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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 elimination: apples and oranges

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

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  • blobby green style avatar for user Lysskiss24
    Why was 11 turned into -11 for the bottom problem? I understand the chages to the 5x & 5y but dont understand the change is it because 10 is less than 11?
    (9 votes)
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    • leafers seedling style avatar for user Fieso Duck
      The 11 was "turned into" -11, by an operation which prepares the whole equation for the elemination process used later on in order to solve this problem. Essentialy, the whole equation 5x+ 5y = 11 is multiplied by -1 and becomes -5x -5y = -11. This "new equation" still has the same meaning it had before you multiplied it by -1, because you did the same thing to both sides. If you had just multiplied the left side (5x + 5y) by -1, you would have changed the meaning of this equation and the results for x any y you got after the elemination process would have been wrong.
      (13 votes)
  • blobby green style avatar for user Røbęrt Ęštrådå
    three times the larger of two numbers is equal to four times the smaller. the sum of the numbers is 21. find the numbers
    (5 votes)
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    • hopper cool style avatar for user Chuck Towle
      Robert,
      You need to convert the words to equations.
      The first sentence is
      "three times the larger of two numbers is equal to four times the smaller."
      First convery the number to number symbols.
      "3 times the larger of two numbers is equal to 4 times the smaller."
      The words times can be changed to "*"
      "3 * the larger of two numbers is equal to 4 * the smaller."
      The words "is equal to" can be replaced with an "=" sign.
      "3 * the larger of two numbers = 4 * the smaller."
      Your unknowns are "the larger of two numbers" and "the smaller"
      Let's use the variable L for "the larger " and S for the smaller
      3 * L = 4 * S

      The second sentence is
      "The sum of the numbers is 21"
      Sum means add and "the numbers are the same unknowns in the first sentence L and S so
      L+S is 21 The is means equals so
      L+S = 21

      So your two equations are
      3L = 4S
      L+S = 21

      I hope that helps make it click for you.
      (6 votes)
  • blobby green style avatar for user mathletejainil
    At , I could not undeerstand why Sal multiplied the bottom equation -1.
    (4 votes)
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    • starky ultimate style avatar for user Ron Joniak
      Sal multiplies the bottom equation by -1 to combine the equations and CANCEL out the x values. When he multiples the entire bottom equation by -1 it does NOT change the value of the equation, so he can do this and then add together the equations.

      Because he now has a -5x value and a 5x value these two variables cancel out when the equations are added together leaving only the y variable. With one variable left, we can now solve for y.

      When you have solved for y using this method, you can then plug that value of y back into an equation and solve for x.

      The key here is to narrow the equation down to only one variable somehow. We cannot solve the equation if there is two unknowns.
      (4 votes)
  • aqualine ultimate style avatar for user Veronica.Garcia
    I still don't get how to find the other answer do we simpfly?
    (4 votes)
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    • duskpin ultimate style avatar for user Sonya
      You plug in what you got for the first answer into the other variable. Say you are solving 3x+y=10 and you know that y=3. All you have to do is this- 3x+3=10 and solve from there. Simplifying is only necessary at the end when you are dividing.
      (2 votes)
  • blobby green style avatar for user Julia Igel
    the sum of the two digits in a two digit number equal 7 when reversed the new number is 45 more than the original number what is the new number
    (3 votes)
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    • mr pants teal style avatar for user Wrath Of Academy
      It isn't straightforward to express that in terms of a linear system of equations. This is a different class of problem. There are only 6 numbers to try though that have two digits with digits adding to 7. Just start at 16 and work your way up from there, testing 16, 25, 34, 43, 52, 61. Of course I immediately see that 16 works, so you only have to try one number.
      (3 votes)
  • aqualine sapling style avatar for user 19LoyolaN
    Lisa weighs 4 times as much as her baby sister. Together they weigh 95 pounds. How much does each person weigh? How would this equation be set up?
    (3 votes)
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  • piceratops ultimate style avatar for user Saikousoku
    Is 5x=6?
    5x+4y=10
    5x+5y=11
    so 5x could be 6 and y could be 1
    (1 vote)
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    • leaf red style avatar for user ls16
      Yes, You can subtract the first problem from the second problem. You get 5x+5y-5x-4y= 11-10. You are left with 0x-1y=1 or simply y=1. Substitute 1 for y in either equation- I will use the 1st for an example- 5x+(4*1)=10 or 5x+4=10. Subtract 4 from either side and yes you are left with 5x=6 as you asked above. . Solving for x you divide each side by 5 and get x=6/5.
      (4 votes)
  • leaf orange style avatar for user Matt Peachey
    Hello, after a student watches a video. where does he go to "try out" some questions that are similiar to the video?
    (1 vote)
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  • starky seedling style avatar for user Brayden Atchley
    so how do I graph this?
    (2 votes)
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  • blobby green style avatar for user washburngs
    How much does it cost to run an 800 watt microwave for 17 hours if the power company charges $.11 per Kilowatt-hour?
    (3 votes)
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Video transcript

You've gone to a fruit stand to get some fresh produce. You notice that the person in front of you gets 5 apples and 4 oranges for $10. You get 5 apples and 5 oranges for $11. Can we solve for the price of an apple and an orange using this information in 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 trying to figure out the price of an apple and the price of an orange. So I would use a for apple, but I don't like using o for orange because o looks too much like a zero. So I'll just say x for apples. Let's let x equal the price of apples. And let's let y equal the price of oranges. So let's describe what happened to the person in line in front of us. They bought 5 apples. So how much did they spend on apples? Well, they bought 5 apples times x dollars per apple, so they spent 5x dollars on their 5 apples. And they bought 4 oranges. They bought 4 oranges times y dollars per orange. So they spent 4y on oranges. So the total amount that they spent is 5x plus 4y. And they tell us that this is $10. This is equal to $10. Now, you get in line, and you buy 5 apples. So you buy 5 apples, just like the guy in front of you. And you paid x dollars per apple. So you're going to pay 5 apples times the price per apple. This is the amount that you spend on apples. And then you buy 5 oranges. So you're going to pay 5 oranges times the price per orange, which is y. So this is how much you spend on oranges. This is how much you spend on apples and oranges, the sum. And they tell us that this is $11. So can we solve for an x and a y? And it looks like we can. And a big giveaway right over here is the ratio between the x's and the y's in these two equations are different. So we're getting some information here. If the ratios were exactly the same, if this was 5x plus 4y right over here, and we got a different number, then we would be in trouble. Because we bought the same combination, but we got a different price. But the good thing is is that we have a different combination here. So let's see if we can work it out. Now, the most obvious thing that jumps out at me is that I have a 5x here, and I have a 5x right over here. So if I could subtract this 5x from that 5x, then I would cancel out all of the x terms. So what I'm going to do is I'm going to multiply this bottom equation by negative 1. So it becomes negative 5x plus negative 5y is equal to negative 11. And then I'm going to essentially add both of these equations. And I could do that because I'm doing the same thing to both sides. I already know that this thing is equal to this thing. So I'm just adding those things to either side. So on the left hand side, I have 5x minus 5x. Well, those cancel out. And then I have 4y minus 5y. Well, that's negative y. And that's going to be equal to 10 minus 11, which is negative 1. And then if we multiply both sides of this times negative 1, or divide both sides by negative 1, we're going to get y is equal to 1. So just like that, we were able to figure out the price of oranges. It's $1 per orange. So this is equal to 1. Now let's figure out the price of apples. So we can go back into either 1 of these equations. I'll go back into this first 1. So 5 times-- so let's go to the person in line in front of us. They bought 5 apples at x dollars per apple, plus 4 oranges at $1 per orange, and they spent a total of $10. So this of course is just 4. Let's subtract 4 from both sides, and we get-- well, 4 times 1 minus 4, that just cancels out. We're just going to be left with 5x on the left hand side. And on the right hand side, we have 10 minus 4, which is equal to 6. And we can just divide both sides by 6 now in order to solve for x. Oh, sorry. We can divide both sides by 5 in order to solve for x. It's late in the day. Brain isn't working. Dividing by 6 wouldn't have done anything. We would have gotten 5/6x. We just want to get an x here. So dividing both sides by 5, we get x is equal to 6/5 dollars. Or you could say that x is equal to 6/5, which is the same thing as 1 and 1/5, which is the same thing as $1.20. So it's $1 per orange, and $1.20 per apple. So we absolutely could figure out the prices of apples and oranges using the information given.