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Equation with variables on both sides: fractions

Sal solves the equation (3/4)x + 2 = (3/8)x - 4. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • aqualine ultimate style avatar for user preety
    can you multiply the denominator on both sides
    (28 votes)
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    • winston default style avatar for user terranpitt
      If you multiply both sides by an integer, it's always multiplying the numerator and therefore making the number larger. If you did otherwise, by multiplying the denominator, then the number would be smaller, which is rather a division. I hope my explanation makes sense and is helpful.
      (2 votes)
  • aqualine sapling style avatar for user Ward
    I am struggling on how to transfer varibles to one side of the equation. Any tips?
    (20 votes)
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    • aqualine ultimate style avatar for user Kaede Katsuki
      Hi, Rebecca;

      Transferring Variables
      Transferring variables might look like a complex subject to tackle at first glance, but it actually proves itself to be much simpler--you just need to understand it.

      In an equation, the left hand side (LHS--the left expression) and the right hand side (RHS--the right expression) are equal. Now, a very significant tip to take note is that since both sides are equal, both must be treated equally. But how do we do that?

      Here is an example:
      Billy has two baskets of equally filled apples. The left basket has 2 packs of 4 apples and 2 pears while the right basket has 3 packs of 2 apples and 2 pears. On the way home, Billy decided to eat 3 fruits from each basket. How many fruits are left in each basket?
      2(4+2)=3(2+2)

      I've done the equation for the original number of fruits in each basket, but after Billy took 3 from each basket, I am left to modify my equation. But how do we do that?
      2(4+2)-3=3(2+2)-3

      We subtract 3 from both sides! This would mean that both baskets, being originally equal, would still be equal when Billy goes home to eat the rest.
      2(4+2) -3 =3(2+2) -3
      2(6)-3=3(4)-3
      12-3=12-3
      9=9
      ∴ There are 9 fruits left in each basket


      Now what does this have to do with transferring variables? Transferring variables are basically like what we did above, except they're not numbers yet.

      This time, let's say we don't actually know how many pears there are in each pack, considering the fact that the number of pears are equal in all of the packs in both baskets.
      2(4+x)=3(2+x)

      First, let us simplify the equation.
      2(4+x)=3(2+x)
      8+2x=6+3x

      Here we are--transferring variables! This time, think about what we did to the equation when Billy decided to eat 3 fruits from both baskets; we subtract the variable from both sides!
      8+2x=6+3x
      8+2x -3x =6+3x -3x
      8+2x-3x=6

      Notice that when we transfered 3x from the RHS to the LHS, it turned negative. When we transfer variables to the other side, its sign becomes opposite! That's how easy it is!
      Now, let's solve the equation!
      8+2x-3x=6
      2x-3x=6-8
      -x=-2
      -1(-x)=-1(-2)
      x=2
      ∴ There are 2 pears in each pack.

      Ta-da! The same equation!

      (Sorry if I was very lengthened about such a simple subject. I like to explain thoroughly)
      (5 votes)
  • winston default style avatar for user Allison Campbell
    in the test i got the problem...

    16 - 2t = 3/2t + 9

    and i converted the fraction to the decimal 1.5
    so...

    16 - 2t = 1.5t + 9
    -16 -16

    -2t = 1.5t - 7
    -1.5t -1.5t

    0.5t = -7
    i devided both sides by 0.5 and got -14, so i punched the answer in and they said the correct answer was actually 2. what did i do wrong?
    (9 votes)
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    • duskpin ultimate style avatar for user Jerusha Curlin
      hello, so what you did wrong was simply a subtracting mistake. you can totally just convert your fraction into a decimal and it will still work. So lets start from the beginning,

      16 - 2t = 3/2t +9

      so you convert the fraction into the decimal

      16 - 2t = 1.5t + 9
      then you subtracted 16 from both sides which is right,

      16 - 2t = 1.5t +9
      -16 -16

      -2t = 1.5t -7

      you were right up to this step. now we subtract 1.5t from both sides

      -2t = 1.5t -7
      -1.5t -1.5

      you get...
      -3.5t = -7
      which equals 2!

      so you only messed up in the step where you add a -2t to a -1.5t.
      if you do not understand why we add these together look at Khan's video "adding negative numbers example"

      hope this helps
      (25 votes)
  • aqualine tree style avatar for user visanshiro
    I still didn't understand where the 8 came from, can someone please explain it again differently? How do you get to that conclusion?
    (6 votes)
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    • stelly blue style avatar for user Kim Seidel
      The 8 is the lowest common multiple of the 2 denominators (4 and 8). Use the same process you would use to select the smalles common denominator for those 2 fractions.
      Multiples of 4: 4, 8, 12, 16, etc.
      Multiples of 8: 8, 16, 24, etc.
      The first multiple in common is 8.

      Hope this helps.
      (13 votes)
  • blobby green style avatar for user jbautista-celada
    my question is how do you solve this problem with one fraction?
    (6 votes)
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    • leaf orange style avatar for user A/V
      The general rule for solving equations with fractions — whether it be only on one side or both — is to try to get rid of all of them. The most common way to find the lowest common multiple (LCM) of all of the fractions, and then multiply the LCM on both sides of the equations.

      hopefully that helps :)
      (5 votes)
  • blobby green style avatar for user Michael meyer
    so what do you do if there is only 1 fraction on one of the sides for example: 2-1/2n = 3n = 16
    (4 votes)
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  • winston default style avatar for user Maxwell Mike
    how do you calculate a equation with a decimal
    (4 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      One way is to use same types of equation solving steps as you would with whole numbers, but you would need to perform the appropriate operations with the decimals.

      Another way is to multiply everything on both sides of the equation by a large enough power of 10 so that all the decimals become whole numbers.
      (For example multiplying both sides by 100 in the equation 0.15x + 0.3 = 0.2x - 0.45 would give 15x + 30 = 20x - 45, eliminating the decimals).

      Have a blessed, wonderful day!
      (4 votes)
  • blobby green style avatar for user cmann
    Is there a video where only one fraction.
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      I don't know if there is a video with only one fraction. However, the concepts are the same. To eliminate the fraction in the equation, you can multiply the entire equation by the denominator of the fraction. This eliminates the denominator. You would then have an equation with only integers to solve.
      (4 votes)
  • blobby green style avatar for user weirdomamliesa
    how do you do it if there's only one fraction?
    (3 votes)
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  • winston default style avatar for user Maxwell Mike
    can you please help me solve 0.5s+1=7+4.5s
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      Subtract 0.5s from both sides
      Subtract 7 from both sides
      Then divide both sides by the coefficient (the number in front of "s".

      Make sure you follow the correct processes for working with decimal numbers to ensure you decimal points end up in the right position after each math operation.

      Give it a try.
      (4 votes)

Video transcript

We have the equation 3/4x plus 2 is equal to 3/8x minus 4. Now, we could just, right from the get go, solve this the way we solved everything else, group the x terms, maybe on the left-hand side, group the constant terms on the right-hand side. But adding and subtracting fractions are messy. So what I'm going to do, right from the start of this video, is to multiply both sides of this equation by some number so I can get rid of the fractions. And the best number to do it by-- what number is the smallest number that if I multiply both of these fractions by it, they won't be fractions anymore, they'll be whole numbers? That smallest number is going to be 8. I'm going to multiply 8 times both sides of this equation. You say, hey, Sal, how did you get 8? And I got 8 because I said, well, what's the least common multiple of 4 and 8? Well, the smallest number that is divisible by 4 and 8 is 8. So when you multiply by 8, it's going to get rid of the fractions. And so let's see what happens. So 8 times 3/4, that's the same thing as 8 times 3 over 4. Let me do it on the side over here. That's the same thing as 8 times 3 over 4, which is equal to 8 divided by 4 is just 2. So it's 2 times 3, which is 6. So the left-hand side becomes 8 times 3/4x is 6x. And then 8 times 2 is 16. You have to remember, when you multiply both sides, or a side, of an equation by a number, you multiply every term by that number. So you have to distribute the 8. So the left-hand side is 6x plus 16 is going to be equal to-- 8 times 3/8, that's pretty easy, the 8's cancel out and you're just left with 3x. And then 8 times negative 4 is negative 32. And now we've cleaned up the equation a good bit. Now the next thing, let's try to get all the x terms on the left-hand side, and all the constant terms on the right. So let's get rid of this 3x from the right. Let's subtract 3x from both sides to do it. That's the best way I can think of of getting rid of the 3x from the right. The left-hand side of this equation, 6x minus 3x is 3x. 6 minus 3 is 3. And then you have a plus 16 is equal to-- 3x minus 3x, that's the whole point of subtracting 3x, is so they cancel out. So those guys cancel out, and we're just left with a negative 32. Now, let's get rid of the 16 from the left-hand side. So to get rid of it, we're going to subtract 16 from both sides of this equation. Subtract 16 from both sides. The left-hand side of the equation just becomes-- you have this 3x here; these 16's cancel out, you don't have to write anything-- is equal to negative 32 minus 16 is negative 48. So we have 3x is equal to negative 48. To isolate the x, we can just divide both sides of this equation by 3. So let's divide both sides of that equation by 3. The left-hand side of the equation, 3x divided by 3 is just an x. That was the whole point behind dividing both sides by 3. And the right-hand side, negative 48 divided by 3 is negative 16. And we are done. x equals negative 16 is our solution. So let's make sure that this actually works by substituting to the original equation up here. And the original equation didn't have those 8's out front. So let's substitute in the original equation. We get 3/4-- 3 over 4-- times negative 16 plus 2 needs to be equal to 3/8 times negative 16 minus 4. So 3/4 of 16 is 12. And you can think of it this way. What's 16 divided by 4? It is 4. And then multiply that by 3, it's 12, just multiplying fractions. So this is going to be a negative 12. So we get negative 12 plus 2 on the left-hand side, negative 12 plus 2 is negative 10. So the left-hand side is a negative 10. Let's see what the right-hand side is. You have 3/8 times negative 16. If you divide negative 16 by 8, you get negative 2, times 3 is a negative 6. So it's a negative 6 minus 4. Negative 6 minus 4 is negative 10. So when x is equal to negative 16, it does satisfy the original equation. Both sides of the equation become negative 10. And we are done.