Systems of equations with elimination: King's cupcakes

After you cross the troll's bridge and you save the prince or princess, you return them back to their father, the King. And he's so excited that you returned their son or daughter to him that he wants to throw a brunch in your honor. But he has a little bit of a conundrum in throwing the brunch. He wants to figure out how many cupcakes should he order? He doesn't want to waste any, but he wants to make sure that everyone has enough to eat. And you say, well what's the problem here? And he says, well I know adults eat a different number of cupcakes than children eat. And I know that in my kingdom, an adult will always eat the same amount and a child will always eat the same amount. And so you say, King, well what information can you give me? I might be able to help you out a little bit. You're feeling very confident after this troll situation. And he says, well I know at the last party we had 500 adults and we had 200 children, and combined they ate 2,900 cupcakes. And you say, OK, that's interesting. But I think I'll need a little bit more information. Have you thrown parties before then? And the king says, of course I have. I like to throw parties. Well what happened at the party before that? And he says, well there we also had 500 adults and we had 300 children. And you say, well how many cupcakes were eaten at that party? And he says, well we know it was 3,100 cupcakes. And so you get a tingling feeling that a little bit of algebra might apply over here. And you say, well let me see. What do we need to figure out? We need to figure out the number of cupcakes on average that an adult will eat. So number of cupcakes for an adult. And we also need to figure out the number of cupcakes for a child. So these are the two things that we need to figure out because then we can know how many adults and children are coming to the next branch that are being held in your honor and get the exact right number of cupcakes. So those are things you're trying to figure out. And we don't know what those things are. Let's define some variables that represent those things. Well let's do a for adults. Let's let a equal the number of cupcakes that, on average, each adult eats. And let's do c for children. So c is the number of cupcakes for a child on average. So given that information, let's see how we can represent what the King has told us algebraically. So let's think about this orange information first. How could we represent this algebraically? Well let's think about how many cupcakes the adults ate at that party. You had 500 adults, and on average, each of them exactly a cupcakes. So the total number of cupcakes that the adults ate were 500 times a. How many did the children eat? Well same logic. You had 200 children and they each ate c cupcakes. So 200 times c is the total number of cupcakes that the children ate. Well how much did they eat in total? Well it's the total number that the adults ate plus the total number that the children ate which is 2,900 cupcakes. So let's do that and apply that same logic to the blue party right over here, this blue information. How can we represent this algebraically? Well once again, how many total cupcakes did the adults eat? Well, you had 500 adults and they each ate a cupcakes, which is an unknown right now. And then what about the children? Well you had 300 children and they each ate c cupcakes. And so if you add up all the cupcakes that the adults ate plus all the cupcakes that the children ate, you get to 3,100 cupcakes. So this is starting to look interesting. I have two equations. I have a system of two equations with two unknowns. And you know from your experience with the troll that you should be able to solve this. You could solve it graphically like you did in the past, but now you feel that there could be another tool in your tool kit which is really just an application of the algebra that you already know. So think a little bit about how you might do this. So let's rewrite this first equation right over here. So we have 500a plus 200c is equal to 2,900 cupcakes. Now, it would be good if we could get rid of this 500a somehow. Well you might say, well let me just subtract 500a. So you might say, oh, I just want to subtract 500a. But if you subtracted 500a from the left hand side, you'd also have to subtract 500a from the right hand side. And so the a wouldn't just disappear. It would just end up on the right hand side, and you would still have one equation with two unknowns which isn't too helpful. But you see something interesting. You're like, well this is a 500a here. What if I subtracted a 500a and this 300c? So if I subtracted the 500a and the 300c from the left hand side. And you're like, well why is that useful? You're going to do the same thing on the right hand side and then you're going to have an a and a c on the right hand. And you just say, hold on, hold on one second here. Hold on, I guess you're talking to yourself. Hold on one second. I'm subtracting the left hand side of this equation, but this left hand side is the exact same thing as this right hand side. So here I could subtract 500a and 300c, and I could do 500a and subtract 300c over here. But we know that subtracting 500a and 300c, that's the exact same thing as subtracting 3,100. Let me make it clear. This is 500a minus 300c is the exact same thing as subtracting 500a plus 300c. And we know that 500a plus 300c is exactly 3,100. This is 3,100. This is what the second information gave us. So instead of subtracting 500a and minus 300c, we can just subtract 3,100. So let me do that. This is exciting. So let me clear that out. So let's clear that out. And so here, instead of doing this, I can subtract the exact same value, which we know is 3,100, subtract 3,100. So looking at it this way, it looks like we're subtracting this bottom equation from the top equation, but we're really just subtracting the same thing from both sides. This is just very basic algebra here. But if we do that, let's see what happens. So on the left hand side, 500a minus 500a, those cancel out. 200c minus 300c, that gives us negative 100c. And on the right hand side, 2,900 minus 3,100 is negative 200. Well now we have one equation with one unknown, and we know how to solve this. We can divide both sides by negative 100. These cancel out. And then over here, you end up with a positive 2. So c is equal to positive 2. So we've solved one of the unknowns, the each child on average drinks two cups. So c is equal to 2. So how can we figure out what a is? Well now we can take this information and go back into either one of these and figure out what a has to be. So let's go back into the orange one right over here and figure out what a has to be. So we had 500a plus 200c, but we know what c is, c is 2. So 200 times 2 is equal to 2,900. And now we just have to solve for a, one equation with one unknown. So we have 500a, 200 times 2 is 400, plus 400 is equal to 2,900. We can subtract 400 from both sides of this equation. Let me do that. Subtracting 400, and we are left with this cancels out. And on the left hand side, we have 500a. This is very exciting. We're in the home stretch. On the right hand side, you have 2,500. 500a equals 2,500. We can divide both sides by 500, and we are left with 2,500 divided by 500 is just 5. So you have a is equal to 5 and you're done. You have solved the King's conundrum. Each child on average drinks 2 cups of water-- sorry, not cups of water. I don't know where I got that from. Each child 2 cupcakes and each adult will eat 5 cupcakes. a is equal to 5. And so based on how many adults and children are coming to the brunch in your honor, you now know exactly the number of cupcakes that the King needs to order.