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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.

## Want to join the conversation?

• so it is x=-16? • 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
• I am struggling on how to transfer varibles to one side of the equation. Any tips? •  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) -32(6)-3=3(4)-312-3=12-39=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+3x8+2x -3x =6+3x -3x8+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=62x-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)
• can you multiply the denominator on both sides • 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? •  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
• But arent what you do to one side you must do to the other?? • 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)

x=-16 • I still didn't understand where the 8 came from, can someone please explain it again differently? How do you get to that conclusion? • 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.   