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Systems of equations with elimination: coffee and croissants

Sal solves a word problem about the price of coffee and croissant by creating a system of equations and solving it. Created by Sal Khan.

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  • leaf red style avatar for user Jack McClelland
    Couldn't you just look at it and say: 5.3 * 2 is not equal to 14!
    (25 votes)
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  • spunky sam blue style avatar for user emipanda
    I've mastered this skill before and watched each video like 5,000,000 times ,but I got it wrong on a mastery and now i can't seem to get ANY of the questions right. Does anyone have any tips ?
    (15 votes)
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    • aqualine ultimate style avatar for user Vinhee Cho
      THIS IS GOING TO BE LONG..... I'll vote up your question.
      So, before I get all scientific. That happened to me too! So, you mastered tis skill already, and watched it like 5,000,000 times, right. Well, what happens in your brain is all that information is getting stored. When, you watched it a lot and mastered the skill, the brain says " Okay, empinda knows this." And stops receiving information, and the the other things that you learned are disappearing, because you are practicing it too much. Now, that doesn't work for everything you study., What do you need too do in order to walk? Well, you have to know how to stand up, and move. That is simple so the brain does not forget about it, even if you practice it 5,00,000 times, you do it every single day, so the brain knows you need this, but solving equations you have to set up the equation, cancel it out, plug it in, and solve, etc. so, everytime you solve equation, your brain goes though all that, and soon your brain can't take it so, it disappears. And you probably learned more stuff after you watched it 5,000,000 times so as new stuff come in old stuff go out. Simple stuff, or important stuff don't go out.
      Tip: don't study it that much, or find out a easier way!
      (20 votes)
  • starky sapling style avatar for user Road Runner
    How did 14 become 10.60?
    (4 votes)
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    • old spice man green style avatar for user Philippe Sawyer
      14 didn't become 10,60.
      We're comparing two situations with the same variable so it can be a little bit tricky but if it can help you :

      Cup of coffee bought by you : X1
      Croissant bought by you : Y1
      Cup of coffee bought by the local : X2
      Croisant bought by the local: Y2

      You know that X2 + Y2 = 5,30
      and : 2x1 + 2y1 = 14

      You want to know if you paid the same price as the local. To verify this, X1 should = x2 and y1 should = y2, or in the video we say 2x1 = 2x2 and 2y1 = 2y2.
      We then put two DIFFERENT equation to see if we can conclude something about it.
      (5 votes)
  • piceratops ultimate style avatar for user Emily
    Can I call this system of equations "inconsistent" ?
    Thanks in advance.
    (3 votes)
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  • blobby green style avatar for user mateodesantafe69
    I’m sad you spent so much time on these really easy problems and hardly any time on the very difficult distance/ & rate problems. I struggle with distance and rate problems but you have many more examples of the much simpler problems. Why is that?
    (5 votes)
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  • marcimus pink style avatar for user rsudhesh75
    In the problem I think it should be worded differently. Like I paused the video and tried it and I said no not possible cause it seems like that it says one coffee and one croissant for 5.3 euros. Not total of 5.3 euros. Or am I reading this wrong?
    (3 votes)
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    • mr pink green style avatar for user David Severin
      So how did you interpret the statement "a cup or coffee and a croissant for 5.3 euros"? I would have seen this and immediately said let f be coffee and c be croissant, so 1f + 1c = 5.3. And often is an addition word in Math. Yes total would have been clearer but I cannot figure out any alternate interpretation.
      (0 votes)
  • piceratops sapling style avatar for user Mahmoud Farrag
    why wouldnt u just ask how much it costed
    (2 votes)
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  • leafers tree style avatar for user Hyper_Claws
    Couldn't you just look at it and say: 5.3 * 2 is not equal to 14!
    (1 vote)
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  • aqualine ultimate style avatar for user su
    You have used substitution and elimination methods to solve the pairs of linear equations but what about the cross-multiplication method? How will I use that?
    (1 vote)
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  • aqualine ultimate style avatar for user ZaneDave01
    Why does Sal write 'let' before he defines his variables?
    (1 vote)
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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.