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## Reducing equations to simpler form

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# Ex 2: Distributive property to simplify

## Video transcript

Solve for x. And we have x minus 8 is
equal to x/3 plus 1/6. Now the first thing
I want to do here-- and there's multiple
ways to do this problem-- but what I want to do is just
to simplify the fraction. I'm going to multiply everything
times the least common multiple of all of these
guys' denominators. This is essentially x/1. This is 8/1, x/3, 1/6. The least common multiple
of 1, 3, and 6 is 6. So if I multiply
everything times 6, what that's going to do is going
to clear out these fractions. So these weren't
fractions to begin with, so we're just
multiplying them by 6. So it becomes 6x minus 6 times
negative 8, or 6 times 8 is 48. And we're subtracting
it right over there. And then we have x/3 times 6. Let me just write it out here. So that's going to be 6
times x/3 plus 6 times 1/6. Or we get 6x minus 48 is equal
to 6 times something divided by 3. That's the same thing
as 6 divided by 3 times that something. That's just going to
be equal to 2x plus 6 times 1/6 or 6 divided by
6 is just going to be 1. So that first step cleared
out all of the fractions and now this is just a
straightforward problem with all integer
coefficients or integers on either side of the equation. And what we want
to do is we want to isolate all of the x's
on one side or the other. And we might as well isolate
them all on the left hand side. So let's subtract
2x from both sides. We want to get rid
of this 2x here. That's why I'm
subtracting the 2x. So let's subtract
2x from both sides. And on the right hand side,
I have 2x plus 1 minus 2x. Those cancel out. That was the whole point. So I'm left with just
this 1 over here. On the left hand side
I have 6x minus 2x. Well, that's just
going to be 4x. If I have 6 of something
minus 2 of that something, I have 4 of that something. Minus 48. And now I can-- let's see,
I want to get rid of this 48 on the left hand side
because I want only x's here. So let me add 48 to both
sides of the equation. I'll do this in a new color. So let me add 48 to both
sides of this equation. And on the left hand
side 4x minus 48 plus 48, those cancel out. I'm left with just a 4x. And on the right hand side,
1 plus 48 is going to be 49. And now I've isolated the x but
it's still multiplied by a 4. So to make that a 1 coefficient,
let's multiply both sides by 1/4. Or you could also say, let's
divide both sides by 4. Anything you do to one side
you have to do to the other. And so you have-- what
do we have over here? 4x/4 is just x. x is equal to 49 over 4. And that's about as far
as we can simplify it because these don't have any
common factors, 49 and 4. Let's check to see whether
49/4 is indeed the answer. So let's put it into
the original equation. Remember, the
original equation is what we have in green here
before we multiplied it by six. But in theory, we
should be able to put it into any of these steps
and the x should satisfy. But let's do it in
our original equation. So we have x minus 8. So we have 49/4 minus 8
should be equal to 49/4 over 3 plus 1/6. So let's see what
we can do here. So we can multiply. Well, like we did before. We can multiply both sides
of this equation by 6. That'll help simplify a
lot of the fractions here. So if we multiply both
sides of this equation by 6-- so we're going to
multiply everything by 6-- what do we get on
the left hand side? 6/4 is the same
thing as 3/2, right? So this is going to
be 3 times 49 over 2. 3 times 49 over 2 minus 48
will be equal to 6 divided by 3 is 2. So it's going to be
2 times 49 over 4, which is the same thing as 49/2. And then 6 times 1/6 is plus 1. And let's see. Well, I'm just going to
actually just evaluate them out. I could just essentially have
to subtract 49/2 from both sides and that'll simplify things. But let me just figure out
what these evaluate to. So 3 times 49, 49 times 3. You could think about it. It's going to be 3
less than 50 times 3. So it's 147. But let's just multiply it out. 9 times 3 is 27 4 times
3 is 12, plus 2 is 14. So 147. So this is 147/2. And then let's put this
over a denominator of 2. So 48 is equal to 96/2, right? 96 divided by 2 is 48 I
just multiplied this by 2. So this is minus 96/2 needs
to be equal to 49/2 plus-- and instead of having this
1 let's write that as 2/2. And what's 47 minus 96? So 147 minus 100
would be 47, but we're going to subtract
4 more than that. So we're going to
have-- we're going to subtract 4 less than 100. So it's going to be 147
minus 96 is going to be 51. 51/2 is equal to--
49 plus 2 is 51/2. So it all checks out.