WEBVTT

00:00:00.660 --> 00:00:03.880
Arbegla starts to feel
angry and embarrassed

00:00:03.880 --> 00:00:08.130
that he was shown up by you and
the bird in front of the King

00:00:08.130 --> 00:00:10.200
and so he storms
out of the room.

00:00:10.200 --> 00:00:12.390
And then a few seconds
later he storms back in.

00:00:12.390 --> 00:00:13.560
He says, my fault.

00:00:13.560 --> 00:00:14.590
My apologies.

00:00:14.590 --> 00:00:18.420
I realize now what
the mistake was.

00:00:18.420 --> 00:00:22.650
There was a slight, I guess,
typing error or writing error.

00:00:22.650 --> 00:00:25.696
In the first week, when
they went to the market

00:00:25.696 --> 00:00:28.070
and bought two pounds of apples
and one pound of bananas,

00:00:28.070 --> 00:00:30.180
it wasn't a $3 cost.

00:00:30.180 --> 00:00:32.515
It was a $5 cost.

00:00:35.720 --> 00:00:40.760
Now surely considering how smart
you and this bird seem to be,

00:00:40.760 --> 00:00:45.550
you surely could figure out what
is the per pound cost of apples

00:00:45.550 --> 00:00:48.140
and what is the per
pound cost of bananas.

00:00:48.140 --> 00:00:51.270
So you think for a
little bit, is there now

00:00:51.270 --> 00:00:54.550
going to be a solution?

00:00:54.550 --> 00:00:57.440
So let's break it down using
the exact same variables.

00:00:57.440 --> 00:01:00.570
You say, well if a is the
cost of apples per pound

00:01:00.570 --> 00:01:05.519
and b is the cost of bananas,
this first constraint tells us

00:01:05.519 --> 00:01:09.416
that two pounds of apples
are going to cost 2a,

00:01:09.416 --> 00:01:11.420
because it's b
dollars per pound.

00:01:11.420 --> 00:01:13.520
And one pound of
bananas is going

00:01:13.520 --> 00:01:17.730
to cost b dollars because
it's one pound times

00:01:17.730 --> 00:01:20.823
b dollars per pound is
now going to cost $5.

00:01:24.180 --> 00:01:27.720
This is the corrected number.

00:01:27.720 --> 00:01:30.050
And we saw from
the last scenario,

00:01:30.050 --> 00:01:31.760
this information hasn't changed.

00:01:31.760 --> 00:01:36.030
Six pounds of apples is
going to cost 6a, six

00:01:36.030 --> 00:01:38.340
pounds times a
dollars per pound.

00:01:38.340 --> 00:01:42.820
And three pounds of bananas is
going to cost 3b, three pounds

00:01:42.820 --> 00:01:44.800
times b dollars per pound.

00:01:44.800 --> 00:01:46.890
The total cost of the
apples and bananas

00:01:46.890 --> 00:01:50.200
in this trip we
are given is $15.

00:01:52.822 --> 00:01:54.280
So once again, you
say, well let me

00:01:54.280 --> 00:01:58.140
try to solve this maybe
through elimination.

00:01:58.140 --> 00:02:00.470
And once again, you say well
let me cancel out the a's.

00:02:00.470 --> 00:02:01.280
I have 2a here.

00:02:01.280 --> 00:02:02.560
I have 6a here.

00:02:02.560 --> 00:02:05.295
If I multiply the 2a
here by negative 3,

00:02:05.295 --> 00:02:06.990
then this will
become a negative 6a.

00:02:06.990 --> 00:02:09.770
And it might be able to cancel
out with all of this business.

00:02:09.770 --> 00:02:10.820
So you do that.

00:02:10.820 --> 00:02:12.340
You multiply this
entire equation.

00:02:12.340 --> 00:02:13.770
You can't just
multiply one term.

00:02:13.770 --> 00:02:17.210
You have to multiply the entire
equation times negative 3

00:02:17.210 --> 00:02:19.270
if you want the
equation to still hold.

00:02:19.270 --> 00:02:21.330
And so we're multiplying
by negative 3

00:02:21.330 --> 00:02:25.790
so 2a times negative
3 is negative 6a.

00:02:25.790 --> 00:02:29.790
b times negative
3 is negative 3b.

00:02:29.790 --> 00:02:34.930
And then 5 times negative
3 is negative 15.

00:02:34.930 --> 00:02:37.220
And now something
fishy starts to look

00:02:37.220 --> 00:02:38.990
like it's about to happen.

00:02:38.990 --> 00:02:40.470
Because when you
add the left hand

00:02:40.470 --> 00:02:43.960
side of this blue equation
or this purplish equation

00:02:43.960 --> 00:02:46.720
to the green one, you get 0.

00:02:46.720 --> 00:02:50.320
All of these things right
over here just cancel out.

00:02:50.320 --> 00:02:53.900
And on the right hand
side, 15 minus 15,

00:02:53.900 --> 00:02:56.450
that is also equal to 0.

00:02:56.450 --> 00:03:00.922
And you get 0 equals 0, which
seems a little bit better

00:03:00.922 --> 00:03:02.630
than the last time
you worked through it.

00:03:02.630 --> 00:03:04.620
Last time we got 0 equals 6.

00:03:04.620 --> 00:03:07.130
But 0 equals 0 doesn't really
tell you anything about

00:03:07.130 --> 00:03:07.890
the x's and y's.

00:03:07.890 --> 00:03:08.890
This is true.

00:03:08.890 --> 00:03:13.040
This is absolutely true that
0 does definitely equals 0,

00:03:13.040 --> 00:03:16.110
but it doesn't tell you any
information about x and y.

00:03:16.110 --> 00:03:18.309
And so then the bird
whispers in the King's ear,

00:03:18.309 --> 00:03:19.850
and then the King
says, well the bird

00:03:19.850 --> 00:03:21.460
says you should graph
it to figure out

00:03:21.460 --> 00:03:23.450
what's actually going on.

00:03:23.450 --> 00:03:26.440
And so you've learned
that listening to the bird

00:03:26.440 --> 00:03:28.620
actually makes a lot of sense.

00:03:28.620 --> 00:03:32.644
So you try to graph
these two constraints.

00:03:32.644 --> 00:03:33.810
So let's do it the same way.

00:03:33.810 --> 00:03:34.643
We'll have a b axis.

00:03:37.290 --> 00:03:39.170
That's our b axis.

00:03:39.170 --> 00:03:42.980
And we will have our a axis.

00:03:42.980 --> 00:03:46.430
Let we mark off some markers
here-- one, two, three, four,

00:03:46.430 --> 00:03:50.190
five and one, two,
three, four, five.

00:03:50.190 --> 00:03:52.390
So this first equation
right over here,

00:03:52.390 --> 00:03:54.610
if we subtract 2a
from both sides,

00:03:54.610 --> 00:03:57.220
I'm just going to put it
into slope intercept form,

00:03:57.220 --> 00:04:03.940
you get b is equal to
negative 2a plus 5.

00:04:03.940 --> 00:04:06.430
All I did is subtract
2a from both sides.

00:04:06.430 --> 00:04:08.700
And if we were to graph
that, our b-intercept when

00:04:08.700 --> 00:04:10.890
a is equal to 0,
b is equal to 5.

00:04:10.890 --> 00:04:12.390
So that's right over here.

00:04:12.390 --> 00:04:13.770
And our slope is negative 2.

00:04:13.770 --> 00:04:18.079
Every time you add 1 to a--
so if a goes from 0 to 1-- b

00:04:18.079 --> 00:04:19.529
is going to go down by 2.

00:04:19.529 --> 00:04:23.510
So go down by two, go down by 2.

00:04:23.510 --> 00:04:26.970
So this first white
equation looks like this

00:04:26.970 --> 00:04:29.270
if we graph the solution set.

00:04:29.270 --> 00:04:34.900
These are all of the prices
for bananas and apples

00:04:34.900 --> 00:04:37.060
that meet this constraint.

00:04:37.060 --> 00:04:39.520
Now let's graph this
second equation.

00:04:39.520 --> 00:04:44.360
If we subtract 6a
from both sides,

00:04:44.360 --> 00:04:51.220
we get 3b is equal to
negative 6a plus 15.

00:04:51.220 --> 00:04:54.950
And now we could divide
both sides by 3, divide

00:04:54.950 --> 00:04:56.760
everything by 3.

00:04:56.760 --> 00:05:03.280
We are left with b is equal
to negative 2a plus 5.

00:05:03.280 --> 00:05:04.750
Well this is interesting.

00:05:04.750 --> 00:05:07.680
This looks very similar, or
it looks exactly the same.

00:05:07.680 --> 00:05:11.360
Our b-intercept is 5 and
our slope is negative 2a.

00:05:11.360 --> 00:05:16.390
So this is essentially
the same line.

00:05:16.390 --> 00:05:19.112
So these are essentially
the same constraints.

00:05:19.112 --> 00:05:21.320
And so you start to look at
it a little bit confused,

00:05:21.320 --> 00:05:25.510
and you say, OK, I see
why we got 0 equals 0.

00:05:25.510 --> 00:05:28.000
There's actually an infinite
number of solutions.

00:05:28.000 --> 00:05:30.750
You pick any x and then
the corresponding y

00:05:30.750 --> 00:05:32.920
for each of these
could be a solution

00:05:32.920 --> 00:05:34.880
for either of these things.

00:05:34.880 --> 00:05:37.040
So there's an infinite
number of solutions.

00:05:37.040 --> 00:05:39.270
But you start to wonder,
why is this happening?

00:05:39.270 --> 00:05:41.400
And so the bird whispers
again into the King's ear

00:05:41.400 --> 00:05:42.941
and the King says,
well the bird says

00:05:42.941 --> 00:05:46.270
this is because in both
trips to the market

00:05:46.270 --> 00:05:49.110
the same ratio of apples
and bananas was bought.

00:05:49.110 --> 00:05:52.520
In the green trip
versus the white trip,

00:05:52.520 --> 00:05:56.610
you bought three times as many
apples, bought three times

00:05:56.610 --> 00:06:00.460
as many bananas, and you
had three times the cost.

00:06:00.460 --> 00:06:04.560
So in any situation for any
per pound prices of apples

00:06:04.560 --> 00:06:07.750
and bananas, if you
buy exactly three times

00:06:07.750 --> 00:06:10.150
the number of apples, three
times the number bananas,

00:06:10.150 --> 00:06:12.750
and have three
times the cost, that

00:06:12.750 --> 00:06:14.810
could be true for any prices.

00:06:14.810 --> 00:06:18.220
And so this is actually
it's consistent.

00:06:18.220 --> 00:06:22.750
We can't say that
Arbegla is lying to us,

00:06:22.750 --> 00:06:25.440
but it's not giving
us enough information.

00:06:25.440 --> 00:06:27.680
This is what we call, this
is a consistent system.

00:06:27.680 --> 00:06:29.430
It's consistent
information here.

00:06:29.430 --> 00:06:31.270
So let me write this down.

00:06:31.270 --> 00:06:34.370
This is consistent.

00:06:34.370 --> 00:06:36.420
And it is consistent,
0 equals 0.

00:06:36.420 --> 00:06:39.300
There's no shadiness
going on here.

00:06:39.300 --> 00:06:41.060
But it's not enough information.

00:06:41.060 --> 00:06:44.260
This system of
equations is dependent.

00:06:44.260 --> 00:06:45.850
It is dependent.

00:06:45.850 --> 00:06:52.190
And you have an infinite
number of solutions.

00:06:52.190 --> 00:06:55.970
Any point this line
represents a solution.

00:06:55.970 --> 00:06:57.970
So you tell Arbegla,
well, if you really

00:06:57.970 --> 00:07:00.050
want us to figure
this out, you need

00:07:00.050 --> 00:07:01.290
to give us more information.

00:07:01.290 --> 00:07:06.328
And preferably buy a different
ratio of apples to bananas.