Infinite solutions to systems

Arbegla starts to feel
angry and embarrassed

that he was shown up by you and
the bird in front of the King

and so he storms
out of the room.

And then a few seconds
later he storms back in.

He says, my fault.

My apologies.

I realize now what
the mistake was.

There was a slight, I guess,
typing error or writing error.

In the first week, when
they went to the market

and bought two pounds of apples
and one pound of bananas,

it wasn't a $3 cost.

It was a $5 cost.

Now surely considering how smart
you and this bird seem to be,

you surely could figure out what
is the per pound cost of apples

and what is the per
pound cost of bananas.

So you think for a
little bit, is there now

going to be a solution?

So let's break it down using
the exact same variables.

You say, well if a is the
cost of apples per pound

and b is the cost of bananas,
this first constraint tells us

that two pounds of apples
are going to cost 2a,

because it's a
dollars per pound.

And one pound of
bananas is going

to cost b dollars because
it's one pound times

b dollars per pound is
now going to cost $5.

This is the corrected number.

And we saw from
the last scenario,

this information hasn't changed.

Six pounds of apples is
going to cost 6a, six

pounds times a
dollars per pound.

And three pounds of bananas is
going to cost 3b, three pounds

times b dollars per pound.

The total cost of the
apples and bananas

in this trip we
are given is $15.

So once again, you
say, well let me

try to solve this maybe
through elimination.

And once again, you say well
let me cancel out the a's.

I have 2a here.

I have 6a here.

If I multiply the 2a
here by negative 3,

then this will
become a negative 6a.

And it might be able to cancel
out with all of this business.

So you do that.

You multiply this
entire equation.

You can't just
multiply one term.

You have to multiply the entire
equation times negative 3

if you want the
equation to still hold.

And so we're multiplying
by negative 3

so 2a times negative
3 is negative 6a.

b times negative
3 is negative 3b.

And then 5 times negative
3 is negative 15.

And now something
fishy starts to look

like it's about to happen.

Because when you
add the left hand

side of this blue equation
or this purplish equation

to the green one, you get 0.

All of these things right
over here just cancel out.

And on the right hand
side, 15 minus 15,

that is also equal to 0.

And you get 0 equals 0, which
seems a little bit better

than the last time
you worked through it.

Last time we got 0 equals 6.

But 0 equals 0 doesn't really
tell you anything about

the x's and y's.

This is true.

This is absolutely true that
0 does definitely equals 0,

but it doesn't tell you any
information about x and y.

And so then the bird
whispers in the King's ear,

and then the King
says, well the bird

says you should graph
it to figure out

what's actually going on.

And so you've learned
that listening to the bird

actually makes a lot of sense.

So you try to graph
these two constraints.

So let's do it the same way.

We'll have a b axis.

That's our b axis.

And we will have our a axis.

Let we mark off some markers
here-- one, two, three, four,

five and one, two,
three, four, five.

So this first equation
right over here,

if we subtract 2a
from both sides,

I'm just going to put it
into slope intercept form,

you get b is equal to
negative 2a plus 5.

All I did is subtract
2a from both sides.

And if we were to graph
that, our b-intercept when

a is equal to 0,
b is equal to 5.

So that's right over here.

And our slope is negative 2.

Every time you add 1 to a--
so if a goes from 0 to 1-- b

is going to go down by 2.

So go down by two, go down by 2.

So this first white
equation looks like this

if we graph the solution set.

These are all of the prices
for bananas and apples

that meet this constraint.

Now let's graph this
second equation.

If we subtract 6a
from both sides,

we get 3b is equal to
negative 6a plus 15.

And now we could divide
both sides by 3, divide

everything by 3.

We are left with b is equal
to negative 2a plus 5.

Well this is interesting.

This looks very similar, or
it looks exactly the same.

Our b-intercept is 5 and
our slope is negative 2a.

So this is essentially
the same line.

So these are essentially
the same constraints.

And so you start to look at
it a little bit confused,

and you say, OK, I see
why we got 0 equals 0.

There's actually an infinite
number of solutions.

You pick any x and then
the corresponding y

for each of these
could be a solution

for either of these things.

So there's an infinite
number of solutions.

But you start to wonder,
why is this happening?

And so the bird whispers
again into the King's ear

and the King says,
well the bird says

this is because in both
trips to the market

the same ratio of apples
and bananas was bought.

In the green trip
versus the white trip,

you bought three times as many
apples, bought three times

as many bananas, and you
had three times the cost.

So in any situation for any
per pound prices of apples

and bananas, if you
buy exactly three times

the number of apples, three
times the number bananas,

and have three
times the cost, that

could be true for any prices.

And so this is actually
it's consistent.

We can't say that
Arbegla is lying to us,

but it's not giving
us enough information.

This is what we call, this
is a consistent system.

It's consistent
information here.

So let me write this down.

This is consistent.

And it is consistent,
0 equals 0.

There's no shadiness
going on here.

But it's not enough information.

This system of
equations is dependent.

It is dependent.

And you have an infinite
number of solutions.

Any point this line
represents a solution.

So you tell Arbegla,
well, if you really

want us to figure
this out, you need

to give us more information.

And preferably buy a different
ratio of apples to bananas.