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The king's advisor, Arbegla, is watching all of this discourse between you, the king, the bird. And he's starting to feel a little bit jealous

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'cause he's supposed to be the wise man in the kingdom, the king's closest advisor.

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So he steps in and says, "okay, so if you and this bird"

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"are so smart, how about you tackle the Riddle of the Fruit Prices?"

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And the king says, "Yes, that is something that we haven't been able to figure out."

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"The fruit prices. Arbegla, tell them the riddle of the fruit prices."

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And so Arbegla says, "Well,"

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"we wanna keep track of how much our fruit costs, but we forgot"

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"to actually log how much it costs when we went to the market but we know how much in total we spent"

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"we know much we got. We know that one week ago, when we went to"

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"the fruit market, we bought two, two, pounds of"

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"apples, we bought two pounds of apples, and one pound of bananas."

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"one pound, I guess, of bananans, bananas. And the total cost"

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"that, time, was three dollars, so there was three dollars, three dollars"

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in total cost. And then when we went the time before that we went the time before that

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we bought six pounds of bananas or six pounds of apples I should say

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Six pounds of apples. And three pounds three pounds of

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bananas. Ba-nanas. And

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the total cost at that point was fifteen dollars. So

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what is the cost of apples and bananas?

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So you look at the bird:

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The bird looks at you, the bird whispers into the king's ear, and the king says

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Well the bird says we'll just start defining some variables here, so we'll start expressing this thing algebraically

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So you go about doing that. What we want to figure out is the cost of apples and the cost of bananas.

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Per pound. So we set some variables. So let's...

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let a= the cost cost of apples, apple per pound. Per pound

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And let's let b = the cost of bananas.

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Ba-nanas. Bananas per pound. So how could we interpret

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this first information right here? Two pounds of apples and a pound of bananas cost

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$3. So how much are the apples going to cost? Well it's going to cost 2, two pounds times

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the cost per pound, times a, that's going to be the total cost of apples

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in this scenarios, and what's the total cost of the banana? Well it's one pound times the cost

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per pound. So, you're just going to have b, that's the total cost

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of the bananas, cause we know we bought one the total cost of the apples and bananas

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are going to be 2a+b and we know what that total cost is

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it is, it is $3. Now let's do the same thing for the other time we went to the market. Simply

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Six pounds of apple, the total cost is going to be six pounds times A dollars

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per pound and the total cost of banas is going to be

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well we bought three poiund of bananas.

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and the cost per pound is b

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and so the total cost of apples and bananas

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this scenario is going to be = to 15

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is going to be = to $15

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so let's think about how we might want to solve it

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we could use elimination we could use substitution

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whatever we want, we might do it graphically

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let's try it first with elimination.

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so the first thing I might want to do is

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is maybe I want to eliminate let's say I want to eliminate

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the a variable right over here so I have two

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a over here, I have six a over here

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so if I multiply this entire right equation by

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-3 then this 2a would become a -6a and then it might

54
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be able to cancel out with that

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so let me do that

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let me multiply this entire equation

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times -3

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times negative three

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so -3 * 2a is -6a

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-3 * b is -3b

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and then -3 * 3 is -9

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is -9

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and now we can essentially add the two equations

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or essentially add the left side of this to the left side of that

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or the right side of this equation to the right side of that

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we're essentially adding the same thing to both sides of this equation

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because we know this is equal to that

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So let's do that

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let's do it

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So on the left hand side, 6a and 6a cancel out.

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But something else interesting happens, the 3b and the 3b cancels out as well.

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So we're just left with 0 on the left hand side.

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And on the right hand side, what do we have?

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15 - 9 = 6.

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So we get this bizarre statement! All of our variables have gone away

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And we're left with this bizarre nonsensical statement that

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0 = 6, which we know is definitely not the case.

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So what's going on over here? What's going on?

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And then, you you you say, what's going on and you look at the bird

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'cause the bird seems to be the most knowledgeable person in the room

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or at least the most knowledgeable vertebrate

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in the room. And so the bird whispers into the king's ear

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and the king says, "Well, he says that there's no solution

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and you should at least try to graph it to see why."

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And you say, well, the bird seems to know what he's talking about

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So let me attempt to graph these two equations and see

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what's going on.

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And so what you do is, you take each of the equation

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and you like, when you graph it, you like to put

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it in kind of the y-intercept form or slope intercept form

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and so you do that, so you say, well let me

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solve both of these for b

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so if you want to solve this first equation for b

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you just subtract 2a from both sides

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if you subtract 2a from both sides of this first equation

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you get b is = to -2a

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+ 3. Now solve this second equation for b.

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So the first thing you might wanna do is subtract 6a from both sides.

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So you would get, you would get, I'll do it right over, let me do it right over here.

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You would get 3b, 3b is = to -6a plus 15

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and then you can divide both sides by 3

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you get b is = to -2a

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plus, plus 5. So the second equation, let me revert back

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to that other shade of green, is b is = to -2a

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plus 5. And we haven't even graphed it yet, but it looks like

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something interesting is going on.

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They both have the exact same slope

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when you solve in terms, when you solve for b

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but they seem to have different, let's call them, b-intercepts

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let's graph it to actually see what's going on

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so let me get, draw some axes over here, let's call that my b-axis

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and then this could be my, a axis

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And this first equation has a b-intercept of positive 3

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so let's see, one, two, three four five

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the first one has a b-intercept of positive three

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and it has a slope of negative 2

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So you go down or you go to the right one

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you go down two. Go to the right one, you go down two.

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So the line looks something like this.

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I'm trying my best to draw it straight. So it looks

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it looks something something like that

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And I'll just draw this green one.

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This green one, our b-intercept is 5

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so it's right over here. but we have the exact same slope

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the slope of -2, so it looks

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it looks something something like that right over there

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and you immediately see now that the bird was right

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There is no solution because these two constraints

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represent or can be represented by lines that

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don't intersect. So the lines don't, don't intersect.

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In-ter-sect.

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They don't intersect, and so the bird is right

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there's no solution, there's no x and y that can make this statement equal true!

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Or that can make 0 = 6, there is no possible, there is no overlap between these two

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things. And so something gets into your brain.

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You realize that Arbegla is trying to stump you.

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And you say, Arbegla, you have given me

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in-con-sistent information!

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This is an in-con-sistent system of equations!

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In. In...con...sistent. Which happens to be the word

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that is sometimes used to refer to a system

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that has no solutions, where the lines do not

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intersect. And there fore this information is incorrect

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We cannot assume that the apple or banana

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Either you are lying, which is possible, or you accounted for it wrong

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Or maybe the prices of apples and bananas actually changed

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between the two visits of the market.

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At which point the bird whispered into the King's ear,

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and says, oh, this character isn't so bad at this algebra stuff.