Number of solutions to systems of equations
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. In-ter-sect. 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, and says, oh, this character isn't so bad at this algebra stuff.