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Systems of equations with elimination: coffee and croissants

Video transcript

you are at a Parisian cafe with a friend a local in front of you buys a cup of coffee and a croissant for $5.00 or 530 euro when you and your friend get two cups of coffee and two croissants you are charged 14 euro can we solve for the price of a cup of coffee and croissant using the information in a system of linear equations in two variables if yes what is the solution if no what is the reason we cannot so we're looking for two things the price of a cup of coffee and the price of a croissant so let's define two variables here let's let since we have all these C's here let me was going to use X's and y so let's let X let's let X be equal to the price of the cup of coffee cup of coffee and let's let Y be equal to the price of a croissant price of a croissant so we first have this information of what the local in front of us did the local in front of us buys one cup of coffee in one croissant for 5:30 euro five five five point three euro you could even say so how would we set that up as an equation well I got one cup of coffee so that's going to be one X or we could just write it X plus 1y because you got one croissant and it cost five thirty so it cost let me write this so this is the amount that he paid five thirty euro five thirty so this is what the local this this equation describes would happen to the local bought one cup of coffee one croissant paid five thirty now when you and your friend get two cups of coffee and two croissants you are charged 14 euro so what's an equation to describe this so we should be charged two times the price of a cup of coffee so it should be 2x and then we should be charged two times the price of a croissant so plus 2y and the sum of these should be the total amount that we're charged we've been charged fourteen fourteen euro 14 euro so let's see if we can solve this system of equations and there's many many many ways to solve this but most obvious way at least looking at this right over here as you we have X we have two X we have Y we have two I let's take this first equation that described the local and multiply it by two so let's just multiply it by two so we're going to multiply both sides otherwise equality won't hold anymore so we would get 2x plus 2y is equal to 2 times 5 30 is $10 60 or 10 euro 60 10 euro 60 now something very interesting is going on here if the local had bought just twice as many twice as many cups of coffee and twice as many croissants he would have paid 10 60 and that would have been the exact amount of coffee and croissants you got a new paid 14 so it looks pretty clear that you got charged a different amount you got the tourist rate for the cup of coffee in the croissant while he got the local rate and we can verify that there's no x and y that's going to satisfy this and even logically it makes sense here two times an x plus two times y is 14 here two times an x plus two times y is $10 10 euros 60 and we can even show that mathematically that this doesn't make sense so if we were to subtract this bottom equation from this top so essentially you can imagine multiplying 2 amiss do it this way multiplying the entire bottom equation times a negative 1 so let's multiply the entire bottom equation by a negative 1 and then we add these two equations remember all we're doing is we're starting with say this equation and we're adding the same thing to both sides we're going to add this to this side and we already know that negative 10 60 is the same thing as this we're going to add it to that side so on the left-hand side this cancels with this this cancels with this we're left with zero and on the right-hand side 14 minus 10 60 will get you to 340 and there's no X&Y that you can think of there's no magical X&Y that can all of a sudden make 0 equal 340 so there is no solution and the only explanation over here is that the local was charged a cheaper rate