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Setting up a system of linear equations example (weight and price)

Practice writing a system of linear equations that fits the constraints in a word problem.

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  • piceratops ultimate style avatar for user 𝓜𝓪𝓱𝓮𝓼𝓱 𝓜𝓪𝓱𝓮𝓷𝓭𝓻𝓪𝓴𝓪𝓻
    In D + L = 80, we have total no. of kilograms of beans but in 3D + 2L = 220 we are representing the total amount of money required to buy the beans, then how can we solve them if both the equations have different meanings?
    (10 votes)
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  • piceratops tree style avatar for user Harmony
    I solved like this.
    We have 2 equations.

    1. D+L= 80
    2. 3D+ 2L= 220

    And find 'D' from the first equation.
    D+L= 80
    D= 80-L

    Then, replace D in equation 2.
    3D+ 2L= 220
    3(80-L) + 2L = 220
    240-3L+2L = 220
    240-L = 220
    -L= 220-240
    L= 20

    If L = 20 and L+D = 80,
    D= 60

    What about you? How do you solve?
    (8 votes)
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  • blobby green style avatar for user flyboy
    For anyone trying to solve this. Here is one approach as mentioned previously; you can solve this by elimination of one variable so

    D + L = 80
    3D + 2L = 220

    Lets try to eliminate one variable so that we can then plug it in and solve for the other variable.

    You can add two equations since they have the same variables and go from there.

    By looking at it you can see that by adding to equations you are not able to eliminate either D or L, for example adding the two gets you to 4D + 3L = 300

    Since that’s the case you can do the following and multiply one or both equations with a constant to the point of being able to eliminate one variable.

    Lets pick first equation and get it to the point of being able to eliminate 2L in the second equation. For that we would need to multiply by -2, which give us:

    -2D -2L = -160

    Now we can add two equations:

    3D+2L=220
    +
    -2D-2L=-160
    =
    D=60


    So D=60, following that

    D + L = 80
    60 + L = 80
    L = 20

    D = 60
    L = 20

    I am surprised elimination is not covered first for this problem. Looks like this topic is covered later however
    (4 votes)
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  • male robot donald style avatar for user DK Lam
    And so how do you solve it (final answer)?
    (3 votes)
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  • mr pink red style avatar for user jkritenbrink
    i got D=60kg and L=20kg
    (3 votes)
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  • duskpin tree style avatar for user Vladimir
    So how would you solve this?
    (2 votes)
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  • mr pants teal style avatar for user Qasim Hashmi
    So there is this question:

    Elliott has some yarn that she wants to use to make hats and scarves. Each hat uses 0.20 kilograms of yarn and each scarf uses 0.10 kilograms of yarn. Elliott wants to use twice as much yarn for scarves as for hats, and she wants to make a total of 20 items.

    Let "h" be the number of hats Elliott makes and "s" be the number of scarves she makes.

    Which system of equations represents this situation?

    So it said these 2 equations:

    H + S = 20
    .1s = 2 x .2h

    So I understand why the first equation makes sense, but I didn't quite understand the second one
    (.1s = 2 x .2h). So in the answer, it said:

    Each hat uses 0.20 kilograms of yarn, and h is the number of hats Elliott makes, so 0.2h represents how much yarn Elliott uses to make hats.
    Similarly, each scarf uses 0.10 kilograms of yarn, and s is the number of scarves Elliott makes, so 0.1s represents how much yarn Elliott uses to make scarves.
    Since she wants to use twice as much yarn for scarves as for hats, we need 0.1s to be twice as much as 0.2h:
    0.1s = 2 x 0.2h

    So my question is this. It says she wants to use TWICE as much yarn for scarves as for hats, so shouldn't the equation by 0.1s x 2 = 0.2h because she is using twice as much yarn for SCARVES as for hats? Makes sense? please answer asap!
    (2 votes)
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  • area 52 blue style avatar for user james potter
    For anyone who wanted to know the solution, I got D = 60 and L = 20 by graphing.
    (2 votes)
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  • primosaur tree style avatar for user Hithanth
    What is the difference between equations? Both equations have a total is there a way be can make a equation with both the weight and cost?
    (0 votes)
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  • leafers ultimate style avatar for user Elli
    you said Karen not Lauren lol!
    (0 votes)
    Default Khan Academy avatar avatar for user

Video transcript

- [Instructor] In this video, we're going to get a little bit of practice setting up systems of linear equations based on a word problem. We're not actually going to end up solving it. You can do that if you like just for kicks. But really we're going to just focus on setting it up. So here we're told Lauren uses a blend of dark roast beans and light roast beans to make coffee at her cafe. She needs 80 kilograms of beans in total for her next order. Dark roast beans cost $3 per kilogram, light roast beans cost $2 per kilogram, and someone wants to spend $220 total. Let D be the number of kilograms of dark roast beans she buys and L be the number of kilograms of light roast beans she buys. All right, so based on this information that we've been given, see if you can pause this video and set up a system of equations. And it's going to have two equations with two unknowns, D and L, that in theory we could solve to figure out the right number of kilograms of dark roast beans and light roast beans that Karen should use. So pause this video and try to work that out. All right, now let's do it together. And what I'm going to do is I'm going to underline. So let's see, we know that D is dark roast beans and the L is the number of kilograms of light roast beans. And then they tell us here, they say she needs 80 kilograms of beans in total. So what we could say is, hey, the number of kilograms of dark roast beans plus the number of kilograms of light roast beans needs to be equal to 80 kilograms. So the number of kilograms of dark roast beans plus the number of kilograms of light roast beans. (laughs) I'm having trouble saying light roast beans. Well, this, what I just underlined here, it says needs to be 80 kilograms in total. So that needs to be 80. So this number of kilograms plus this number of kilograms is going to be equal to your total number of kilograms. All right, so I have one equation with two unknowns. Let's see if we can get another one. So next, they say dark roast beans cost $3 per kilogram, light roast beans cost $2 per kilogram, and she wants to spend $220 total. So what I just underlined in this aquamarine color we can set up another equation with. And if you haven't already set up your system of equations, see if you can now do that. See if you can set up this second equation. Pause the video. All right, well, the way to think about it is we just have to have an expression for how much did she spend on dark roast beans, how much did she spend on light roast beans, and then we need to add those two together, and that needs to be equal to $220 'cause that's how much she wants to spend in total. So how much does she spend on dark roast beans? Well, it's going to be the number of kilograms of dark roast beans that she buys, and it says that it cost $3 per kilogram, so we're gonna multiply it by three. $3 per kilogram times the number of kilograms of dark roast beans. This is how much she spends on dark roast beans. And so how much is she going to spend on light roast beans? Well, she buys L kilograms of light roast beans. They told us that there. And they cost $2 per kilogram, so $2 per kilogram times the number of kilograms. This is how much she spends on light roast beans. So you add how much she spends on dark roast to how much she spends on light roast, and so this is going to be $220 in total. And there you have it. We have our two equations with two unknowns, and so now we could go and solve it, but you can do that outside of this video. But the whole point of this video is to understand how to construct these based on the constraints, based on the information that we see in this. So typically when you're trying to set these up, there's often a sentence or two that will focus on one equation. So this first one is saying, hey, the kilograms, let's add those up for the total number of kilograms. And then there's another sentence or two that'll focus on, in this case, some other equation. In this case, it's the price. So the price of the dark plus the price of the light is going to be equal to the total amount she wants to spend.