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8th grade (Eureka Math/EngageNY)

Unit 4: Lesson 4

Topic D: Systems of linear equations and their solutions

Systems of equations with substitution: shelves

Sal solves a system of equations to figure out how long a couple of different shelves will be. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user audrapierceall
    At about , I don't understand how -1-1/2 becomes -3/2. I have tried to do it over and over in my head and it still makes no sense to me as to why 1/2 - 1/2 equals 3/2. Can someone please explain, thoroughly.
    (72 votes)
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  • blobby green style avatar for user Jonah  Smith
    Jenny mixes a 30% saline solution with a 50% saline solution to make 800 milliliters of a 45% saline solution. How many milliliters of each solution does she use?
    (4 votes)
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  • primosaur ultimate style avatar for user Jordankfoltz
    Is there a quicker way to do this?
    (3 votes)
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  • blobby green style avatar for user Devlina Bhattacharjee
    By selling 12 dozen pencils, Alice lost an amount equal to the selling price of 6 pencils. Find the lost percent.
    (2 votes)
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    • orange juice squid orange style avatar for user lichr19
      Ok. So how many pencils do you have in total? If a dozen is 12, and you have 12 dozens, you can multiply it out and reach 144. Now she lost a certain percentage of money that is equal to 6 pencils. Now, you just find out how much 6/144 is, but with a denominator of 100. If you simplify this fraction (6/144), you will get 1/24. So, you would end up with 0.04166666 (you get the idea, it's infinite 6's), and you could say that it is approximately 4.16666 %. Hope this helped!
      (2 votes)
  • marcimus purple style avatar for user Navya
    This is for those of you that need a challenge :D Here is a systems of equations word problem that I found online.

    The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number.
    (2 votes)
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    • mr pink red style avatar for user andrewp18
      The digits of a two digit number must be integers so this greatly reduces the number of possibilities. First, what pairs add up to 7?
      1 + 6
      2 + 5
      3 + 4
      And the reverses of these. Immediately we can see that the number is either 16, 25, or 34, and cannot be 61, 52, or 43 because the question says when the digits are reversed the number INCREASES so the original number must have a smaller tens digit than units digit.
      61 is 45 greater than 16.
      52 is 27 greater than 25.
      So the number is 25.
      (1 vote)
  • mr pink red style avatar for user Olivia is a kid named Olivia
    Is there any practice for this topic?
    (2 votes)
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  • blobby green style avatar for user Beth Rivera Dungca
    a small plane leave san jose airport and flies to los angeles at 240 miles per hour. a jet plane leaves san jose airport 30 minutes later to follow small plane at 360 miles per hour. how long does it take the jet to overtake to the small plane
    (3 votes)
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    • leafers ultimate style avatar for user Derek Muller
      I got a different answer being one hour. So let us say that y is the distance from the airport and x the amount of hours so for the small plane the equation would be 240x + 240 * 0.5 = y (i got 0.5 because 30 minutes = 0.5 hours) and the equation for the jet being 360x = y. So then you set them equal so 360x = 240x +240 * 0.5. simplify it 360x = 240x + 120. get x on one side 120x = 120. and get x alone. x=1 so just one hour to pass it.
      (0 votes)
  • aqualine seed style avatar for user milesabennett45
    why are there no practice problems for the word problem
    (2 votes)
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  • blobby green style avatar for user joann909
    help me solve this problem two linear problem 3x+y =2 3x+2y=7
    (1 vote)
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    • hopper cool style avatar for user Chuck Towle
      Joann,
      When you have two equations like
      3x+y =2 and
      3x+2y=7
      And you want to use the substitution method, you first need to isolate either variable by itself on one side of one of the equations.
      In this case, the easiest way is to isolate the y on the right in the first equation.
      3x+y =2
      Subtract 3x from both sides.
      3x-3x+y = 2-3x
      The 3x-3x becomes 0 and disappears
      y=2-3x
      Now you can use that information and just
      substitute (2-3x) for the y in the second equation
      3x+2y=7
      Substitute (2-3x) for the y
      3x + 2(2-3x) = 7
      Now you can find the solution for x

      Then you can use that answer for x by putting it in either equation for the x and solve that equation for y

      Then you can check you answer by putting the answers for both x and y into the other equation and see that it balances correctly.

      I hope that helps make it click for you.
      (2 votes)
  • piceratops tree style avatar for user A.J.011312
    Please break this down for me, I am stuck.
    You invest a total of $5800 in two investments earning 3.5% and 5.5% simple interest. Your goal is to have a total annual interest income of $283. Write a system of linear equations that represents this situation where x represents the amount invested in the 3.5% fund and y represents the amount invested in the 5.5% fund. Solve this system of to determine the smallest amout that you can invest at 5.5% in order to meet your objective.
    (1 vote)
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    • leaf green style avatar for user reliew
      Step 1: Write down what you know:
      1. We know there are 2 investments
      2. We know that the total dollars invested in those investments = 5,800
      3. We know that one investment, represented by the letter x earns 3.5% interest
      4. We know that the other investment is represented by the letter y, and earns 5.5% interest
      5. We know that they are asking us to solve for the total dollars invested in investment y

      Step 2: Come up with equations with the information that was given:
      Dollars invested in first investment = x
      Dollars invested in second investment = y
      Equation 1: x + y = total dollars invested = 5,800
      Equation 2: 0.035x + 0.055y = 283,
      I find it easier working with whole numbers, so I will multiplying both sides of
      equation 2 by 1000 to get 35x + 55y = 283,000

      Step 3: Isolate one of the variables, then use substitution to solve:
      We are looking for y, so we need to substitute x for y, using the first equation, we get
      x = 5800 - y, substituting that value for x into the second equation, we get
      35(5800 - y) + 55y = 283,000, or 203,000 -35y + 55y = 283,000, or
      20y = 80,000, y = $4000. So, $4000 was invested at 5.5%
      and $1800 (5,800 - 4,000) was invested at 3.5%.
      4000 * 0.055 + 1800 * 0.035 = 283. It checks.
      Hope that helps and good luck with your studies!
      (2 votes)

Video transcript

So we have the question, Devon is going to make 3 shelves for her father. She has a piece of lumber that is 12 feet long. She wants the top shelf to be half a foot shorter than the middle shelf. So let me do this in different colors. She wants the top shelf to be half a foot shorter than the middle shelf. So I'll just-- let's just read the whole thing first. And the bottom shelf to be half a foot shorter than twice the length of the top shelf. Let me do that in a different color. I'll do that in blue. The bottom shelf to be half a foot shorter than twice the length of the top shelf. How long will each shelf be if she uses the entire 12 feet of wood? So let's define some variables for our different shelves, because that's what we have to figure out. We have the top shelf, the middle shelf, and the bottom shelf. So let's say that t is equal to length of top shelf-- t for top-- of top shelf. Let's make m equal the length of the middle shelf. m for middle. And then let's make b equal to the length of the bottom shelf-- b for bottom-- bottom shelf. So let's see what these different statements tell us. So this first statement, she says she wants the top shelf-- and I'll do it in that same color-- she wants the top shelf to be 1/2 a foot shorter than the middle shelf. So she wants the length of the top shelf to be-- so this is equal to 1/2 a foot shorter than the middle shelf. So, if we're doing everything in feet, it's going to be the length of the middle shelf in feet minus 1/2, minus 1/2 feet. So that's what that sentence in orange is telling us. The top shelf needs to be 1/2 a foot shorter than the length of the middle shelf. Now, what does the next statement tell us? And the bottom shelf to be-- so the bottom shelf needs to be equal to 1/2 a foot shorter than-- so it's 1/2 a foot shorter than twice the length of the top shelf. So it's 1/2 a foot shorter than twice the length of the top shelf. These are the two statements interpreted in equal equation form. The top shelf's length has to be equal to the middle shelf's length minus 1/2. It's 1/2 foot shorter than the middle shelf. And the bottom shelf needs to be 1/2 a foot shorter than twice the length of the top shelf. And so how do we solve this? Well, you can't just solve it just with these two constraints, but they gave us more information. They tell us how long will each shelf be if she uses the entire 12 feet of wood? So the length of all of the shelves have to add up to 12 feet. She's using all of it. So t plus m, plus b needs to be equal to 12 feet. That's the length of each of them. She's using all 12 feet of the wood. So the lengths have to add to 12. So what can we do here? Well, we can get everything here in terms of one variable, maybe we'll do it in terms of m, and then substitute. So we already have t in terms of m. We could, everywhere we see a t, we could substitute with m minus 1/2. But here we have b in terms of t. So how can we put this in terms of m? Well, we know that t is equal to m minus 1/2. So let's take, everywhere we see a t, let's substitute it with this thing right here. That is what t is equal to. So we can rewrite this blue equation as, the length of the bottom shelf is 2 times the length of the top shelf, t, but we know that t is equal to m minus 1/2. And if we wanted to simplify that a little bit, this would be that the bottom shelf is equal to-- let's distribute the 2-- 2 times m is 2m. 2 times negative 1/2 is negative 1. And then minus another 1/2. Or, we could rewrite this as b is equal to 2 times the middle shelf minus 3/2. Right? 1/2 is 2/2 minus another 1/2 is negative 3/2, just like that. So now we have everything in terms of m, and we can substitute back here. So the top shelf-- instead of putting a t there, we could put m minus 1/2. So we put m minus 1/2, plus the length of the middle shelf, plus the length of the bottom shelf. Well, we already put that in terms of m. That's what we just did. This is the length of the bottom shelf in terms of m. So instead of writing b there, we could write 2m minus 3/2. Plus 2m minus 3/2, and that is equal to 12. All we did is substitute for t. We wrote t in terms of m, and we wrote b in terms of m. Now let's combine the m terms and the constant terms. So if we have, we have one m here, we have another m there, and then we have a 2m there. They're all positive. So 1 plus 1, plus 2 is 4m. So we have 4m. And then what do our constant terms tell us? We have a negative 1/2, and then we have a negative 3/2. So negative 1/2 minus 3/2, that is negative 4/2 or negative 2. So we have 4m minus 2. And, of course, we still have that equals 12. Now, we want to isolate just the m variable on one side of the equation. So let's add 2 to both sides to get rid of this 2 on the left-hand side. So if we add 2 to both sides of this equation, the left-hand side, we're just left with 4m-- these guys cancel out-- is equal to 14. Now, divide both sides by 4, we get m is equal to 14 over 4, or we could call that 7/2 feet, because we're doing everything in feet. So we solved for m, but now we still have to solve for t and b. So let's do that. Let's solve for t. t is equal to m minus 1/2. So it's equal to-- our m is 7/2 minus 1/2, which is equal to 6/2, or 3 feet. Everything is in feet, so that's how we know it's feet there. So that's the top shelf is 3 feet. The middle shelf is 7/2 feet, which is the same thing as 3 and 1/2 feet. And then the bottom shelf is 2 times the top shelf, minus 1/2. So what's that going to be equal to? That's going to be equal to 2 times 3 feet-- that's what the length of the top shelf is-- minus 1/2, which is equal to 6 minus 1/2, or 5 and 1/2 feet. And we're done. And you can verify that these definitely do add up to 12. 5 and 1/2 plus 3 and 1/2 is 9, plus 3 is 12 feet, and it meets all of the other constraints. The top shelf is 1/2 a foot shorter than the middle shelf, and the bottom shelf is 1/2 a foot shorter than 2 times the top shelf. And we are done. We know the lengths of the shelves that Devon needs to make.