1
00:00:00,660 --> 00:00:03,880
Arbegla starts to feel
angry and embarrassed

2
00:00:03,880 --> 00:00:08,130
that he was shown up by you and
the bird in front of the King

3
00:00:08,130 --> 00:00:10,200
and so he storms
out of the room.

4
00:00:10,200 --> 00:00:12,390
And then a few seconds
later he storms back in.

5
00:00:12,390 --> 00:00:13,560
He says, my fault.

6
00:00:13,560 --> 00:00:14,590
My apologies.

7
00:00:14,590 --> 00:00:18,420
I realize now what
the mistake was.

8
00:00:18,420 --> 00:00:22,650
There was a slight, I guess,
typing error or writing error.

9
00:00:22,650 --> 00:00:25,696
In the first week, when
they went to the market

10
00:00:25,696 --> 00:00:28,070
and bought two pounds of apples
and one pound of bananas,

11
00:00:28,070 --> 00:00:30,180
it wasn't a $3 cost.

12
00:00:30,180 --> 00:00:32,515
It was a $5 cost.

13
00:00:35,720 --> 00:00:40,760
Now surely considering how smart
you and this bird seem to be,

14
00:00:40,760 --> 00:00:45,550
you surely could figure out what
is the per pound cost of apples

15
00:00:45,550 --> 00:00:48,140
and what is the per
pound cost of bananas.

16
00:00:48,140 --> 00:00:51,270
So you think for a
little bit, is there now

17
00:00:51,270 --> 00:00:54,550
going to be a solution?

18
00:00:54,550 --> 00:00:57,440
So let's break it down using
the exact same variables.

19
00:00:57,440 --> 00:01:00,570
You say, well if a is the
cost of apples per pound

20
00:01:00,570 --> 00:01:05,519
and b is the cost of bananas,
this first constraint tells us

21
00:01:05,519 --> 00:01:09,416
that two pounds of apples
are going to cost 2a,

22
00:01:09,416 --> 00:01:11,420
because it's b
dollars per pound.

23
00:01:11,420 --> 00:01:13,520
And one pound of
bananas is going

24
00:01:13,520 --> 00:01:17,730
to cost b dollars because
it's one pound times

25
00:01:17,730 --> 00:01:20,823
b dollars per pound is
now going to cost $5.

26
00:01:24,180 --> 00:01:27,720
This is the corrected number.

27
00:01:27,720 --> 00:01:30,050
And we saw from
the last scenario,

28
00:01:30,050 --> 00:01:31,760
this information hasn't changed.

29
00:01:31,760 --> 00:01:36,030
Six pounds of apples is
going to cost 6a, six

30
00:01:36,030 --> 00:01:38,340
pounds times a
dollars per pound.

31
00:01:38,340 --> 00:01:42,820
And three pounds of bananas is
going to cost 3b, three pounds

32
00:01:42,820 --> 00:01:44,800
times b dollars per pound.

33
00:01:44,800 --> 00:01:46,890
The total cost of the
apples and bananas

34
00:01:46,890 --> 00:01:50,200
in this trip we
are given is $15.

35
00:01:52,822 --> 00:01:54,280
So once again, you
say, well let me

36
00:01:54,280 --> 00:01:58,140
try to solve this maybe
through elimination.

37
00:01:58,140 --> 00:02:00,470
And once again, you say well
let me cancel out the a's.

38
00:02:00,470 --> 00:02:01,280
I have 2a here.

39
00:02:01,280 --> 00:02:02,560
I have 6a here.

40
00:02:02,560 --> 00:02:05,295
If I multiply the 2a
here by negative 3,

41
00:02:05,295 --> 00:02:06,990
then this will
become a negative 6a.

42
00:02:06,990 --> 00:02:09,770
And it might be able to cancel
out with all of this business.

43
00:02:09,770 --> 00:02:10,820
So you do that.

44
00:02:10,820 --> 00:02:12,340
You multiply this
entire equation.

45
00:02:12,340 --> 00:02:13,770
You can't just
multiply one term.

46
00:02:13,770 --> 00:02:17,210
You have to multiply the entire
equation times negative 3

47
00:02:17,210 --> 00:02:19,270
if you want the
equation to still hold.

48
00:02:19,270 --> 00:02:21,330
And so we're multiplying
by negative 3

49
00:02:21,330 --> 00:02:25,790
so 2a times negative
3 is negative 6a.

50
00:02:25,790 --> 00:02:29,790
b times negative
3 is negative 3b.

51
00:02:29,790 --> 00:02:34,930
And then 5 times negative
3 is negative 15.

52
00:02:34,930 --> 00:02:37,220
And now something
fishy starts to look

53
00:02:37,220 --> 00:02:38,990
like it's about to happen.

54
00:02:38,990 --> 00:02:40,470
Because when you
add the left hand

55
00:02:40,470 --> 00:02:43,960
side of this blue equation
or this purplish equation

56
00:02:43,960 --> 00:02:46,720
to the green one, you get 0.

57
00:02:46,720 --> 00:02:50,320
All of these things right
over here just cancel out.

58
00:02:50,320 --> 00:02:53,900
And on the right hand
side, 15 minus 15,

59
00:02:53,900 --> 00:02:56,450
that is also equal to 0.

60
00:02:56,450 --> 00:03:00,922
And you get 0 equals 0, which
seems a little bit better

61
00:03:00,922 --> 00:03:02,630
than the last time
you worked through it.

62
00:03:02,630 --> 00:03:04,620
Last time we got 0 equals 6.

63
00:03:04,620 --> 00:03:07,130
But 0 equals 0 doesn't really
tell you anything about

64
00:03:07,130 --> 00:03:07,890
the x's and y's.

65
00:03:07,890 --> 00:03:08,890
This is true.

66
00:03:08,890 --> 00:03:13,040
This is absolutely true that
0 does definitely equals 0,

67
00:03:13,040 --> 00:03:16,110
but it doesn't tell you any
information about x and y.

68
00:03:16,110 --> 00:03:18,309
And so then the bird
whispers in the King's ear,

69
00:03:18,309 --> 00:03:19,850
and then the King
says, well the bird

70
00:03:19,850 --> 00:03:21,460
says you should graph
it to figure out

71
00:03:21,460 --> 00:03:23,450
what's actually going on.

72
00:03:23,450 --> 00:03:26,440
And so you've learned
that listening to the bird

73
00:03:26,440 --> 00:03:28,620
actually makes a lot of sense.

74
00:03:28,620 --> 00:03:32,644
So you try to graph
these two constraints.

75
00:03:32,644 --> 00:03:33,810
So let's do it the same way.

76
00:03:33,810 --> 00:03:34,643
We'll have a b axis.

77
00:03:37,290 --> 00:03:39,170
That's our b axis.

78
00:03:39,170 --> 00:03:42,980
And we will have our a axis.

79
00:03:42,980 --> 00:03:46,430
Let we mark off some markers
here-- one, two, three, four,

80
00:03:46,430 --> 00:03:50,190
five and one, two,
three, four, five.

81
00:03:50,190 --> 00:03:52,390
So this first equation
right over here,

82
00:03:52,390 --> 00:03:54,610
if we subtract 2a
from both sides,

83
00:03:54,610 --> 00:03:57,220
I'm just going to put it
into slope intercept form,

84
00:03:57,220 --> 00:04:03,940
you get b is equal to
negative 2a plus 5.

85
00:04:03,940 --> 00:04:06,430
All I did is subtract
2a from both sides.

86
00:04:06,430 --> 00:04:08,700
And if we were to graph
that, our b-intercept when

87
00:04:08,700 --> 00:04:10,890
a is equal to 0,
b is equal to 5.

88
00:04:10,890 --> 00:04:12,390
So that's right over here.

89
00:04:12,390 --> 00:04:13,770
And our slope is negative 2.

90
00:04:13,770 --> 00:04:18,079
Every time you add 1 to a--
so if a goes from 0 to 1-- b

91
00:04:18,079 --> 00:04:19,529
is going to go down by 2.

92
00:04:19,529 --> 00:04:23,510
So go down by two, go down by 2.

93
00:04:23,510 --> 00:04:26,970
So this first white
equation looks like this

94
00:04:26,970 --> 00:04:29,270
if we graph the solution set.

95
00:04:29,270 --> 00:04:34,900
These are all of the prices
for bananas and apples

96
00:04:34,900 --> 00:04:37,060
that meet this constraint.

97
00:04:37,060 --> 00:04:39,520
Now let's graph this
second equation.

98
00:04:39,520 --> 00:04:44,360
If we subtract 6a
from both sides,

99
00:04:44,360 --> 00:04:51,220
we get 3b is equal to
negative 6a plus 15.

100
00:04:51,220 --> 00:04:54,950
And now we could divide
both sides by 3, divide

101
00:04:54,950 --> 00:04:56,760
everything by 3.

102
00:04:56,760 --> 00:05:03,280
We are left with b is equal
to negative 2a plus 5.

103
00:05:03,280 --> 00:05:04,750
Well this is interesting.

104
00:05:04,750 --> 00:05:07,680
This looks very similar, or
it looks exactly the same.

105
00:05:07,680 --> 00:05:11,360
Our b-intercept is 5 and
our slope is negative 2a.

106
00:05:11,360 --> 00:05:16,390
So this is essentially
the same line.

107
00:05:16,390 --> 00:05:19,112
So these are essentially
the same constraints.

108
00:05:19,112 --> 00:05:21,320
And so you start to look at
it a little bit confused,

109
00:05:21,320 --> 00:05:25,510
and you say, OK, I see
why we got 0 equals 0.

110
00:05:25,510 --> 00:05:28,000
There's actually an infinite
number of solutions.

111
00:05:28,000 --> 00:05:30,750
You pick any x and then
the corresponding y

112
00:05:30,750 --> 00:05:32,920
for each of these
could be a solution

113
00:05:32,920 --> 00:05:34,880
for either of these things.

114
00:05:34,880 --> 00:05:37,040
So there's an infinite
number of solutions.

115
00:05:37,040 --> 00:05:39,270
But you start to wonder,
why is this happening?

116
00:05:39,270 --> 00:05:41,400
And so the bird whispers
again into the King's ear

117
00:05:41,400 --> 00:05:42,941
and the King says,
well the bird says

118
00:05:42,941 --> 00:05:46,270
this is because in both
trips to the market

119
00:05:46,270 --> 00:05:49,110
the same ratio of apples
and bananas was bought.

120
00:05:49,110 --> 00:05:52,520
In the green trip
versus the white trip,

121
00:05:52,520 --> 00:05:56,610
you bought three times as many
apples, bought three times

122
00:05:56,610 --> 00:06:00,460
as many bananas, and you
had three times the cost.

123
00:06:00,460 --> 00:06:04,560
So in any situation for any
per pound prices of apples

124
00:06:04,560 --> 00:06:07,750
and bananas, if you
buy exactly three times

125
00:06:07,750 --> 00:06:10,150
the number of apples, three
times the number bananas,

126
00:06:10,150 --> 00:06:12,750
and have three
times the cost, that

127
00:06:12,750 --> 00:06:14,810
could be true for any prices.

128
00:06:14,810 --> 00:06:18,220
And so this is actually
it's consistent.

129
00:06:18,220 --> 00:06:22,750
We can't say that
Arbegla is lying to us,

130
00:06:22,750 --> 00:06:25,440
but it's not giving
us enough information.

131
00:06:25,440 --> 00:06:27,680
This is what we call, this
is a consistent system.

132
00:06:27,680 --> 00:06:29,430
It's consistent
information here.

133
00:06:29,430 --> 00:06:31,270
So let me write this down.

134
00:06:31,270 --> 00:06:34,370
This is consistent.

135
00:06:34,370 --> 00:06:36,420
And it is consistent,
0 equals 0.

136
00:06:36,420 --> 00:06:39,300
There's no shadiness
going on here.

137
00:06:39,300 --> 00:06:41,060
But it's not enough information.

138
00:06:41,060 --> 00:06:44,260
This system of
equations is dependent.

139
00:06:44,260 --> 00:06:45,850
It is dependent.

140
00:06:45,850 --> 00:06:52,190
And you have an infinite
number of solutions.

141
00:06:52,190 --> 00:06:55,970
Any point this line
represents a solution.

142
00:06:55,970 --> 00:06:57,970
So you tell Arbegla,
well, if you really

143
00:06:57,970 --> 00:07:00,050
want us to figure
this out, you need

144
00:07:00,050 --> 00:07:01,290
to give us more information.

145
00:07:01,290 --> 00:07:06,328
And preferably buy a different
ratio of apples to bananas.