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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 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.