Systems of equations with elimination: coffee and croissants

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 or 5.30 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 into 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. Since we have all these C's here, I'm just going to use x's and y's. So let's let x be equal to the price of the cup of coffee. And let's let y be equal to the 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 and one croissant for 5.30 Euro. So how would we set that up as an equation? Well, we got one cup of coffee. So that's going to be one x, or we could just write x, plus one y because he got one croissant, and it cost 5.30 Euro. So this equation describes what happened to the local-- bought one cup of coffee, one croissant, paid 5.30 Euro. 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. So we've been charged 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 the most obvious way at least looking at this right over here is you have x, we have 2x, we have y, we have 2y. Let's take this first equation that described 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 euro 60. Now, something very interesting is going on here. If the local had bought 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 and you paid 14. So it looks pretty clear that you got charged a different amount. You got the tourist rate for the cup of coffee and 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. 2x plus 2y is 14. Here, 2x plus 2y is 10 euro 60. And we could even show that mathematically this doesn't make sense. So if we were to subtract this bottom equation from this top, so essentially you could imagine multiplying the entire bottom equation times negative 1. So let's multiply the entire bottom equation by 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 to that side. So on the left hand side, this cancels with this, this cancels with this, we're left with 0. And on the right hand side, 14 minus 10.60 will get you to 3.40. And there's no x and y that you can think of that can all of a sudden make 0 equal 3.40. So there is no solution. And the only explanation over here is that the local was charged a cheaper rate.