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Systems of linear equations word problems — Harder example

Watch Sal work through a harder Systems of linear equations word problem.

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  • aqualine ultimate style avatar for user Jashan Nagra
    Agnes has 23 collectible stones, all of which are labradorite crystals or galena crystals. Labradorite crystals are worth $20 each, while galena crystals are worth $13 each. Agnes earns $439 by selling her entire collection. How many stones of each type did she sell?


    I'm still stuck on this one problem....please help!
    (17 votes)
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  • female robot amelia style avatar for user amna amna
    good concept but a little bit confusing
    (25 votes)
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  • duskpin sapling style avatar for user Paulina B.
    Why are we given harder practice problems then what the example shows? This one looks more simple, but the other practice problems are so complicated sometimes. The SAT is coming soon, I just feel so under prepared..
    (15 votes)
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  • blobby green style avatar for user najatmasri5
    you can solve it with less time and effort. you can substract 4 children and 1 adult from each choice and see if the remaining children are double the adults since each of the remaining adults brought 2 children. think smarter not harder haha
    (12 votes)
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  • duskpin seedling style avatar for user Charlotte Dyball
    how can we use this for graphs?
    (6 votes)
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  • blobby green style avatar for user danielzutto
    Hi, is this problem you immediately knew that there will be only 1 adult with two kids so you did the math of 4+2x2=8.
    I thought, however, that there will be two adults (parents) as my fist logical thing to think.
    how could one understand that there is only one adult for each 2 kids?

    thanks!
    (4 votes)
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  • blobby green style avatar for user ayesha
    I just solved this question by constructing the following equations and got the same answer

    4 + 2(a-1)= c
    2c + 4a= 60
    (5 votes)
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  • scuttlebug green style avatar for user Tasya Adzkiya
    Here is my way of solving it:
    We know that they charge $2 for each child and $4 for each adult and the total ticket sales from the children and adults was $60.

    2c + 4a = 60

    We know that there is an adult that brought 4 children and the remaining adults brought 2 children each.

    c = 4 + 2 (a-1)

    Let's substitute that into the first equation.
    => 2c + 4a = 60
    => 2 (4 + 2[a-1]) + 4a = 60
    => 2 (4 + 2a - 2) + 4a = 60
    => 2 (2 + 2a) + 4a = 60
    => 4 + 4a + 4a = 60
    => 4 + 8a = 60 Subtract 4 from both side of the equation to isolate the term with the variable
    => 8a = 56 Divide both side of the equation by 8
    => a = 7

    Now substitute that information to find out the number of tickets for children.
    2c + 4a = 60
    2c + 4(7) = 60
    2c + 28 = 60 Subtract 28 from both side of the equation
    2c = 32 Divide both side of the equation by 2
    c = 16
    (5 votes)
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  • aqualine ultimate style avatar for user AngelicaR
    Hello,

    Why is 4 + 4 x 2 is 12? If 4 plus 4 = 8, 8 times 2 is 16?
    (2 votes)
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  • blobby green style avatar for user elaninichole1
    Agnes has 23 collectible stones, all of which are labradorite crystals or galena crystals. Labradorite crystals are worth $20 each, while galena crystals are worth $13 each. Agnes earns $439 by selling her entire collection. How many stones of each type did she sell?

    im stuck on this .
    (1 vote)
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    • piceratops ultimate style avatar for user Hecretary Bird
      This type of problem is pretty common on the SAT, and test to see if you can set up and solve a system of equations. What you want to do is group the information you have into two groups, from which you can create two equations. In this problem, you are given the total number of stones, the price per stone of each stone, and the total price of the collection. Based on the last sentence, you know that your variables should be the number of stones. Everything that talks about the prices seems related. If L is the number of Labradorite crystals and G is the number of galena crystals, you can represent the prices as this:
      20L + 13G = 439
      This is because multiplying the unit price per crystal by the amount of crystal will give you the money you would get for all the crystals of that type. Add this to the amount of money you would get for the other crystal type and you have the total.
      We can set up another equation because we know how many crystals in total Agnes has, which is just x + y = 23.
      Now you have the two equations set up, and it's time to solve them. This is a pretty necessary skill for the SAT, so if you don't know it already, I'd recommend you check out some videos on how the process works (https://www.khanacademy.org/math/algebra-basics/alg-basics-systems-of-equations). But, as always on SAT math, setting up this problem and figuring out how to do it is the actual hard part, and the actual doing it is more automatic.
      You should end up getting 20 Labradorite crystals and 3 Galena crystals.
      (5 votes)

Video transcript

- [Instructor] Tickets for a play were $2 for each child and $4 for each adult. At one showing of the play, one adult brought four children and the remaining adults brought two children each. The total ticket sales from the children and adults was $60. How many children and adults attended the play? Alright, this is an interesting one. Okay, so let's just think about how much we've spent at the play and we know it has to add up to $60. Let's think about it in terms of the children and the adults and their admissions. So you have this one adult right over here that brought four children. So how much is that adult, how much is this family? Let's just assume it's a family. How much are they going to spend? Well, that one adult is going to spend $4 for their own ticket and then four children at $2 each. So plus four children times $2 per child, this is going to be $8 for the children's tickets plus $4 on theirs. They're going to spend $12. So that adult is going to spend $12. And then there's some remaining number of adults that brought two children. So let's just say r is I could say the remaining number of adults or the number of adults with two children. Adults with two children, that's r. So each of these adults with two children, how much are they going to spend? Well, they're each going to spend $4 on their own ticket for the adult and then they're gonna have two children at $2 each, so they're gonna spend $4 on the children's tickets. So they're gonna spend $8 in total. So each of these adults with two children is gonna spend $8 at the play and there's r of them. So they're going to spend $8 for each of these adults with two children and there are r of them. So this is the total amount of ticket sales from the adults with two children. And we add that to the ticket sales from this one adult with the four children and they're gonna have to add up to $60. So this is gonna have to add up to $60. Let's see, we can subtract 12 from both sides and so on the left, we'll be left with 8r is equal to 60 minus 12 is 48. Divide both sides by eight and you get r is equal to six. So we wanna be very careful, you might say, okay there was six adults these are just the adults with two children. There's six adults with two children, but there's another adult. There's another adult who brought four children. So there's a total of seven adults, seven adults total. Pardon my handwriting. Seven adults total. So we could just look at these choices, only one of these choices have seven adults. And we could verify that this would also amount to 16 children because this person up here, in magenta, they would bring four children, so you would have four children plus six adults brought two children, so six adults bringing two children each, that would amount to 12 children. And that indeed, that indeed does add up to be 16. And if you're under time pressure, you can see there is only one choice that has seven adults, so you could just pick that one.