Main content

# 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.

## Want to join the conversation?

- so it is x=-16?(131 votes)
- 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(66 votes)

- I am struggling on how to transfer varibles to one side of the equation. Any tips?(31 votes)
- 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)(86 votes)

- can you multiply the denominator on both sides(32 votes)
- 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)

- 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?(16 votes)- 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(27 votes)

- I still didn't understand where the 8 came from, can someone please explain it again differently? How do you get to that conclusion?(11 votes)
- 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.(20 votes)

- But arent what you do to one side you must do to the other??(16 votes)
- 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(13 votes)- 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)

- how is this going to help us in the real world(7 votes)
- It probably won't, it probably will. It depends what you are doing in life and what job you have(9 votes)

- what if in a question there is only one fraction on one side.(4 votes)
- 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.(11 votes)

- I don't get why he multiplied by 8 on both sides. Don't you just subtract each fraction normally?(5 votes)
- 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.(8 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.