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Equation with variables on both sides: fractions

To solve the equation (3/4)x + 2 = (3/8)x - 4, we first eliminate fractions by multiplying both sides by the least common multiple of the denominators. Then, we add or subtract terms from both sides of the equation to group the x-terms on one side and the constants on the other. Finally, we solve and check as normal. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • duskpin ultimate style avatar for user ♉ flamebird360 ♉
    so it is x=-16?
    (136 votes)
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    • starky sapling style avatar for user Oreo
      Yes, but as I saw the comment from a year ago the same thought popped up in my head. Where are you know. Your comment was from 9 years ago, and your description saws you were 14 so you would have to be 23, or so. Time flies, wow
      (77 votes)
  • aqualine sapling style avatar for user Ward
    I am struggling on how to transfer varibles to one side of the equation. Any tips?
    (30 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?

      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?

      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
      ∴ 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.

      First, let us simplify the equation.

      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 -3x =6+3x -3x

      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!
      ∴ 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)
      (91 votes)
  • aqualine ultimate style avatar for user preety
    can you multiply the denominator on both sides
    (32 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.
      (7 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

    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?
    (16 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
      (28 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?
    (12 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 smallest 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.
      (21 votes)
  • piceratops tree style avatar for user nanotyrannus
    But arent what you do to one side you must do to the other??
    (16 votes)
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  • leaf yellow style avatar for user Tiago
    Another way to solve this "quickly" is this: 3/4x+2=3/8x-4, what number you need to multiply 3/4x for so denominator becomes 8 as well?

    You multiply by 2 and get 6/8x+2=3/8x-4

    1st step 6/8x-3/8+2=3/8x-3/8x-4
    2nd step 3/8x+2-2=-4-2
    3/8x/3/8=-6/3/8 (0.375)

    (14 votes)
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    • mr pink green style avatar for user David Severin
      The third step is much easier to multiply by the reciprocal rather than dividing, so
      3/8 x • 8/3 = 6 • 8/3, since 6/3 =2, you get 16 faster.
      If you are going to do it quickly, try doing everything as simply as possible, but this is great for those not afraid of fractions.
      (2 votes)
  • starky seedling style avatar for user Maurice Thornton
    how is this going to help us in the real world
    (9 votes)
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  • blobby green style avatar for user Ceasar Queen
    I don't get why he multiplied by 8 on both sides. Don't you just subtract each fraction normally?
    (6 votes)
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    • stelly blue style avatar for user Kim Seidel
      Yes, the equation can be solved using the fractions. However, many people dislike working with fractions. Multiplying by the LCM of the denominators (the 8) eliminates the fractions.

      There will also be more complicated equations
      that have fractions with variables in the both the numerator and denominator. For those equations, you will need to eliminate the fractions, not keep them. It will require a lot less work. So, learning the basic concepts now will help you later in higher level math lessons.
      (9 votes)
  • primosaur ultimate style avatar for user Tater Kilgore
    what if in a question there is only one fraction on one side.
    (4 votes)
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    • stelly blue style avatar for user Kim Seidel
      That's ok. If you want to eliminate the fraction, you can multiply the equation by the denominator of the fraction. Or, equations can be solved just doing the math with the fraction. Most people choose to eliminate the fraction because the work steps are then a little simpler to complete.
      (12 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.