# A clever application of the distributive property to solve a multi-step equation | Khan Academy

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.

this is so well instructed!

THANK YOU!!!

epic.

except 3/4x = 3/(4x). If you mean, "3 quarters of x" you have to write (3/4)x, otherwise you have 3 Γ· 4x as written. You mean (3/4)x or 3/4 * x. Small yet important detail

thanx this helped me with my homework so much

One hour of math class = 5 minutes of a Khan Academy video.

your the best

You saved my life

Noice M8

HELLLPPPPPPPP!!!!!!! Wat if its a fraction on only one side I don't want to ask In class cuz I mean scared they'll get mad if its a silly question so PLLZZZZZ SOMEONE HELLLP!!!!! ππππ

not to be negative but the audio sounds a little off

I'm not sure how he knew to multiply both sides by eight at the beginning,

So many people that want to learn so cooooooool Khan i luv u so much u help me alot

god bless you im too stupid

could someone help?if it's a single variable on one side, do i also multiply that by the LCM?

(for example: x/3 + 2/5 = 2/3 – x)

But how do you know to multiply both sides by 8 π€π€π€

I need practice in solving math equations. thanks.

How do you know what number to subtract?

You might consider expanding on your lessons. How does the 8 not effect the end result? Why does the 8 cancel out the second 8?

borinng