Inconsistent systems of equations Systems of equations without a solution
Inconsistent systems of equations
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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
- 'cause he's supposed to be the wise man in the kingdom, the king's closest advisor.
- So he steps in and says, "okay, so if you and this bird"
- "are so smart, how about you tackle the Riddle of the Fruit Prices?"
- And the king says, "Yes, that is something that we haven't been able to figure out."
- "The fruit prices. Arbegla, tell them the riddle of the fruit prices."
- And so Arbegla says, "Well,"
- "we wanna keep track of how much our fruit costs, but we forgot"
- "to actually log how much it costs when we went to the market but we know how much in total we spent"
- "we know much we got. We know that one week ago, when we went to"
- "the fruit market, we bought two, two, pounds of"
- "apples, we bought two pounds of apples, and one pound of bananas."
- "one pound, I guess, of bananans, bananas. And the total cost"
- "that, time, was three dollars, so there was three dollars, three dollars"
- in total cost. And then when we went the time before that we went the time before that
- we bought six pounds of bananas or six pounds of apples I should say
- Six pounds of apples. And three pounds three pounds of
- bananas. Ba-nanas. And
- the total cost at that point was fifteen dollars. So
- what is the cost of apples and bananas?
- So you look at the bird:
- The bird looks at you, the bird whispers into the king's ear, and the king says
- Well the bird says we'll just start defining some variables here, so we'll start expressing this thing algebraically
- So you go about doing that. What we want to figure out is the cost of apples and the cost of bananas.
- Per pound. So we set some variables. So let's...
- let a= the cost cost of apples, apple per pound. Per pound
- And let's let b = the cost of bananas.
- Ba-nanas. Bananas per pound. So how could we interpret
- this first information right here? Two pounds of apples and a pound of bananas cost
- $3. So how much are the apples going to cost? Well it's going to cost 2, two pounds times
- the cost per pound, times a, that's going to be the total cost of apples
- in this scenarios, and what's the total cost of the banana? Well it's one pound times the cost
- per pound. So, you're just going to have b, that's the total cost
- of the bananas, cause we know we bought one the total cost of the apples and bananas
- are going to be 2a+b and we know what that total cost is
- it is, it is $3. Now let's do the same thing for the other time we went to the market. Simply
- Six pounds of apple, the total cost is going to be six pounds times A dollars
- per pound and the total cost of banas is going to be
- well we bought three poiund of bananas.
- and the cost per pound is b
- and so the total cost of apples and bananas
- this scenario is going to be = to 15
- is going to be = to $15
- so let's think about how we might want to solve it
- we could use elimination we could use substitution
- whatever we want, we might do it graphically
- let's try it first with elimination.
- so the first thing I might want to do is
- is maybe I want to eliminate let's say I want to eliminate
- the a variable right over here so I have two
- a over here, I have six a over here
- so if I multiply this entire right equation by
- -3 then this 2a would become a -6a and then it might
- be able to cancel out with that
- so let me do that
- let me multiply this entire equation
- times -3
- times negative three
- so -3 * 2a is -6a
- -3 * b is -3b
- and then -3 * 3 is -9
- is -9
- and now we can essentially add the two equations
- or essentially add the left side of this to the left side of that
- or the right side of this equation to the right side of that
- we're essentially adding the same thing to both sides of this equation
- because we know this is equal to that
- So let's do that
- let's do it
- So on the left hand side, 6a and 6a cancel out.
- But something else interesting happens, the 3b and the 3b cancels out as well.
- So we're just left with 0 on the left hand side.
- And on the right hand side, what do we have?
- 15 - 9 = 6.
- So we get this bizarre statement! All of our variables have gone away
- And we're left with this bizarre nonsensical statement that
- 0 = 6, which we know is definitely not the case.
- So what's going on over here? What's going on?
- And then, you you you say, what's going on and you look at the bird
- 'cause the bird seems to be the most knowledgeable person in the room
- or at least the most knowledgeable vertebrate
- in the room. And so the bird whispers into the king's ear
- and the king says, "Well, he says that there's no solution
- and you should at least try to graph it to see why."
- And you say, well, the bird seems to know what he's talking about
- So let me attempt to graph these two equations and see
- what's going on.
- And so what you do is, you take each of the equation
- and you like, when you graph it, you like to put
- it in kind of the y-intercept form or slope intercept form
- and so you do that, so you say, well let me
- solve both of these for b
- so if you want to solve this first equation for b
- you just subtract 2a from both sides
- if you subtract 2a from both sides of this first equation
- you get b is = to -2a
- + 3. Now solve this second equation for b.
- So the first thing you might wanna do is subtract 6a from both sides.
- So you would get, you would get, I'll do it right over, let me do it right over here.
- You would get 3b, 3b is = to -6a plus 15
- and then you can divide both sides by 3
- you get b is = to -2a
- plus, plus 5. So the second equation, let me revert back
- to that other shade of green, is b is = to -2a
- plus 5. And we haven't even graphed it yet, but it looks like
- something interesting is going on.
- They both have the exact same slope
- when you solve in terms, when you solve for b
- but they seem to have different, let's call them, b-intercepts
- let's graph it to actually see what's going on
- so let me get, draw some axes over here, let's call that my b-axis
- and then this could be my, a axis
- And this first equation has a b-intercept of positive 3
- so let's see, one, two, three four five
- the first one has a b-intercept of positive three
- and it has a slope of negative 2
- So you go down or you go to the right one
- you go down two. Go to the right one, you go down two.
- So the line looks something like this.
- I'm trying my best to draw it straight. So it looks
- it looks something something like that
- And I'll just draw this green one.
- This green one, our b-intercept is 5
- so it's right over here. but we have the exact same slope
- the slope of -2, so it looks
- it looks something something like that right over there
- and you immediately see now that the bird was right
- There is no solution because these two constraints
- represent or can be represented by lines that
- don't intersect. So the lines don't, don't intersect.
- They don't intersect, and so the bird is right
- there's no solution, there's no x and y that can make this statement equal true!
- Or that can make 0 = 6, there is no possible, there is no overlap between these two
- things. And so something gets into your brain.
- You realize that Arbegla is trying to stump you.
- And you say, Arbegla, you have given me
- in-con-sistent information!
- This is an in-con-sistent system of equations!
- In. In...con...sistent. Which happens to be the word
- that is sometimes used to refer to a system
- that has no solutions, where the lines do not
- intersect. And there fore this information is incorrect
- We cannot assume that the apple or banana
- Either you are lying, which is possible, or you accounted for it wrong
- Or maybe the prices of apples and bananas actually changed
- between the two visits of the market.
- At which point the bird whispered into the King's ear,
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