:13:30.230 --> 00:13:36.170
So if you do 8 minus 4, that
right there would be

00:13:36.170 --> 00:13:38.960
8 minus 4 is 4.

00:13:38.960 --> 00:13:43.240
And then 4 times 5 would be
20, and then 12 minus 20--

00:13:43.240 --> 00:13:44.270
yeah, that works.

00:13:44.270 --> 00:13:46.090
So let me confirm that.

00:13:46.090 --> 00:13:49.370
So I'm saying I'm going to put
parentheses right there and

00:13:49.370 --> 00:13:51.740
right there and let's
work it out.

00:13:51.740 --> 00:13:56.370
You would get 8 minus 4 is 4.

00:13:56.370 --> 00:13:59.190
So this whole thing was
simplified to 12

00:13:59.190 --> 00:14:01.990
minus 4 times 5.

00:14:01.990 --> 00:14:03.890
And you just do order of
operations, you do

00:14:03.890 --> 00:14:07.360
multiplication first.
So that is just 20.

00:14:07.360 --> 00:14:09.370
And if I wanted to make it very
clear I could actually

00:14:09.370 --> 00:14:09.890
write it like this.

00:14:09.890 --> 00:14:11.480
I could actually put
another round of

00:14:11.480 --> 00:14:13.040
parentheses right like that.

00:14:13.040 --> 00:14:15.450
But order of operations would
tell us to do it anyway.

00:14:15.450 --> 00:14:20.870
So it becomes 12 minus 20, which
is, indeed, minus 8 or

00:14:20.870 --> 00:14:22.540
negative 8.


Equations and Inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.680
Let's do some problems that
deal with equations and

00:00:03.680 --> 00:00:04.405
inequalities.

00:00:04.405 --> 00:00:07.300
So let's start with this
problem 2 here.

00:00:07.300 --> 00:00:09.850
So it says define the variables
and translate the

00:00:09.850 --> 00:00:13.430
following expressions
into inequalities.

00:00:13.430 --> 00:00:18.990
Let's see, part a: A bus can
seat 65 passengers or fewer.

00:00:18.990 --> 00:00:29.860
So in Part a, let's let x be
the number bus can seat.

00:00:29.860 --> 00:00:32.150
So the number of people you
can sit in the bus, we're

00:00:32.150 --> 00:00:33.350
going to call that x.

00:00:33.350 --> 00:00:36.010
And they're saying that
the bus can fit

00:00:36.010 --> 00:00:38.800
65 people or fewer.

00:00:38.800 --> 00:00:47.610
So x has to be less than
or equal to 65.

00:00:47.610 --> 00:00:49.740
And we put that equal there,
that less than or equal,

00:00:49.740 --> 00:00:51.740
because it could be 65.

00:00:51.740 --> 00:00:53.810
It doesn't say definitely
less than 65.

00:00:53.810 --> 00:00:56.490
It could be 65 passengers
or fewer.

00:00:56.490 --> 00:00:59.150
So that's why it's less
than or equal to.

00:00:59.150 --> 00:01:05.770
b: the sum of two consecutive
integers is less than 54.

00:01:05.770 --> 00:01:12.800
So if you say x is first
integer, and then x

00:01:12.800 --> 00:01:16.300
plus 1 is the next.

00:01:16.300 --> 00:01:19.280
They're saying the sum of these
two is less than 54.

00:01:19.280 --> 00:01:27.520
So x plus 1 is less than 54.

00:01:27.520 --> 00:01:29.330
And if you actually wanted to
figure out what x is, what the

00:01:29.330 --> 00:01:32.090
first integer is, you could just
solve this inequality.

00:01:32.090 --> 00:01:38.340
c: an amount of money is
invested at 5% annual

00:01:38.340 --> 00:01:48.970
interest. Let x be the
amount of money.

00:01:48.970 --> 00:01:52.530
It's invested at 5% annual
interest. The interest earned

00:01:52.530 --> 00:01:56.310
at the end of the year is
greater than or equal to $250.

00:01:56.310 --> 00:01:58.870
So what's the interest earned
at the end of the year?

00:01:58.870 --> 00:02:02.830
It's the amount of money you
invest times your annual

00:02:02.830 --> 00:02:07.000
interest rate, times 5% or, if
you write this as a decimal ,

00:02:07.000 --> 00:02:15.700
it's 0.05 So 0.05 or 5% times
the amount of money invested,

00:02:15.700 --> 00:02:20.030
that is going to be greater than
or equal to-- this is the

00:02:20.030 --> 00:02:28.960
amount of interest earned --that
is going to be greater

00:02:28.960 --> 00:02:33.340
than or equal to $250.

00:02:33.340 --> 00:02:34.460
So that's what we did.

00:02:34.460 --> 00:02:37.650
We set up the problem
into an inequality.

00:02:37.650 --> 00:02:38.900
And part d.

00:02:42.140 --> 00:02:44.700
You buy hamburgers at a
fast food restaurant.

00:02:44.700 --> 00:02:47.280
A hamburger costs $0.49.

00:02:47.280 --> 00:02:50.150
You have at most
$3.00 to spend.

00:02:50.150 --> 00:02:51.410
Write an inequality
for the number of

00:02:51.410 --> 00:02:53.130
hamburgers you can buy.

00:02:53.130 --> 00:02:55.280
So let's change the letters.

00:02:55.280 --> 00:03:00.235
Let h be equal to the number
of hamburgers.

00:03:02.890 --> 00:03:05.280
So the total I'm going to
spend is the number of

00:03:05.280 --> 00:03:11.830
hamburgers times the amount
per hamburger, times 0.49.

00:03:11.830 --> 00:03:14.430
That's the total amount I'm
going to spend, and that can

00:03:14.430 --> 00:03:16.050
at most be $3.00.

00:03:16.050 --> 00:03:23.520
So it has to be less than
or equal to $3.00.

00:03:23.520 --> 00:03:27.520
So that is our inequality.

00:03:27.520 --> 00:03:29.150
Let's do problem 4 here.

00:03:32.500 --> 00:03:35.280
Check that the given number
is a solution to the

00:03:35.280 --> 00:03:37.670
corresponding inequality.

00:03:37.670 --> 00:03:39.030
All right.

00:03:39.030 --> 00:03:41.100
Part a here.

00:03:41.100 --> 00:03:45.840
So the inequality is 2 times
x plus 6 is less

00:03:45.840 --> 00:03:47.700
than or equal to 8x.

00:03:47.700 --> 00:03:50.810
And they're saying that x is
equal to 12 is a solution.

00:03:50.810 --> 00:03:52.290
Let's verify that.

00:03:52.290 --> 00:03:55.835
So if you put a 12 here-- do it
in a different color --if

00:03:55.835 --> 00:03:59.450
this is a 12 and then
this would be a 12.

00:03:59.450 --> 00:04:02.140
So you get 2 times 12 plus 6.

00:04:02.140 --> 00:04:08.860
So this is 2 times 18 is less
than or equal to 8 times 12.

00:04:08.860 --> 00:04:13.080
8 times 12 is 96.

00:04:13.080 --> 00:04:16.800
2 times 18 is 36.

00:04:16.800 --> 00:04:20.360
So 36 is definitely less
than or equal to 96.

00:04:20.360 --> 00:04:25.050
So this x is equal to
12 is a solution.

00:04:25.050 --> 00:04:26.300
Part b.

00:04:28.450 --> 00:04:38.990
We have 1.4 times z plus 5.2
is greater than 0.4z.

00:04:42.010 --> 00:04:46.550
For the solution, if z is equal
to minus 9-- negative 9

00:04:46.550 --> 00:04:47.110
I should say.

00:04:47.110 --> 00:04:47.995
I always say minus 9.

00:04:47.995 --> 00:04:50.580
The correct wording
is negative 9.

00:04:50.580 --> 00:04:51.910
So let's put a negative
9 here.

00:04:51.910 --> 00:04:54.590
So it's 1.4-- they're saying
that that's one of the

00:04:54.590 --> 00:05:01.760
solutions --times negative 9
plus 5.2 is greater than 0.4

00:05:01.760 --> 00:05:07.930
times negative 9.

00:05:07.930 --> 00:05:11.060
And I'm just going to use the
calculator on this one just

00:05:11.060 --> 00:05:12.130
for the sake of time.

00:05:12.130 --> 00:05:15.480
You could do that in your
head if you like.

00:05:15.480 --> 00:05:27.940
So we have on the left-hand
side, 1.4 times negative 9.

00:05:27.940 --> 00:05:29.830
That's equal to minus 12.6.

00:05:29.830 --> 00:05:40.200
So this is minus is 12.6 plus
5.2 is equal to negative 7.4.

00:05:40.200 --> 00:05:46.770
So this left-hand side
is negative 7.4.

00:05:46.770 --> 00:05:49.990
Then they're claiming that that
is greater than-- This I

00:05:49.990 --> 00:05:51.020
can do in my head.

00:05:51.020 --> 00:05:55.790
4 times 9 is 36.

00:05:55.790 --> 00:05:57.480
I'm going to have to put a
negative sign because it's a

00:05:57.480 --> 00:05:58.860
positive times a negative.

00:05:58.860 --> 00:06:01.930
And I have one number behind
the decimal point.

00:06:01.930 --> 00:06:11.040
So this is saying negative 7.4
is greater than negative 3.6.

00:06:11.040 --> 00:06:13.510
This isn't true.

00:06:13.510 --> 00:06:15.910
If you draw a number
line right here.

00:06:15.910 --> 00:06:20.530
So if this is 0, this
is negative 3.6,

00:06:20.530 --> 00:06:22.520
this is negative 7.4.

00:06:22.520 --> 00:06:25.700
It's less than negative 3.6.

00:06:25.700 --> 00:06:27.820
So this is not true.

00:06:27.820 --> 00:06:30.480
I'll do it in a big red color.

00:06:30.480 --> 00:06:32.170
That is not true.

00:06:32.170 --> 00:06:36.840
z is equal to negative 9 is
not a solution of this

00:06:36.840 --> 00:06:38.220
inequality right there.

00:06:38.220 --> 00:06:41.760
It does not satisfy
that inequality.

00:06:41.760 --> 00:06:43.010
Part c.

00:06:46.390 --> 00:06:57.220
They have minus 5/2 y plus 1/2
is less than negative 18.

00:06:57.220 --> 00:06:59.640
And they're claiming y is
equal 40 is a solution.

00:06:59.640 --> 00:07:01.670
Let's try that out.

00:07:01.670 --> 00:07:05.840
So minus 5/2-- instead of y I
can write 40 just to see if it

00:07:05.840 --> 00:07:10.120
works --plus 1/2 is less
than minus 18.

00:07:10.120 --> 00:07:11.610
That's their claim.

00:07:11.610 --> 00:07:15.220
So 5/2 times 40-- Divide
the numerator and

00:07:15.220 --> 00:07:15.950
denominator by 2.

00:07:15.950 --> 00:07:18.450
So 1 becomes a 20.

00:07:18.450 --> 00:07:23.740
So it becomes minus 5 times 20
is minus 100 plus 1/2 is less

00:07:23.740 --> 00:07:26.110
than minus 18.

00:07:26.110 --> 00:07:33.580
Well, you could view this
as negative 99 and 1/2.

00:07:33.580 --> 00:07:35.160
You could do that in your head
if you like, or you could view

00:07:35.160 --> 00:07:41.090
this as 99.5-- you're adding
0.5 to minus 100 --is less

00:07:41.090 --> 00:07:44.290
than negative 18, which
is definitely true.

00:07:44.290 --> 00:07:46.630
This is more negative
than this is.

00:07:46.630 --> 00:07:49.620
So this is correct.

00:07:49.620 --> 00:07:53.720
And then finally part d.

00:07:53.720 --> 00:07:58.785
Let me scroll down or let me
clear some space for myself.

00:08:03.560 --> 00:08:04.810
Part d.

00:08:09.130 --> 00:08:14.000
They're saying 80 is greater
than or equal to 10

00:08:14.000 --> 00:08:17.210
times 3t plus 2.

00:08:17.210 --> 00:08:20.770
And they're claiming that t is
equal to 0.4 as a solution.

00:08:20.770 --> 00:08:22.040
Let's try that out.

00:08:22.040 --> 00:08:31.860
80 is greater than or equal
to 3 times 0.4 plus 2.

00:08:31.860 --> 00:08:34.380
So that's saying that 80 is
greater than or equal to 10

00:08:34.380 --> 00:08:41.210
times-- 3 times 0.4 is 1.2
--plus 2, or 80 is greater

00:08:41.210 --> 00:08:46.540
than or equal to 10 times--
This is 3.2.

00:08:46.540 --> 00:08:54.040
Or that 80 is greater than
or equal to 32, which is

00:08:54.040 --> 00:08:55.220
absolutely right.

00:08:55.220 --> 00:08:59.860
So d is also a valid solution.

00:08:59.860 --> 00:09:01.110
Problem 5.

00:09:03.830 --> 00:09:08.270
The cost of a Ford Focus
is 27% of the

00:09:08.270 --> 00:09:11.880
price of a Lexus GS450H.

00:09:11.880 --> 00:09:18.650
If the price of a Ford is
$15,000-- So the Ford is equal

00:09:18.650 --> 00:09:23.150
to $15,000, what is the
price of a Lexus?

00:09:23.150 --> 00:09:28.690
So they tell us that the Ford
is equal to 27% of the price

00:09:28.690 --> 00:09:29.260
of the Lexus.

00:09:29.260 --> 00:09:36.485
Is equal to 0.27, or 27%, times
the price of a Lexus.

00:09:39.210 --> 00:09:40.920
I could write F and L,
but I'll just write

00:09:40.920 --> 00:09:42.430
out the words there.

00:09:42.430 --> 00:09:44.420
So the Ford we know
is $15,000.

00:09:44.420 --> 00:09:52.240
So we know that $15,000 is equal
to 27%, 0.27, times the

00:09:52.240 --> 00:09:55.560
price of a Lexus.

00:09:55.560 --> 00:09:58.030
So to figure out the price of
a Lexus we just divide both

00:09:58.030 --> 00:10:07.170
sides by 0.27, and divide
this side by 0.27.

00:10:07.170 --> 00:10:09.040
That just becomes a 1.

00:10:09.040 --> 00:10:15.220
And so you have the price of
your Lexus is equal to $15,000

00:10:15.220 --> 00:10:17.990
divided by 0.27.

00:10:17.990 --> 00:10:20.020
See what we get.

00:10:20.020 --> 00:10:21.270
So let me clear it.

00:10:21.270 --> 00:10:30.370
We have 15-- 1, 2, 3, $15,000
divided by 0.27 is equal to

00:10:30.370 --> 00:10:34.370
$55,555.55.

00:10:34.370 --> 00:10:39.000
So we'll say $55,556
just to round up.

00:10:39.000 --> 00:10:46.920
So this is equal to $55,556.

00:10:46.920 --> 00:10:49.800
So you're spending a lot more on
the Lexus than you would on

00:10:49.800 --> 00:10:51.120
the Ford Focus.

00:10:51.120 --> 00:10:54.520
It better be a much
better car.

00:10:54.520 --> 00:10:56.860
Finally, number 6.

00:10:56.860 --> 00:11:00.520
On your new job you can be
paid in one of two ways.

00:11:00.520 --> 00:11:01.580
All right.

00:11:01.580 --> 00:11:04.130
You can either be paid--
So this is option one.

00:11:07.350 --> 00:11:08.930
And let's write option 2 here.

00:11:12.160 --> 00:11:16.350
You can either be paid $1,000
per month plus 6% commission

00:11:16.350 --> 00:11:18.340
of total sales.

00:11:18.340 --> 00:11:28.050
So let's let s is equal
to total sales.

00:11:28.050 --> 00:11:32.410
So your first option you'll be
paid $1,000 per month plus 6%

00:11:32.410 --> 00:11:34.230
of total sales.

00:11:34.230 --> 00:11:39.510
So on a monthly basis you'll
be paid $1,000 plus 6% of

00:11:39.510 --> 00:11:40.320
total sales.

00:11:40.320 --> 00:11:46.100
So plus 6% times your
total sales.

00:11:46.100 --> 00:11:47.020
That's option 1.

00:11:47.020 --> 00:11:48.470
That's right there.

00:11:48.470 --> 00:11:52.290
You can either be paid $1,000
per month plus 6% commission

00:11:52.290 --> 00:11:53.730
of total sales.

00:11:53.730 --> 00:11:57.185
And then option 2 is
$1,200 per month--

00:11:57.185 --> 00:12:00.420
Let me switch colors.

00:12:00.420 --> 00:12:07.890
Option 2-- I'll do it in yellow
--paid $1,200 per month

00:12:07.890 --> 00:12:14.135
plus 5% commission on
sales over $2,000.

00:12:14.135 --> 00:12:17.120
Let me scroll over
a little bit.

00:12:17.120 --> 00:12:23.090
Plus 5% of sales over $2,000.

00:12:23.090 --> 00:12:28.820
So if my sales are s, my sales
over $2,000 are going to be s

00:12:28.820 --> 00:12:34.390
minus $2,000.

00:12:34.390 --> 00:12:37.720
If my sales are, let's say
they're $3,000 in a month.

00:12:37.720 --> 00:12:39.440
The sales over $2,000
are going to be

00:12:39.440 --> 00:12:42.280
$3,000 minus $2,000.

00:12:42.280 --> 00:12:44.610
It's going to be $1,000
over $2,000.

00:12:44.610 --> 00:12:46.800
So these are our two options.

00:12:46.800 --> 00:12:53.040
Let me highlight that
right over there.

00:12:53.040 --> 00:12:55.950
And they say for what amount of
sales is the first option

00:12:55.950 --> 00:12:58.040
better than the second option?

00:12:58.040 --> 00:13:00.490
And assume they're always
sales over $2,000.

00:13:00.490 --> 00:13:02.670
So they're essentially saying,
OK, this is always going to be

00:13:02.670 --> 00:13:03.850
non-0 right here.

00:13:03.850 --> 00:13:05.570
So we want to know the situation
where the first

00:13:05.570 --> 00:13:07.210
option is better.

00:13:07.210 --> 00:13:09.960
Better, probably meaning
for me, meaning that

00:13:09.960 --> 00:13:10.910
I'll get more money.

00:13:10.910 --> 00:13:13.970
So when is option 1 going to
be, we could either say

00:13:13.970 --> 00:13:17.690
greater than or greater than
or equal to, option 2.

00:13:17.690 --> 00:13:19.350
And then we could solve
this equation.

00:13:19.350 --> 00:13:20.600
So let's see what we can do.

00:13:24.710 --> 00:13:25.750
The first thing, let's subtract

00:13:25.750 --> 00:13:27.300
$1,000 from both sides.

00:13:27.300 --> 00:13:30.280
If you subtract $1,000 from the
left-hand side you're just

00:13:30.280 --> 00:13:35.420
left with 0.06s is greater than
or equal to-- So subtract

00:13:35.420 --> 00:13:37.120
$1,000 from the left-hand
side.

00:13:37.120 --> 00:13:38.380
I'll take the $1,000
from here.

00:13:38.380 --> 00:13:47.740
You're left with 200 plus 0.05
times s minus 2,000.

00:13:47.740 --> 00:13:51.380
And then let's see, I probably
want to multiply that out.

00:13:51.380 --> 00:14:01.240
This is going to be 200 plus
0.05s minus-- This 0.05,

00:14:01.240 --> 00:14:05.400
that's the same 5%, that's
the same thing as 1/20.

00:14:05.400 --> 00:14:11.110
1/20 of 2,000, that's 100.

00:14:11.110 --> 00:14:14.600
So this is minus 100.

00:14:14.600 --> 00:14:16.920
That times that is 100.

00:14:16.920 --> 00:14:21.830
I'm just distributing this
5% or this 0.05.

00:14:21.830 --> 00:14:23.450
So that's the right-hand
side of the equation.

00:14:23.450 --> 00:14:28.810
On the left-hand side we have
0.06, and we just simplify the

00:14:28.810 --> 00:14:29.890
right-hand side a little bit.

00:14:29.890 --> 00:14:35.680
We have 200 minus 100, so that
just becomes 200 minus 100, we

00:14:35.680 --> 00:14:39.070
could just write as plus
100 right there.

00:14:39.070 --> 00:14:43.790
So our equation is 0.06 is
greater than or equal to

00:14:43.790 --> 00:14:51.540
0.05-- Sorry, this is 0.06s is
greater than or equal to-- I

00:14:51.540 --> 00:14:58.040
don't want to lose that s over
there --0.05s plus 100.

00:14:58.040 --> 00:15:04.640
Now let's subtract 0.05s from
both sides of the equation.

00:15:04.640 --> 00:15:12.250
So 0.06 minus 0.05s is going to
be 0.01s is greater than or

00:15:12.250 --> 00:15:14.630
equal to-- I subtracted this
from both sides, so it's not

00:15:14.630 --> 00:15:21.020
going to be here anymore --is
greater than or equal to 100.

00:15:21.020 --> 00:15:25.860
And now I just divide
both sides by 0.01.

00:15:25.860 --> 00:15:31.120
So 0.01, 0.01.

00:15:31.120 --> 00:15:32.800
That becomes a 1.

00:15:32.800 --> 00:15:36.670
And then we are left with s is
to going to be greater than or

00:15:36.670 --> 00:15:40.910
equal to-- What's 100
divided by 0.01?

00:15:40.910 --> 00:15:45.880
This is the same thing as 100
divided by 1 over 100, which

00:15:45.880 --> 00:15:48.910
is the same thing as
100 times 100.

00:15:48.910 --> 00:15:53.220
100 times 100 is 10,000.

00:15:53.220 --> 00:15:57.680
So option 1 is definitely better
for you if your total

00:15:57.680 --> 00:16:03.350
sales for the month,
if s is greater

00:16:03.350 --> 00:16:05.370
than or equal to $10,000.

00:16:05.370 --> 00:16:10.250
If it's less than that you're
better with option 2.


Bunch of examples

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.950
Let's do a couple of example
problems where we use

00:00:03.950 --> 00:00:07.190
information to set up equations
to solve our

00:00:07.190 --> 00:00:09.690
problem, and then we'll actually
solve the equation.

00:00:09.690 --> 00:00:11.660
So let's do problem number 1.

00:00:11.660 --> 00:00:17.280
Write a mathematical equation
that describes the

00:00:17.280 --> 00:00:20.380
relationship between the
variables in the table.

00:00:20.380 --> 00:00:22.890
So you have your days
and your profit.

00:00:22.890 --> 00:00:27.820
And if we look at them, on day
1 profit of 20, on day 2

00:00:27.820 --> 00:00:31.290
profit of 40, day
3 profit of 60.

00:00:31.290 --> 00:00:35.690
It looks like no matter what
happens the profit is always

00:00:35.690 --> 00:00:41.370
going to be equal to-- let's
see, it's 20, 40, 60, 80.

00:00:41.370 --> 00:00:45.300
It looks like every day
it increases by 20.

00:00:45.300 --> 00:00:49.910
This is plus 20, to go from 20
to 40, then it's plus 20, plus

00:00:49.910 --> 00:00:50.900
20, plus 20.

00:00:50.900 --> 00:00:52.850
So every day it increases
by 20.

00:00:52.850 --> 00:01:00.810
So maybe the formula here is
that our profit is equal to

00:01:00.810 --> 00:01:06.890
the day times 20, is equal
to 20 times the day.

00:01:06.890 --> 00:01:08.460
Does that fit the data?

00:01:08.460 --> 00:01:11.120
Let's see, you could try it with
all of them, but when the

00:01:11.120 --> 00:01:14.670
day is 5, 20 times 5 is 100.

00:01:14.670 --> 00:01:16.560
5 times 20 is 100.

00:01:16.560 --> 00:01:18.140
Yeah, it looks like it actually
fits the data.

00:01:18.140 --> 00:01:20.790
And if we wanted to write it a
little bit more algebraically,

00:01:20.790 --> 00:01:26.630
we could say that profit is p
and day is d, and we would

00:01:26.630 --> 00:01:32.260
write our equation as profit is
equal to 20 times d or 20d.

00:01:32.260 --> 00:01:33.140
So that's part a.

00:01:33.140 --> 00:01:35.840
I wrote a mathematical equation
that describes the

00:01:35.840 --> 00:01:36.920
relationship.

00:01:36.920 --> 00:01:40.290
And then part b, they say what
is the profit on day 10?

00:01:40.290 --> 00:01:44.070
Well lucky for us we've written
our equation, so

00:01:44.070 --> 00:01:46.370
they're essentially just saying
what happens to p,

00:01:46.370 --> 00:01:50.640
profit, what happens to profit
when d or day is equal to 10?

00:01:50.640 --> 00:01:54.780
So profit is going to be equal
to 20 times the day

00:01:54.780 --> 00:01:57.630
we're on, day 10.

00:01:57.630 --> 00:02:02.230
20 times 10, which
is equal to 200.

00:02:02.230 --> 00:02:03.430
That's it.

00:02:03.430 --> 00:02:05.420
Problem 3.

00:02:05.420 --> 00:02:08.320
Write a mathematical equation
for the following

00:02:08.320 --> 00:02:10.270
situations and solve.

00:02:10.270 --> 00:02:13.500
All right, let's do part a.

00:02:13.500 --> 00:02:16.400
7 times a number is 35.

00:02:16.400 --> 00:02:17.610
What is the number?

00:02:17.610 --> 00:02:19.500
So whenever someone says a
number, I'll just use the

00:02:19.500 --> 00:02:20.350
variable x.

00:02:20.350 --> 00:02:22.270
I could use any variable,
I could say y

00:02:22.270 --> 00:02:24.640
or b or z or anything.

00:02:24.640 --> 00:02:27.280
But I'll go with x because
that's what I'm most used to

00:02:27.280 --> 00:02:30.310
and what's most typically
done unless you've

00:02:30.310 --> 00:02:31.690
already used your x.

00:02:31.690 --> 00:02:41.690
So 7 times a number, I'll say
7 times x is equal to 35.

00:02:41.690 --> 00:02:42.990
So that's our equation.

00:02:42.990 --> 00:02:44.980
We've written our mathematical
equation.

00:02:44.980 --> 00:02:47.040
And now it's, what
is the number?

00:02:47.040 --> 00:02:50.730
So to solve for x you could
it one of two ways.

00:02:50.730 --> 00:02:54.950
You could divide both sides
of the equation by 7.

00:02:54.950 --> 00:02:57.930
So if you divide that side by 7
and that side by 7-- And the

00:02:57.930 --> 00:03:00.270
whole reason I'm doing that
is because I want this

00:03:00.270 --> 00:03:02.080
coefficient to become 1.

00:03:02.080 --> 00:03:04.600
And anything I do to one side
of the equation I have to do

00:03:04.600 --> 00:03:05.780
to the other side.

00:03:05.780 --> 00:03:10.360
So 7 divided by 7 is 1, so you
just have 1x or you just have

00:03:10.360 --> 00:03:14.200
x is equal to 35 divided by 7.

00:03:14.200 --> 00:03:15.980
x is equal to 5.

00:03:15.980 --> 00:03:17.480
The other way you could have
thought about this, you could

00:03:17.480 --> 00:03:19.670
have said that this is
multiplying both sides of the

00:03:19.670 --> 00:03:21.140
equation times 1/7.

00:03:21.140 --> 00:03:29.180
So 1/7 times 7x is equal
to 35 times 1/7.

00:03:29.180 --> 00:03:30.830
These are equivalent
ways of doing it.

00:03:30.830 --> 00:03:32.830
Either way these would
have canceled out.

00:03:32.830 --> 00:03:36.420
You would have gotten
1x is equal to 5.

00:03:36.420 --> 00:03:37.670
Part b.

00:03:40.040 --> 00:03:45.690
One number is 25 more than
2 times another number.

00:03:45.690 --> 00:03:49.330
All right, so let's say
that x is equal to a

00:03:49.330 --> 00:03:52.220
number, the one number.

00:03:52.220 --> 00:03:55.875
And let's say that y is equal
to the other number.

00:03:59.750 --> 00:04:02.440
So they're introducing
more variables here.

00:04:02.440 --> 00:04:08.250
So one number, let's say x,
is-- is, that's like equal

00:04:08.250 --> 00:04:17.279
--it's equal to 25 more than
2 times another number.

00:04:17.279 --> 00:04:18.760
Well, this is the
other number.

00:04:18.760 --> 00:04:21.440
So 2 times our other number.

00:04:21.440 --> 00:04:26.400
That's what I got out of this
first line right there.

00:04:26.400 --> 00:04:31.310
One number is 25 more than
2 times another number.

00:04:31.310 --> 00:04:34.580
2 times another number plus 25
is going to be equal to my

00:04:34.580 --> 00:04:35.780
first number.

00:04:35.780 --> 00:04:37.460
All right, now what is the
second [UNINTELLIGIBLE].

00:04:37.460 --> 00:04:48.760
If each number is multiplied by
5, their sum would be 350.

00:04:48.760 --> 00:04:51.900
So that means 5 times this plus
5 times that is going to

00:04:51.900 --> 00:04:53.580
be equal to 350.

00:04:53.580 --> 00:05:05.500
So 5 times x plus 5 times y is
going to be equal to 350.

00:05:05.500 --> 00:05:08.230
And they're asking what
are the numbers?

00:05:08.230 --> 00:05:11.050
Well, here we can do something
called substitution.

00:05:11.050 --> 00:05:15.230
We already see that x is equal
to this whole thing over here.

00:05:15.230 --> 00:05:18.490
It's just like saying x is equal
to 5 or x is equal to 9

00:05:18.490 --> 00:05:19.272
or whatever.

00:05:19.272 --> 00:05:20.830
x is equal to something.

00:05:20.830 --> 00:05:24.050
So every time we see an
x, we can replace

00:05:24.050 --> 00:05:25.940
it with this something.

00:05:25.940 --> 00:05:29.640
So we see an x right here, so
we can replace it with this

00:05:29.640 --> 00:05:33.550
something, and then we'll have
an equation that only has a y

00:05:33.550 --> 00:05:35.670
in it and then we'll be
able to solve for y.

00:05:35.670 --> 00:05:41.590
So this can be written as 5
times, instead of an x we have

00:05:41.590 --> 00:05:45.650
all this business, 25 plus 2y.

00:05:45.650 --> 00:05:48.310
I substituted for x, because
they told us x is equal to

00:05:48.310 --> 00:05:50.440
this thing.

00:05:50.440 --> 00:05:56.860
25 plus 2y plus 5y
is equal to 350.

00:05:56.860 --> 00:05:59.110
Use the distributive property.

00:05:59.110 --> 00:06:07.360
5 times 25 is 125 plus 5 times
2y is 10y, plus all of this,

00:06:07.360 --> 00:06:13.670
5y is equal to 350.

00:06:13.670 --> 00:06:17.940
10y plus 5y is 15y.

00:06:17.940 --> 00:06:26.070
That's 15y plus 125
is equal to 350.

00:06:26.070 --> 00:06:32.510
Now we can subtract 125 from
both sides, so minus 125--

00:06:32.510 --> 00:06:36.900
Sorry, minus 125.

00:06:36.900 --> 00:06:39.040
I'm doing that to get
rid of the 125 from

00:06:39.040 --> 00:06:40.460
the left-hand side.

00:06:40.460 --> 00:06:44.950
So our equation simplifies to--
I'll continue it up here.

00:06:44.950 --> 00:06:51.700
We get 15y is equal
to 350 minus 125.

00:06:51.700 --> 00:06:55.130
That is equal to 225.

00:06:55.130 --> 00:06:59.120
And you may or may not already
know that 15 times 15 is 225,

00:06:59.120 --> 00:07:00.210
so we already know what y is.

00:07:00.210 --> 00:07:02.990
Let's divide both sides
by 15 just to make it

00:07:02.990 --> 00:07:04.510
clear what I'm doing.

00:07:04.510 --> 00:07:09.280
You divide both sides by 15 you
get y is equal to 225 over

00:07:09.280 --> 00:07:11.730
15, which is equal to 15.

00:07:11.730 --> 00:07:13.300
Now we're not completely
done yet.

00:07:13.300 --> 00:07:16.860
We know what y is, but we still
have to solve for x.

00:07:16.860 --> 00:07:20.260
So x is equal to 25
plus 2 times y.

00:07:20.260 --> 00:07:22.720
y is equal to 15.

00:07:22.720 --> 00:07:29.090
2 times 15, that
is equal to 30.

00:07:29.090 --> 00:07:36.750
25 plus 30 is 55.

00:07:36.750 --> 00:07:39.420
Let me do part c.

00:07:39.420 --> 00:07:41.110
Let me clear this
out of the way.

00:07:47.150 --> 00:07:51.170
Clear this out of
the way as well.

00:07:51.170 --> 00:07:52.420
Part c.

00:07:55.620 --> 00:07:58.490
The sum of two consecutive
integers is 35.

00:07:58.490 --> 00:08:00.060
What are the numbers?

00:08:00.060 --> 00:08:03.020
Well, let's say that x is equal
to the first integer.

00:08:05.540 --> 00:08:07.690
Then what's the second
integer going to be?

00:08:07.690 --> 00:08:08.710
Well the second integer's
just going to be

00:08:08.710 --> 00:08:09.970
one more than that.

00:08:09.970 --> 00:08:11.610
They're consecutive, they're
one after another.

00:08:11.610 --> 00:08:17.730
So x plus 1 is equal to
the second integer.

00:08:17.730 --> 00:08:22.280
And they're saying the sum of
both of these numbers is 35.

00:08:22.280 --> 00:08:27.360
So x plus the number that comes
right after x, x plus 1,

00:08:27.360 --> 00:08:30.490
is equal to 35.

00:08:30.490 --> 00:08:35.440
Add these together, you get
2x plus 1 is equal to 35.

00:08:35.440 --> 00:08:39.600
Subtract 1 from both sides of
the equation, you get 2x is

00:08:39.600 --> 00:08:43.919
equal to 34-- just subtracted
1 from both sides.

00:08:43.919 --> 00:08:48.560
Divide both sides by 2 and
you get x is equal to 17.

00:08:48.560 --> 00:08:52.240
So the first number is 17, and
the second number is just one

00:08:52.240 --> 00:08:54.140
more than that, it's 18.

00:08:54.140 --> 00:08:58.270
And notice, two consecutive
integers, 17 then 18, add them

00:08:58.270 --> 00:09:01.740
together, you get 35.

00:09:01.740 --> 00:09:02.990
Part d.

00:09:05.580 --> 00:09:09.050
Peter is 3 times as old
as he was 6 years ago.

00:09:09.050 --> 00:09:10.690
How old is Peter?

00:09:10.690 --> 00:09:15.130
So Peter-- I'll say p for
Peter's age --is equal to 3

00:09:15.130 --> 00:09:19.540
times as old as he
was 6 years ago.

00:09:19.540 --> 00:09:21.620
How old was he 6 years ago?

00:09:21.620 --> 00:09:25.430
Well, if he's p years old
now, he was p minus 6

00:09:25.430 --> 00:09:27.420
years old 3 years ago.

00:09:27.420 --> 00:09:31.360
So let me write this as p minus
6-- that's as old as he

00:09:31.360 --> 00:09:32.630
was 6 years ago.

00:09:32.630 --> 00:09:35.120
That's what p minus 6 is.

00:09:35.120 --> 00:09:38.020
This 3 times-- that's the
3 times as old as

00:09:38.020 --> 00:09:39.980
he was 6 years ago.

00:09:39.980 --> 00:09:43.010
And, of course, this is how
old he is right now.

00:09:43.010 --> 00:09:45.710
So we just have to
solve for p.

00:09:45.710 --> 00:09:50.350
So we get p is equal to,
distribute the 3, 3 times p is

00:09:50.350 --> 00:09:56.140
3p minus 3 times 6 is 18.

00:09:56.140 --> 00:10:00.310
Now we can subtract 3p from both
sides of the equation.

00:10:00.310 --> 00:10:05.040
We get minus 2p, right? p
minus 3p is minus, or

00:10:05.040 --> 00:10:10.240
negative, 2p is equal
to negative 18.

00:10:10.240 --> 00:10:15.450
And divide both sides by
negative 2 and you get p is

00:10:15.450 --> 00:10:17.050
equal to 9.

00:10:17.050 --> 00:10:18.910
Just multiply both sides
by negative 2.

00:10:18.910 --> 00:10:20.230
And you can verify it.

00:10:20.230 --> 00:10:21.810
Right now he's 9.

00:10:21.810 --> 00:10:27.120
6 years ago he was 3, and he
is 3 times as old as that.

00:10:27.120 --> 00:10:31.690
Right, 6 years ago he was 3,
and at 9 years old he is 3

00:10:31.690 --> 00:10:34.250
times as old as he
was 6 years ago.

00:10:34.250 --> 00:10:35.820
So it definitely works out.

00:10:35.820 --> 00:10:37.460
That's the cool thing about
algebra, you can

00:10:37.460 --> 00:10:38.400
always check your work.

00:10:38.400 --> 00:10:41.080
You always know when you
got the right answer.

00:10:41.080 --> 00:10:43.010
Problem 6.

00:10:43.010 --> 00:10:46.600
The price of an MP3 player
decreased by 20% from last

00:10:46.600 --> 00:10:48.300
year to this year.

00:10:48.300 --> 00:10:51.670
This year the price of
the player is $120.

00:10:51.670 --> 00:10:53.820
What was the price last year?

00:10:53.820 --> 00:11:03.760
So let me say, p is equal
to the price last year.

00:11:03.760 --> 00:11:07.420
And they're telling us that
the price of an MP3 player

00:11:07.420 --> 00:11:10.910
decreased by 20% from last
year to this year.

00:11:10.910 --> 00:11:13.240
So let's take 20%
away from this.

00:11:13.240 --> 00:11:20.230
So price minus 20% of the
price-- this is decreasing by

00:11:20.230 --> 00:11:23.880
20% --is equal to the price
this year, which they're

00:11:23.880 --> 00:11:29.950
telling us is $120.

00:11:29.950 --> 00:11:33.570
Well what's p minus 0.2
or minus 20% of p?

00:11:33.570 --> 00:11:35.840
We decreased by 20%.

00:11:35.840 --> 00:11:39.360
Well you're going to
be left with 80%.

00:11:39.360 --> 00:11:44.560
So we could say 80% of the price
last year is equal to

00:11:44.560 --> 00:11:46.950
the price this year.

00:11:46.950 --> 00:11:50.180
Like we did before, we divide
both sides by the coefficient

00:11:50.180 --> 00:11:51.510
on our variable.

00:11:51.510 --> 00:11:57.485
Divide both sides by 0.8, or
0.80, depending how you want

00:11:57.485 --> 00:11:58.450
to view it.

00:11:58.450 --> 00:12:00.460
0.80.

00:12:00.460 --> 00:12:02.420
These cancel out.

00:12:02.420 --> 00:12:11.980
So we have to divide
0.8 into $120.

00:12:11.980 --> 00:12:14.870
A little exercise in
dividing decimals.

00:12:14.870 --> 00:12:18.840
Let me put some zeroes there.

00:12:18.840 --> 00:12:22.870
We could multiply the 0.8
and the 120 by 10.

00:12:22.870 --> 00:12:25.460
That's essentially moving
the decimals over to

00:12:25.460 --> 00:12:26.990
the right by one.

00:12:26.990 --> 00:12:33.510
So this is the same as dividing
8 into 1,200.

00:12:33.510 --> 00:12:35.520
So this decimal isn't there
anymore, it's over here.

00:12:35.520 --> 00:12:37.810
This decimal is now over here.

00:12:37.810 --> 00:12:42.060
8 goes into 12 one time,
1 times 8 is 8.

00:12:42.060 --> 00:12:46.140
12 minus 8 is 4, bring
down the zero.

00:12:46.140 --> 00:12:48.930
8 goes into 40 five times.

00:12:48.930 --> 00:12:54.700
5 times 8 is 40, 40 minus 40
is 0, bring down another 0.

00:12:54.700 --> 00:12:59.470
8 goes into 0 zero times, 0
times 8-- you get the idea.

00:12:59.470 --> 00:13:00.430
0 times 8 is 0.

00:13:00.430 --> 00:13:01.730
We have no remainder.

00:13:01.730 --> 00:13:06.170
So our answer is $150.

00:13:06.170 --> 00:13:09.810
So it was $150 last year,
it decreased by 20%.

00:13:09.810 --> 00:13:13.720
20% of $150's $30, so it
decreased by $30, and it

00:13:13.720 --> 00:13:15.960
definitely got us to $120.


Examples of evaluating variable expressions

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.500
Let's do some practice problems
dealing with variable

00:00:03.500 --> 00:00:04.190
expressions.

00:00:04.190 --> 00:00:07.240
So these first problems say
write the following in a more

00:00:07.240 --> 00:00:11.185
condensed form by leaving out
the multiplication symbol or

00:00:11.185 --> 00:00:14.130
leaving out a multiplication
symbol.

00:00:14.130 --> 00:00:19.900
So here we have 2 times 11x, so
if we have 11 x's and then

00:00:19.900 --> 00:00:23.100
we're going to have 2 times
those 11 x's, we're going to

00:00:23.100 --> 00:00:24.970
have 22 x's.

00:00:24.970 --> 00:00:32.400
So another way you could view
this, 2 times 11x, you could

00:00:32.400 --> 00:00:36.960
view this as being equal to 2
times 11, and all of that

00:00:36.960 --> 00:00:41.120
times x, and that's going
to be equal to 22 x's.

00:00:41.120 --> 00:00:44.065
You had 11 x's, you're going to
have 2 times as many x's,

00:00:44.065 --> 00:00:46.500
so you're going to
have 22 x's.

00:00:46.500 --> 00:00:50.700
Let's see, you have
1.35 times y.

00:00:50.700 --> 00:00:53.510
Now here we're just going to do
a straight simplifying how

00:00:53.510 --> 00:00:54.180
we write it.

00:00:54.180 --> 00:01:00.630
So 1.35 times y-- I'll do it
in a different color-- 1.35

00:01:00.630 --> 00:01:03.720
times y-- that's a
little dot there.

00:01:03.720 --> 00:01:06.710
In algebra we can just get
rid of that dot symbol.

00:01:06.710 --> 00:01:09.760
If we have a variable following
a number, we know

00:01:09.760 --> 00:01:13.220
that means 1.35 times
that variable.

00:01:13.220 --> 00:01:19.560
So that, we could rewrite as
just being equal to 1.35y.

00:01:19.560 --> 00:01:21.350
We've condensed it by
getting rid of the

00:01:21.350 --> 00:01:23.360
multiplication sign.

00:01:23.360 --> 00:01:26.540
Let's see, here we
have 3 times 1/4.

00:01:26.540 --> 00:01:30.000
Well, this is just straight
up multiplying a fraction.

00:01:30.000 --> 00:01:34.020
So in problem 3-- this was
problem 1, this is problem 2,

00:01:34.020 --> 00:01:39.190
problem 3-- 3 times 1/4, that's
the same thing as 3

00:01:39.190 --> 00:01:41.500
over 1 times 1/4.

00:01:41.500 --> 00:01:44.050
Multiply the numerators,
you get 3.

00:01:44.050 --> 00:01:47.790
Multiply the denominators,
1 times 4, you get 4.

00:01:47.790 --> 00:01:50.560
So number 3, I got 3/4.

00:01:50.560 --> 00:01:53.740
And then finally, you
have 1/4 times z.

00:01:53.740 --> 00:01:55.650
We could do the exact same
thing we did up here in

00:01:55.650 --> 00:01:57.030
problem number 2.

00:01:57.030 --> 00:01:59.950
This was the same
thing as 1.35y.

00:01:59.950 --> 00:02:03.130
That's the same thing
as 1.35 times y.

00:02:03.130 --> 00:02:08.180
So down here we could rewrite
this as either being equal to

00:02:08.180 --> 00:02:15.230
1/4z, or we could view this as
being equal to 1 over 4 times

00:02:15.230 --> 00:02:21.260
z over 1, which is the same
thing as z times 1, over 4

00:02:21.260 --> 00:02:25.980
times 1, or the same
thing as z over 4.

00:02:25.980 --> 00:02:28.560
So all of these are
equivalent.

00:02:28.560 --> 00:02:30.440
Now, what do they want
us to do down here?

00:02:30.440 --> 00:02:33.610
Evaluate the following
expressions for a is equal to

00:02:33.610 --> 00:02:47.320
3, b is equal to 2, c is equal
to 5, and d is equal to minus

00:02:47.320 --> 00:02:50.860
4-- or, actually, I should say
negative 4 is the correct

00:02:50.860 --> 00:02:51.540
terminology.

00:02:51.540 --> 00:02:53.160
Negative 4.

00:02:53.160 --> 00:02:55.110
So we just substitute.

00:02:55.110 --> 00:02:57.930
Every time we see an a, we're
going to put a minus 3 there,

00:02:57.930 --> 00:02:59.660
or a negative 3 there.

00:02:59.660 --> 00:03:03.410
Every time we see a b, we'll
put a positive 2 there.

00:03:03.410 --> 00:03:06.180
Every time we see a c,
we'll put a 5 there.

00:03:06.180 --> 00:03:08.750
And every time we see a d, we'll
put a minus 4 there.

00:03:08.750 --> 00:03:09.710
And I'll do a couple of these.

00:03:09.710 --> 00:03:13.540
I won't do all of them, just
for the sake of time.

00:03:13.540 --> 00:03:17.240
So let's say problem number 5.

00:03:17.240 --> 00:03:22.640
They gave us 2 times
a plus 3 times b.

00:03:22.640 --> 00:03:26.340
Well, this is the same thing as
2 times-- instead of an a,

00:03:26.340 --> 00:03:30.340
we know that a is going to
be equal to negative 3.

00:03:30.340 --> 00:03:36.930
So 2 times minus 3, plus
3 times b-- what's b?

00:03:36.930 --> 00:03:41.960
They're telling us that b is
equal to 2-- so 3 times 2.

00:03:41.960 --> 00:03:43.510
And what is this equal to?

00:03:43.510 --> 00:03:50.840
2 times minus 3-- let me do it
in a different color-- 2 times

00:03:50.840 --> 00:03:55.050
negative 3 is negative
6, plus 3 times 2.

00:03:55.050 --> 00:03:56.580
3 times 2 is 6.

00:03:56.580 --> 00:03:58.230
That's positive 6.

00:03:58.230 --> 00:04:00.180
So that is equal to 0.

00:04:00.180 --> 00:04:01.390
And notice the order
of operations.

00:04:01.390 --> 00:04:05.360
We did the multiplications, we
did the two multiplications

00:04:05.360 --> 00:04:07.680
before we added the
two numbers.

00:04:07.680 --> 00:04:11.510
Multiplication and division
takes precedence over addition

00:04:11.510 --> 00:04:13.030
and subtraction.

00:04:13.030 --> 00:04:16.160
Let's do problem 6.

00:04:16.160 --> 00:04:17.850
I'll do that right here.

00:04:17.850 --> 00:04:20.320
So you have 4 times c.

00:04:20.320 --> 00:04:23.180
4 times-- now what's
c equal to?

00:04:23.180 --> 00:04:25.260
They tell us c is equal to 5.

00:04:25.260 --> 00:04:30.080
So 4 times 5, that's
our c, plus d.

00:04:30.080 --> 00:04:33.375
d is minus or negative 4.

00:04:33.375 --> 00:04:39.090
So we have 4 times 5 is 20, plus
negative 4-- that's the

00:04:39.090 --> 00:04:44.000
same thing as minus 4, so
that is equal to 16.

00:04:44.000 --> 00:04:45.280
Problem 6.

00:04:45.280 --> 00:04:47.910
Now, let's do one of the
harder ones down here.

00:04:47.910 --> 00:04:52.690
This problem 10 looks a little
bit more daunting.

00:04:52.690 --> 00:04:55.650
Problem 10 right there.

00:04:55.650 --> 00:04:59.395
So we have a minus 4b in the
numerator, if you can read it,

00:04:59.395 --> 00:05:00.730
it's kind of small.

00:05:00.730 --> 00:05:02.180
a is minus 3.

00:05:02.180 --> 00:05:09.570
So we have minus 3-- or negative
3-- minus 4 times b.

00:05:09.570 --> 00:05:11.620
b is 2.

00:05:11.620 --> 00:05:13.520
So 4 times 2.

00:05:13.520 --> 00:05:19.240
Remember, this right here is
a, that right there is b.

00:05:19.240 --> 00:05:19.940
How do I know?

00:05:19.940 --> 00:05:21.730
They're telling me up here.

00:05:21.730 --> 00:05:26.780
And then all of that over--
all of that is

00:05:26.780 --> 00:05:30.670
over 3c plus 2d.

00:05:30.670 --> 00:05:41.630
So 3 times-- what was c?
c is 5 plus 2 times d.

00:05:41.630 --> 00:05:42.910
What is d?

00:05:42.910 --> 00:05:44.350
d is negative 4.

00:05:47.070 --> 00:05:48.550
So let's figure this out.

00:05:48.550 --> 00:05:50.440
So we have to do order
of operations.

00:05:50.440 --> 00:05:53.310
Multiplication comes first
before addition and

00:05:53.310 --> 00:05:54.560
subtraction.

00:05:56.910 --> 00:06:01.900
So this is going to be equal
to minus 3 minus 4 times 2,

00:06:01.900 --> 00:06:09.710
minus 8, all of that over-- 3
times 5 is 15, plus 2 times

00:06:09.710 --> 00:06:13.590
negative 4 is negative 8,
or 15 plus negative

00:06:13.590 --> 00:06:16.750
8 is 15 minus 8.

00:06:16.750 --> 00:06:23.330
And now, the numerator becomes
negative 3 minus 8, which is

00:06:23.330 --> 00:06:25.870
negative 11.

00:06:25.870 --> 00:06:32.850
And the denominator is 15
minus 8, which is 7.

00:06:32.850 --> 00:06:39.510
So problem 10, we simplified
it to negative 11 over 7.

00:06:39.510 --> 00:06:41.920
Right there.

00:06:41.920 --> 00:06:43.450
Let's do a couple of
these over here.

00:06:43.450 --> 00:06:46.400
OK, we see some exponents.

00:06:46.400 --> 00:06:49.350
I'll pick one of the
harder ones.

00:06:49.350 --> 00:06:56.640
Let's do this one over
here, problem 18.

00:06:56.640 --> 00:07:00.850
So 2x squared minus 3x squared,
plus 5x minus 4.

00:07:00.850 --> 00:07:02.030
OK, well, this wasn't
that hard.

00:07:02.030 --> 00:07:03.940
All of them are dealing
with x.

00:07:03.940 --> 00:07:06.100
But what we could do here--
let me write this down.

00:07:06.100 --> 00:07:12.680
2x squared minus 3x squared,
plus 5x minus 4.

00:07:12.680 --> 00:07:17.400
And they tell us that x is
equal to negative 1.

00:07:17.400 --> 00:07:19.720
One thing we could do is
simplify this before we even

00:07:19.720 --> 00:07:22.700
substitute for negative 1.

00:07:22.700 --> 00:07:25.340
So what's 2 of something
minus 3 of something?

00:07:25.340 --> 00:07:25.460
Right?

00:07:25.460 --> 00:07:30.750
This is 2x squareds
minus 3x squareds.

00:07:30.750 --> 00:07:33.780
So 2 of something minus 3 of
something, that's going to be

00:07:33.780 --> 00:07:35.730
minus 1 of that something.

00:07:35.730 --> 00:07:39.050
So that right there-- or
negative 1 of that something--

00:07:39.050 --> 00:07:45.450
that would be negative 1x
squared plus 5x minus 4.

00:07:45.450 --> 00:07:47.880
And they tell us x is
equal to negative 1.

00:07:47.880 --> 00:07:53.220
So this is negative 1 times x
squared, negative 1 squared,

00:07:53.220 --> 00:07:58.500
plus 5 times x, which is
negative 1, minus 4.

00:07:58.500 --> 00:07:59.120
So what is this?

00:07:59.120 --> 00:08:02.180
Negative 1 squared is just 1.

00:08:02.180 --> 00:08:03.510
That's just 1.

00:08:03.510 --> 00:08:08.780
So this whole expression
simplifies to negative 1 plus

00:08:08.780 --> 00:08:11.140
5 times negative 1-- we do the

00:08:11.140 --> 00:08:12.880
multiplication first, of course.

00:08:12.880 --> 00:08:17.690
So that's minus 5, or
negative 5 minus 4.

00:08:17.690 --> 00:08:22.840
So negative 1 minus 5 is
negative 6, minus 4 is equal

00:08:22.840 --> 00:08:27.710
to negative 10.

00:08:27.710 --> 00:08:32.369
And I'll do these last two just
to get a sample of all of

00:08:32.369 --> 00:08:38.299
the types of problems in this
variable expression section.

00:08:38.299 --> 00:08:42.380
The weekly cost, c, of
manufacturing x remote

00:08:42.380 --> 00:08:47.050
controls-- so the cost is c, x
is the remote controls-- is

00:08:47.050 --> 00:08:48.480
given by this formula.

00:08:48.480 --> 00:08:53.260
The cost is equal to 2,000 plus
3 times the number of

00:08:53.260 --> 00:08:56.840
remote controls, where cost
is given in dollars.

00:08:56.840 --> 00:08:58.880
Question a, what is the
cost of producing

00:08:58.880 --> 00:09:03.510
1,000 remote controls?

00:09:03.510 --> 00:09:06.330
Well, the number of remote
controls is x.

00:09:06.330 --> 00:09:10.010
So for part a-- I could write
it over here-- the cost is

00:09:10.010 --> 00:09:15.090
going to be equal to-- just use
this formula-- 2,000 plus

00:09:15.090 --> 00:09:17.840
3 times the number of
remote controls.

00:09:17.840 --> 00:09:19.880
x is the number of
remote controls.

00:09:19.880 --> 00:09:23.695
So 3 times 1,000.

00:09:23.695 --> 00:09:28.430
So it's going to be equal to
2,000 plus 3 times 1,000 is

00:09:28.430 --> 00:09:33.810
3,000, which is equal
to $5,000.

00:09:33.810 --> 00:09:38.500
So that's part a. $5,000.

00:09:38.500 --> 00:09:39.530
Now part b.

00:09:39.530 --> 00:09:42.870
What is the cost of producing
2,000 remote controls?

00:09:42.870 --> 00:09:45.450
Well, the cost-- just use the
same formula-- is equal to

00:09:45.450 --> 00:09:51.330
2,000 plus 3 times the number
of remote controls.

00:09:51.330 --> 00:09:56.350
So 3 times 2,000.

00:09:56.350 --> 00:10:02.050
So that's equal to 2,000 plus
3 times 2,000 is 6,000.

00:10:02.050 --> 00:10:06.900
So that's equal to $8,000.

00:10:06.900 --> 00:10:09.770
Now we're at problem 22.

00:10:09.770 --> 00:10:13.720
The volume of a box without a
lid is given by the formula,

00:10:13.720 --> 00:10:21.240
volume is equal to 4x times 10
minus x squared, where x is a

00:10:21.240 --> 00:10:25.700
length in inches, and v is the
volume in cubic inches.

00:10:25.700 --> 00:10:28.700
What is the volume when
x is equal to 2?

00:10:28.700 --> 00:10:32.570
So part a, x is equal to 2.

00:10:32.570 --> 00:10:34.790
We just substitute 2 wherever
we see an x here.

00:10:34.790 --> 00:10:39.170
So the volume is going to be
equal to 4 times x, which is

00:10:39.170 --> 00:10:47.330
2, times 10 minus 2 squared.

00:10:47.330 --> 00:10:53.980
And so this is going to be equal
to 4 times 2 is 8, times

00:10:53.980 --> 00:10:57.950
10 minus 2 is 8 squared.

00:10:57.950 --> 00:11:02.550
So that's 8 times-- so this
is equal to 8 times--

00:11:02.550 --> 00:11:04.460
8 squared is 64.

00:11:04.460 --> 00:11:07.000
You could say this is going to
be 8 to the third power.

00:11:07.000 --> 00:11:12.380
And 64 times 8-- 4
times 8 is 32.

00:11:12.380 --> 00:11:21.070
6 times 8 is 48, plus 3 is 51.

00:11:21.070 --> 00:11:27.430
So it's 512.

00:11:27.430 --> 00:11:31.500
Now, what is the volume
when x is equal to 3?

00:11:31.500 --> 00:11:36.310
I'll do it here in pink. b, when
is equal to 3, then the

00:11:36.310 --> 00:11:42.480
volume is equal to 4 times 3-- x
is equal to 3 now-- times 10

00:11:42.480 --> 00:11:46.410
minus 3 squared.

00:11:46.410 --> 00:11:52.230
4 times 3 is 12, times 10
minus 3 is 7 squared.

00:11:52.230 --> 00:11:56.390
So it's equal to 12 times 49.

00:11:56.390 --> 00:11:58.810
And just to get the exact
answer, let's multiply that.

00:11:58.810 --> 00:12:05.740
So 49 times 12-- 2 times
9 is 18, 2 times 4 is

00:12:05.740 --> 00:12:08.335
8, plus 1 is 9.

00:12:08.335 --> 00:12:09.240
Put a 0.

00:12:09.240 --> 00:12:11.180
1 times 9 is 9.

00:12:11.180 --> 00:12:13.630
1 times 4 is 4.

00:12:13.630 --> 00:12:15.912
And then we add the two.

00:12:15.912 --> 00:12:23.390
We get 8, 9 plus 9 is 18, and
then 1 plus 4 is 588.


Napoleon and the War of the Third Coalition

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:04.630
Let's review a little bit of
what Napoleon was up to going

00:00:04.630 --> 00:00:08.960
into the war of the Third
Coalition, which really does

00:00:08.960 --> 00:00:13.340
establish Napoleon as the
dominant figure in Europe.

00:00:13.340 --> 00:00:19.300
So in 1799, he takes power.

00:00:19.300 --> 00:00:22.770
First with two other consuls,
but then he declares himself

00:00:22.770 --> 00:00:23.595
First Consul.

00:00:23.595 --> 00:00:25.500
So he takes power.

00:00:25.500 --> 00:00:30.260
He becomes First-- let me put
that in capital letters--

00:00:30.260 --> 00:00:35.820
First Consul.

00:00:35.820 --> 00:00:38.610
Then in 1802, actually
before I get to 1802.

00:00:38.610 --> 00:00:41.640
Let me say what he
did in 1800.

00:00:41.640 --> 00:00:43.710
Remember when he took power,
we were still in the war of

00:00:43.710 --> 00:00:44.560
the Second Coalition.

00:00:44.560 --> 00:00:47.610
We talked about that
a little bit.

00:00:47.610 --> 00:00:50.580
France had lost ground, they
had lost a lot of what they

00:00:50.580 --> 00:00:53.780
had gained in the First
Coalition in Italy.

00:00:53.780 --> 00:00:55.820
The Austrians had
taken it back.

00:00:55.820 --> 00:01:00.120
So Napoleon decides to take
charge, cross the Alps.

00:01:00.120 --> 00:01:03.465
This is a picture of Napoleon
crossing the Alps.

00:01:09.610 --> 00:01:13.290
Leading the troops into the Alps
to take back what he felt

00:01:13.290 --> 00:01:16.700
needed to be taken back
from Austria.

00:01:16.700 --> 00:01:19.040
And then we learned in the
last video that that

00:01:19.040 --> 00:01:21.785
essentially is what ended
the First Coalition.

00:01:21.785 --> 00:01:30.330
So in 1800 he takes back or
leaves to take back Italy.

00:01:40.280 --> 00:01:42.120
And when we talk about Italy,
we're really talking about the

00:01:42.120 --> 00:01:44.500
Kingdom of Italy, which
was northern Italy.

00:01:44.500 --> 00:01:46.920
We're not talking about the
Kingdom of Naples, which was

00:01:46.920 --> 00:01:48.090
southern Italy.

00:01:48.090 --> 00:01:49.270
Or the Papal States.

00:01:49.270 --> 00:01:51.610
Actually let me show you that.

00:01:51.610 --> 00:01:53.690
If we go all the
way down here.

00:01:53.690 --> 00:01:56.220
You'll see when people talk
about the Kingdom of Italy in

00:01:56.220 --> 00:01:58.550
this period of time, they're
really talking about this

00:01:58.550 --> 00:02:01.470
region up here, which is
really northern Italy.

00:02:01.470 --> 00:02:03.030
The Papal States
are right here.

00:02:03.030 --> 00:02:05.650
And you had your Kingdom
of Naples down there.

00:02:05.650 --> 00:02:08.900
Compared to modern Europe,
the two countries that we

00:02:08.900 --> 00:02:11.120
associate with Europe today that
really didn't exist in a

00:02:11.120 --> 00:02:15.940
unified form in the early 1800s
were Italy and Germany.

00:02:15.940 --> 00:02:19.840
They were really just broken up
into a bunch of kingdoms.

00:02:19.840 --> 00:02:22.530
And as I mentioned before,
Germany at this point, that

00:02:22.530 --> 00:02:24.550
confederation of kingdoms was
kind of referred to as the

00:02:24.550 --> 00:02:30.130
Holy Roman Empire where the King
of Austria had the title

00:02:30.130 --> 00:02:32.210
of Holy Roman Emperor.

00:02:32.210 --> 00:02:39.790
But he didn't control the Holy
Roman Empire in a very

00:02:39.790 --> 00:02:40.930
centralized fashion.

00:02:40.930 --> 00:02:44.930
It was actually controlled by
a bunch of smaller kings.

00:02:44.930 --> 00:02:48.990
But anyway, let's go
back to Napoleon.

00:02:48.990 --> 00:02:51.850
So that's him crossing
the Alps in 1800.

00:02:51.850 --> 00:02:54.170
That ends when he
takes it back.

00:02:54.170 --> 00:02:55.630
He defeats Austria.

00:02:55.630 --> 00:02:58.950
That is essentially the end
of the Second Coalition.

00:02:58.950 --> 00:03:00.900
We learned that in
the last video.

00:03:00.900 --> 00:03:02.320
1801.

00:03:02.320 --> 00:03:04.165
You had your Treaty
of Luneville.

00:03:04.165 --> 00:03:09.590
End of Second Coalition.

00:03:09.590 --> 00:03:12.470
They don't have a treaty with
Great Britain until the next

00:03:12.470 --> 00:03:13.800
year, really just out
of attrition.

00:03:13.800 --> 00:03:16.760
But that doesn't matter, it was
a very short-lived treaty.

00:03:16.760 --> 00:03:23.030
And then in 1802, before we
start talking about the Third

00:03:23.030 --> 00:03:31.160
Coalition, Napoleon gets
the Constitution

00:03:31.160 --> 00:03:36.790
of the Year X passed.

00:03:36.790 --> 00:03:38.030
And why do you say year 10?

00:03:38.030 --> 00:03:40.820
Remember, they had this whole
revolutionary calendar going.

00:03:40.820 --> 00:03:43.640
So in the revolutionary calendar
this was the 10th

00:03:43.640 --> 00:03:45.930
year of the Revolution.

00:03:45.930 --> 00:03:50.310
But what's really relevant from
Napoleon's point of view

00:03:50.310 --> 00:03:52.390
is article one.

00:03:52.390 --> 00:03:55.890
It says, the French people name
and the senate proclaims

00:03:55.890 --> 00:03:59.655
Napoleon Bonaparte First
Consul for life.

00:04:06.060 --> 00:04:10.350
So, if you want to view this
is kind of legally giving

00:04:10.350 --> 00:04:18.670
Napoleon almost a king-like
status, it writes it in words

00:04:18.670 --> 00:04:22.750
that he will rule France for
the rest of his life.

00:04:22.750 --> 00:04:24.020
Then we get to 1803.

00:04:24.020 --> 00:04:25.720
And actually, before I get
to 1803, remember what's

00:04:25.720 --> 00:04:27.060
happening at the
same time here.

00:04:27.060 --> 00:04:39.150
You had your Revolution in
Haiti, which, if you remember

00:04:39.150 --> 00:04:43.700
from those videos, Haiti was
the most profitable slave

00:04:43.700 --> 00:04:45.210
colony in the world.

00:04:45.210 --> 00:04:49.030
Once the slaves revolted, got
their freedom, all of a sudden

00:04:49.030 --> 00:04:53.310
not as profitable to France
as it was before.

00:04:53.310 --> 00:04:55.600
And remember, the whole French
Revolution started off because

00:04:55.600 --> 00:04:57.290
France was broke.

00:04:57.290 --> 00:05:00.200
So Napoleon tries to
reinstate slavery.

00:05:00.200 --> 00:05:02.140
That essentially backfires
on him.

00:05:02.140 --> 00:05:05.330
Dessalines catches wind of
it, him and the other

00:05:05.330 --> 00:05:06.190
revolutionaries.

00:05:06.190 --> 00:05:10.540
So they essentially
start antagonizing

00:05:10.540 --> 00:05:12.350
Leclerc even more.

00:05:12.350 --> 00:05:17.750
Leclerc and the French occupiers
get ravaged by

00:05:17.750 --> 00:05:18.530
yellow fever.

00:05:18.530 --> 00:05:21.230
And Dessalines is a very
aggressive general.

00:05:21.230 --> 00:05:23.010
So he starts losing Haiti.

00:05:23.010 --> 00:05:25.510
You have Rochambeau, if you
remember from those videos,

00:05:25.510 --> 00:05:29.400
he's this hugely brutal figure
who takes over after Leclerc.

00:05:29.400 --> 00:05:31.230
But we have this revolution
in Haiti.

00:05:31.230 --> 00:05:32.140
And you can kind of say they're

00:05:32.140 --> 00:05:33.680
starting to lose Haiti.

00:05:38.470 --> 00:05:42.200
And at the same time, and this
is 1802, and as we're entering

00:05:42.200 --> 00:05:50.370
into 1803, Napoleon begins to
realize that Great Britain

00:05:50.370 --> 00:05:52.410
controls the seas.

00:05:52.410 --> 00:06:02.580
Great Britain dominant navy.

00:06:02.580 --> 00:06:04.900
And because they're losing
Haiti, which is really their

00:06:04.900 --> 00:06:08.890
main profit center, and Great
Britain is dominant, Napoleon

00:06:08.890 --> 00:06:11.190
essentially gives up
on North America.

00:06:23.210 --> 00:06:26.830
So outside of Haiti, which was
this major profit center for

00:06:26.830 --> 00:06:29.630
the French Empire, they also
had the territory of

00:06:29.630 --> 00:06:32.780
Louisiana, which is essentially
the middle 1/3 of

00:06:32.780 --> 00:06:35.580
what we now consider to
be the United States.

00:06:35.580 --> 00:06:38.120
And Napoleon figures out,
gee, Great Britain

00:06:38.120 --> 00:06:39.530
has a dominant navy.

00:06:39.530 --> 00:06:41.190
I'm giving up on Haiti.

00:06:41.190 --> 00:06:45.030
I'm probably going to lose
it anyway to Dessalines.

00:06:45.030 --> 00:06:47.080
Let me just give up on
the entire continent.

00:06:47.080 --> 00:06:50.540
If I don't sell Louisiana,
either Great Britain or the

00:06:50.540 --> 00:06:54.020
United States will probably
be in a good position to

00:06:54.020 --> 00:06:55.770
take it from me.

00:06:55.770 --> 00:06:59.585
So he decides to
sell Louisiana.

00:07:06.560 --> 00:07:09.540
So just as someone, or as an
American, and especially an

00:07:09.540 --> 00:07:13.480
American who was born in
Louisiana, it's fascinating

00:07:13.480 --> 00:07:15.450
the chain of events that
led up to this.

00:07:15.450 --> 00:07:17.320
Because when you learn it from
an American history point of

00:07:17.320 --> 00:07:19.700
view, you just say hey Thomas
Jefferson, he was able to get

00:07:19.700 --> 00:07:21.800
Louisiana for $15 million.

00:07:21.800 --> 00:07:24.250
Why would someone sell,
all of a sudden, this

00:07:24.250 --> 00:07:26.470
huge amount of territory?

00:07:26.470 --> 00:07:28.985
And the reason is because
Napoleon figured he was going

00:07:28.985 --> 00:07:30.210
to lose it no matter what.

00:07:30.210 --> 00:07:32.005
Great Britain had this
dominant navy.

00:07:32.005 --> 00:07:34.810
And at the same time, he
had nothing else to

00:07:34.810 --> 00:07:35.800
protect in the area.

00:07:35.800 --> 00:07:39.340
The other major valuable
asset there was Haiti.

00:07:39.340 --> 00:07:42.710
So a combination of Great
Britain having this navy and

00:07:42.710 --> 00:07:46.300
of the slave revolt in Haiti
ends up within the United

00:07:46.300 --> 00:07:51.780
States being able to acquire
what's now the middle 1/3 of

00:07:51.780 --> 00:07:52.570
the continent.

00:07:52.570 --> 00:07:54.610
You can kind of wonder, if they
never did that, would

00:07:54.610 --> 00:07:59.480
they have ever a gone on to
capture the entire west coast?

00:07:59.480 --> 00:08:01.890
So these small little
things in history.

00:08:01.890 --> 00:08:04.630
And I wouldn't necessarily call
these little things, but

00:08:04.630 --> 00:08:09.040
they lead to very tremendous
changes in our modern world.

00:08:09.040 --> 00:08:10.390
Anyway, enough about that.

00:08:10.390 --> 00:08:12.490
So he sells Louisiana.

00:08:12.490 --> 00:08:14.890
Great Britain has
a dominant navy.

00:08:14.890 --> 00:08:20.380
At first, Napoleon was amassing
troops up here, he

00:08:20.380 --> 00:08:22.780
was thinking about an invasion
of Great Britain.

00:08:22.780 --> 00:08:25.430
But more and more it dawns on
him that Great Britain has a

00:08:25.430 --> 00:08:26.310
dominant navy.

00:08:26.310 --> 00:08:28.360
Great Britain, I think, starts
to feel a little

00:08:28.360 --> 00:08:29.190
self-confident.

00:08:29.190 --> 00:08:32.669
And they don't like all of the
gains that Napoleon has gotten

00:08:32.669 --> 00:08:34.429
in the last two wars.

00:08:34.429 --> 00:08:45.780
So, Great Britain declares
war in 1803, in May 1803.

00:08:45.780 --> 00:08:49.180
Or I could say United Kingdom of
Great Britain and Ireland,

00:08:49.180 --> 00:08:49.955
the same thing.

00:08:49.955 --> 00:08:52.425
U.K.

00:08:52.425 --> 00:08:54.320
declares war.

00:08:54.320 --> 00:08:57.450
And you could view this as the
beginning of the war of the

00:08:57.450 --> 00:08:58.270
Third Coalition.

00:08:58.270 --> 00:08:59.570
But it's not a coalition yet.

00:08:59.570 --> 00:09:01.120
It's just the United Kingdom.

00:09:01.120 --> 00:09:03.120
Or it's just Great Britain.

00:09:03.120 --> 00:09:08.540
And it really gets
heated in 1805.

00:09:08.540 --> 00:09:10.240
Everyone else jumps in.

00:09:10.240 --> 00:09:12.880
And now we're talking about
a real coalition.

00:09:12.880 --> 00:09:26.320
Now Great Britain is joined
by Austria and Russia.

00:09:26.320 --> 00:09:29.180
And Austria, all the time in
every one of these coalitions,

00:09:29.180 --> 00:09:32.780
they lose territory to France
and especially Napoleon.

00:09:32.780 --> 00:09:34.140
They just want to get revenge.

00:09:34.140 --> 00:09:36.630
United Kingdom kind of senses
that they control the water,

00:09:36.630 --> 00:09:38.050
they want to get revenge.

00:09:38.050 --> 00:09:40.400
Russia doesn't like this
upstart Napoleon.

00:09:40.400 --> 00:09:43.150
And so they all jump in.

00:09:43.150 --> 00:09:46.380
And so there's two interesting
angles of the War

00:09:46.380 --> 00:09:47.260
of the Third Coalition.

00:09:47.260 --> 00:09:49.510
Let me write this right here.

00:09:49.510 --> 00:09:52.080
This is the Third Coalition.

00:09:52.080 --> 00:09:54.110
And there were other people who
jumped in, but these were

00:09:54.110 --> 00:09:56.050
the major powers.

00:09:56.050 --> 00:10:02.420
So this right here is
the Third Coalition.

00:10:02.420 --> 00:10:05.960
So the first thing that happens,
or really the two

00:10:05.960 --> 00:10:08.150
almost happen simultaneously.

00:10:08.150 --> 00:10:16.650
In October of 1805, the French
navy gets destroyed by the

00:10:16.650 --> 00:10:19.230
British in the Battle
of Trafalgar.

00:10:19.230 --> 00:10:21.010
This is Trafalgar, it's
actually written here.

00:10:23.700 --> 00:10:27.900
And they get destroyed by
Napoleon's good old friend,

00:10:27.900 --> 00:10:31.500
Admiral Horatio Nelson.

00:10:31.500 --> 00:10:34.190
So this guy was definitely a
thorn in Napoleon's side.

00:10:38.000 --> 00:10:43.510
He destroyed Napoleon's ships in
the Battle of the Nile that

00:10:43.510 --> 00:10:45.930
stranded their troops
in Egypt.

00:10:45.930 --> 00:10:49.530
And now he comes and essentially
destroys the

00:10:49.530 --> 00:10:52.550
French navy at the Battle
of Trafalgar.

00:10:52.550 --> 00:10:54.155
And this is an image of it.

00:10:54.155 --> 00:10:59.620
And I guess the only redeeming
fact of this battle for

00:10:59.620 --> 00:11:03.370
Napoleon was that near the end
of the battle, Horatio Nelson

00:11:03.370 --> 00:11:06.940
actually gets stricken
by a bullet

00:11:06.940 --> 00:11:08.190
and he dies at Trafalgar.

00:11:14.260 --> 00:11:20.300
So the Royal Navy, the British
navy was already dominant,

00:11:20.300 --> 00:11:23.080
destroying the French navy,
which was the only navy that

00:11:23.080 --> 00:11:26.500
could in some way compete
on any level.

00:11:26.500 --> 00:11:29.470
Destroying them at Trafalgar
really established British

00:11:29.470 --> 00:11:31.010
domination of the seas.

00:11:31.010 --> 00:11:34.540
This is actually an image of
the Battle of Trafalgar.

00:11:34.540 --> 00:11:36.780
This is the battle
right there.

00:11:36.780 --> 00:11:38.840
A bunch of ships just shooting
at each other, you can't

00:11:38.840 --> 00:11:39.980
really make out what's
happening.

00:11:39.980 --> 00:11:45.350
And just to connect that to
modern day Great Britain, or

00:11:45.350 --> 00:11:48.590
the modern day London, this
right here is an image of

00:11:48.590 --> 00:11:49.320
Trafalgar Square.

00:11:49.320 --> 00:11:51.960
It is named after the Battle
of Trafalgar, which

00:11:51.960 --> 00:11:54.910
established Britain is
dominant in the seas.

00:11:54.910 --> 00:11:56.890
That is Trafalgar Square
in London.

00:12:00.800 --> 00:12:04.050
And if you go there, you will
see Nelson's column.

00:12:04.050 --> 00:12:07.090
This is Nelson's column
right there, named

00:12:07.090 --> 00:12:10.650
after Horatio Nelson.

00:12:10.650 --> 00:12:13.170
And at the very top of it,
there's a little figure, or

00:12:13.170 --> 00:12:16.390
it's probably pretty large,
but little relative to the

00:12:16.390 --> 00:12:19.060
column, of Horatio
Nelson himself.

00:12:19.060 --> 00:12:21.840
So that was a bit of an aside,
but this occurred during the

00:12:21.840 --> 00:12:24.720
war of the Third Coalition.

00:12:24.720 --> 00:12:26.930
So Napoleon's navy
is destroyed.

00:12:26.930 --> 00:12:27.920
But he doesn't give up.

00:12:27.920 --> 00:12:30.910
He's just like, OK, oh well, I
can't do much beyond Europe,

00:12:30.910 --> 00:12:34.300
but in Europe I can still
do a lot of damage.

00:12:34.300 --> 00:12:37.640
So what he does is he
takes his troops.

00:12:37.640 --> 00:12:40.120
He had given up on actually
attacking Britain by sea.

00:12:40.120 --> 00:12:42.670
He realized that he could never
realistically do that.

00:12:42.670 --> 00:12:45.320
Let me show you the troop
movements that were going on

00:12:45.320 --> 00:12:46.780
at this time.

00:12:46.780 --> 00:12:49.850
So he had troops here that he
had given up on actually

00:12:49.850 --> 00:12:52.090
attacking Great Britain.

00:12:52.090 --> 00:12:55.750
And what he does is he
meets the Austrians.

00:12:55.750 --> 00:12:58.480
So the Austrians are amassing
their troops here.

00:12:58.480 --> 00:13:00.610
They actually thought that there
was a possibility that

00:13:00.610 --> 00:13:02.110
the main campaign would
go in Italy.

00:13:02.110 --> 00:13:04.830
So they actually focused most
of their troops there, that

00:13:04.830 --> 00:13:06.830
we'll see as a major mistake.

00:13:06.830 --> 00:13:11.940
But they have troops in what is
now southern Germany near

00:13:11.940 --> 00:13:15.360
the Black Forest. And then they
also amassed some troops

00:13:15.360 --> 00:13:16.190
near Italy.

00:13:16.190 --> 00:13:18.240
And then the Russians were
coming up behind.

00:13:18.240 --> 00:13:19.840
They weren't able to meet.

00:13:19.840 --> 00:13:23.000
And we're dealing with the fall,
this is now September,

00:13:23.000 --> 00:13:24.840
October of 1805.

00:13:24.840 --> 00:13:28.180
And the Russians were back here
trying to bring their

00:13:28.180 --> 00:13:34.060
troops to assist the Austrians
in meeting Napoleon.

00:13:34.060 --> 00:13:38.120
And all of these, it's always
unclear on how much of it was

00:13:38.120 --> 00:13:40.640
tactical genius versus
pure luck.

00:13:40.640 --> 00:13:42.496
Because they didn't have
satellites, they didn't know

00:13:42.496 --> 00:13:43.250
where the other troops were.

00:13:43.250 --> 00:13:45.880
They didn't necessarily know
exactly what's happening on a

00:13:45.880 --> 00:13:49.510
minute-by-minute basis,
like armies do today.

00:13:49.510 --> 00:13:54.750
But what happened, and it's one
of the pivotal battles in

00:13:54.750 --> 00:13:57.480
European history, is that
Napoleon meets the

00:13:57.480 --> 00:14:00.220
Austrians at Ulm.

00:14:00.220 --> 00:14:03.980
Let me draw it on this
map right here.

00:14:03.980 --> 00:14:07.290
So on this map they just do
a very general-- This is

00:14:07.290 --> 00:14:10.670
Napoleon meeting the Austrians
right here at Ulm.

00:14:10.670 --> 00:14:11.880
There's actually a
whole campaign of

00:14:11.880 --> 00:14:13.330
Ulm, several battles.

00:14:13.330 --> 00:14:15.940
But it was a decisive
victory at Ulm.

00:14:15.940 --> 00:14:21.270
And the crux of that victory was
that Napoleon was able to

00:14:21.270 --> 00:14:24.030
fool the Austrians.

00:14:24.030 --> 00:14:27.760
So this is a zoomed-in version
of what was going on here.

00:14:27.760 --> 00:14:29.110
This is France.

00:14:29.110 --> 00:14:33.730
This is the Black Forest. This
is Austria right there.

00:14:33.730 --> 00:14:37.720
Blue is Napoleon troops, red
are the Austrian troops.

00:14:37.720 --> 00:14:40.860
And you saw, Austria had amassed
most of their troops

00:14:40.860 --> 00:14:42.480
on the Italian front.

00:14:42.480 --> 00:14:46.410
So here in Black Forest, you
have your Austrian troops.

00:14:46.410 --> 00:14:50.570
What Napoleon did is he had
General Marat make it look

00:14:50.570 --> 00:14:54.720
like the main force of the
French were coming straight

00:14:54.720 --> 00:14:58.210
from the West. And they did
that by having cavalry

00:14:58.210 --> 00:15:01.740
essentially giving the
appearance of a major force

00:15:01.740 --> 00:15:03.830
coming in from here.

00:15:03.830 --> 00:15:06.110
So the Austrians were thinking,
OK, this is the

00:15:06.110 --> 00:15:12.780
direction that we're going to
face our main antagonism from.

00:15:12.780 --> 00:15:14.860
But at the same time, the bulk
of the Napoleon's army--

00:15:14.860 --> 00:15:17.340
remember he doesn't see exactly
how many people are

00:15:17.340 --> 00:15:18.950
here, they just made it
look like this is

00:15:18.950 --> 00:15:19.670
the bulk of the army.

00:15:19.670 --> 00:15:23.810
But the bulk of the army
actually did what they call a

00:15:23.810 --> 00:15:29.760
wheeling manoeuvre, where they
went around the Austrian army.

00:15:29.760 --> 00:15:33.440
And they were able to do it much
faster and in a much more

00:15:33.440 --> 00:15:35.480
nimble way than anyone
would have predicted.

00:15:35.480 --> 00:15:37.320
Because they essentially
didn't have to

00:15:37.320 --> 00:15:38.680
carry as many supplies.

00:15:38.680 --> 00:15:40.940
They were doing this during
the harvest season.

00:15:40.940 --> 00:15:42.450
And this was by design.

00:15:42.450 --> 00:15:47.060
If you capture enemy territory
during the harvest season, and

00:15:47.060 --> 00:15:49.140
these are things that you
usually don't think about when

00:15:49.140 --> 00:15:51.150
you learn about military
battles.

00:15:51.150 --> 00:15:53.890
But you've got troops, you've
got to feed the troops.

00:15:53.890 --> 00:15:58.570
If they can't capture food from
the land, then you've got

00:15:58.570 --> 00:15:59.800
to carry supplies with you.

00:15:59.800 --> 00:16:02.190
And supplies are heavy it
slows down your army.

00:16:02.190 --> 00:16:05.080
But if you do the attack during
the harvest season, you

00:16:05.080 --> 00:16:06.460
don't have to carry food.

00:16:12.080 --> 00:16:14.810
You can just take it from the
farmers that you pass by.

00:16:14.810 --> 00:16:18.230
They're harvesting food as you
pass by, you just take it, you

00:16:18.230 --> 00:16:19.600
can be fast and nimble.

00:16:19.600 --> 00:16:22.750
And he essentially went around
the army and was able to

00:16:22.750 --> 00:16:25.210
attack the Austrians from
this direction.

00:16:25.210 --> 00:16:27.920
And then essentially
destroy them.

00:16:27.920 --> 00:16:30.780
And this is why I hinted in the
last video that Napoleon

00:16:30.780 --> 00:16:32.570
starts to view himself
as invincible.

00:16:32.570 --> 00:16:35.830
And this was before the
Russians could come to

00:16:35.830 --> 00:16:38.350
actually reinforce
the Austrians.

00:16:38.350 --> 00:16:40.990
So they did it-- by being able
to do it so quickly, they were

00:16:40.990 --> 00:16:44.650
able to just only tackle
the Austrians.

00:16:44.650 --> 00:16:47.890
So it was a decisive
victory at Ulm.

00:16:47.890 --> 00:16:50.640
Let me go to the painting
right there.

00:16:50.640 --> 00:16:52.550
This is a painting of the
surrender at Ulm.

00:16:57.140 --> 00:17:02.420
And then the combined Russian
and Austrian forces, they

00:17:02.420 --> 00:17:07.960
essentially regrouped, but
they re-met Napoleon's

00:17:07.960 --> 00:17:10.540
forces after Ulm.

00:17:10.540 --> 00:17:14.849
In another several weeks,
at Austerlitz.

00:17:14.849 --> 00:17:18.450
And once again, Napoleon
just routed them.

00:17:18.450 --> 00:17:19.579
And there's always a debate.

00:17:19.579 --> 00:17:22.829
Some of it might have been
tactical genius, some of it

00:17:22.829 --> 00:17:24.990
might have been pure luck.

00:17:24.990 --> 00:17:30.390
But after routing them at
Austerlitz-- so this is

00:17:30.390 --> 00:17:36.660
Austerlitz-- and taking so much
I guess both in terms of

00:17:36.660 --> 00:17:42.110
casualties and prisoners from
the enemy without incurring so

00:17:42.110 --> 00:17:45.680
much themselves, this really was
the high point in terms of

00:17:45.680 --> 00:17:48.680
a battle for Napoleon.

00:17:48.680 --> 00:17:52.010
Austerlitz is what really
convinced Napoleon that he is

00:17:52.010 --> 00:17:53.910
truly, truly a military
genius.

00:17:53.910 --> 00:17:55.220
And that's not saying
that he's not.

00:17:55.220 --> 00:17:58.520
But it was such a rout that
he started to imagine that

00:17:58.520 --> 00:18:00.860
he's on some level invincible.

00:18:00.860 --> 00:18:04.600
And so the outcome, this
essentially ends the war of

00:18:04.600 --> 00:18:05.850
the Third Coalition.

00:18:11.530 --> 00:18:16.610
And with that ending, Austria
had to give its territory that

00:18:16.610 --> 00:18:18.740
it had gotten in Italy.

00:18:18.740 --> 00:18:23.210
So Austria had to cede the
territory that it had in

00:18:23.210 --> 00:18:27.510
Italy, some of the territory
that it had in Bavaria.

00:18:27.510 --> 00:18:30.230
And from, I guess just a general
point of view, this

00:18:30.230 --> 00:18:31.550
was historically significant.

00:18:31.550 --> 00:18:35.130
Because until this point in
time, this whole area was

00:18:35.130 --> 00:18:38.470
called the Holy Roman Empire,
with the King of Austria

00:18:38.470 --> 00:18:41.300
calling himself the Holy
Roman Emperor.

00:18:41.300 --> 00:18:43.520
Even though he didn't have
direct control over all of

00:18:43.520 --> 00:18:45.120
these Germanic regions.

00:18:48.030 --> 00:18:49.340
Let me go back up here.

00:18:49.340 --> 00:18:53.890
This is in 1805, or we can even
say we're entering 1806,

00:18:53.890 --> 00:19:00.300
but in late 1805, with the end
of the Third Coalition, now

00:19:00.300 --> 00:19:02.130
the Austrian King no longer
calls himself

00:19:02.130 --> 00:19:03.050
the Holy Roman Emperor.

00:19:03.050 --> 00:19:04.340
He gives up the title.

00:19:04.340 --> 00:19:12.000
And this region right here is
no longer called the Holy

00:19:12.000 --> 00:19:12.840
Roman Empire.

00:19:12.840 --> 00:19:14.905
It is called the Confederation
of the Rhine.

00:19:23.910 --> 00:19:27.390
And it's essentially under the
protection of Napoleon.

00:19:27.390 --> 00:19:29.990
And this is the first time--
we're starting to get to a

00:19:29.990 --> 00:19:32.720
point, Germany won't
be unified for

00:19:32.720 --> 00:19:34.930
another 60 or 70 years.

00:19:34.930 --> 00:19:37.490
But we're starting to get to
the point that the German

00:19:37.490 --> 00:19:39.610
nation is escaping from
the bounds of being

00:19:39.610 --> 00:19:40.920
the Holy Roman Empire.

00:19:40.920 --> 00:19:44.670
And so it's getting
in that direction.

00:19:44.670 --> 00:19:49.190
So the Holy Roman
Empire is gone.

00:19:49.190 --> 00:19:52.230
France gets significant
territory from

00:19:52.230 --> 00:19:54.120
Austria, once again.

00:19:54.120 --> 00:19:57.810
Russia on some level is just
humbled a little bit.

00:19:57.810 --> 00:20:00.200
And then we end in 1806,
or at the end of

00:20:00.200 --> 00:20:05.060
1805, with France dominant.

00:20:05.060 --> 00:20:07.370
And while all of that was
happening, it was really in

00:20:07.370 --> 00:20:10.050
1804 when France was only at
war with Great Britain.

00:20:10.050 --> 00:20:12.040
The whole alliance hadn't
formed yet.

00:20:12.040 --> 00:20:16.950
It 1804, in December-- and this
is relevant that it's a

00:20:16.950 --> 00:20:19.140
peek into his ego.

00:20:19.140 --> 00:20:22.690
So this is the year
before Austerlitz.

00:20:22.690 --> 00:20:27.800
In 1804, Napoleon declares
himself, or he

00:20:27.800 --> 00:20:29.095
crowns himself Emperor.

00:20:39.340 --> 00:20:41.550
And this is a picture
of him as Emperor.

00:20:41.550 --> 00:20:43.380
And after this point, instead
of being called Napoleon

00:20:43.380 --> 00:20:45.920
Bonaparte, he is called
Napoleon I.

00:20:49.560 --> 00:20:51.850
And the idea of crowning
yourself Emperor, you might

00:20:51.850 --> 00:20:52.780
say, what's the point?

00:20:52.780 --> 00:20:54.800
You already were Consul for
life, you have power for your

00:20:54.800 --> 00:20:55.550
whole life.

00:20:55.550 --> 00:20:59.020
But other than you get this
neat title emperor, which

00:20:59.020 --> 00:21:02.610
probably feeds your ego quite
well, but beyond that, this

00:21:02.610 --> 00:21:05.200
essentially establishes
a dynasty.

00:21:05.200 --> 00:21:07.080
It now says, I'm not
just a First

00:21:07.080 --> 00:21:09.510
Consul, I am the emperor.

00:21:09.510 --> 00:21:10.590
I'm Napoleon I.

00:21:10.590 --> 00:21:14.950
And it implies that his
descendants will continue to

00:21:14.950 --> 00:21:17.710
reign over the French Empire.

00:21:17.710 --> 00:21:19.590
So on some level, he
kind of destroys

00:21:19.590 --> 00:21:21.260
the notion of a republic.

00:21:21.260 --> 00:21:25.020
So we're ending the Third
Coalition with France is

00:21:25.020 --> 00:21:29.470
dominant in Europe.

00:21:29.470 --> 00:21:34.270
It has gotten all of this
territory mainly from Austria.

00:21:34.270 --> 00:21:37.210
The Holy Roman Empire
no longer exists.

00:21:37.210 --> 00:21:40.380
It's now the Confederation
of the Rhine.

00:21:40.380 --> 00:21:44.110
Napoleon, after Austerlitz,
thinks that he is invincible.

00:21:44.110 --> 00:21:46.580
And he also is now Emperor.

00:21:46.580 --> 00:21:51.120
So all of his visions of
grandeur are coming true.