Systems of equations number of solutions: fruit prices (2 of 2)

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 b 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.