Distributive Property

Let's do some problems with
the distributive property.

And the distributive property
just essentially reminds us

that if we have, let's say, a
times b plus c, and then we

need to multiply a times this,
we have to multiply a times

both of these numbers.

So this is going to be equal to
a times b plus a times c.

It will not be just a
times b then plus c.

And that makes complete sense.

Let me give you an example.

If I had said 5 times 3 plus
7, now, if you were to work

this out using order of
operations, you'd say, this is

5 times 10.

So you'd say, this is 5 times
10, which is equal to 50.

And we know that that's
the right answer.

Now, use the distributive
property, that tells us that

this is going to be equal to 5
times 3, which is 15, plus 5

times 7, which is 35.

And 15 plus 35 is
definitely 50.

If you only multiplied the 5
times the 3, you'd have 15,

and then plus the seven, you'd
get the wrong answer.

You're multiplying 5 times
these things, you have to

multiply 5 times both
of these things.

Because you're multiplying
the sum of these guys.

Anyway.

Let's just apply that to a
sampling of these problems.

Let's do A.

So we have 1/2 times
x minus y minus 4.

Well, we multiply 1/2
times both of these.

So it's going to be 1/2
x minus 1/2 y minus

4, and we're done.

Let's do C.

We have 6 plus x
minus 5 plus 7.

Well, here there's actually
no distributive

property to even do.

We can actually just remove
the parentheses.

6 plus this thing, that's the
same thing as 6 plus x plus

negative 5 plus 7.

Or you could view this
as 6 plus-- So this

right here is 2, right?

Negative 5 plus 7 is 2,
2 plus 6 is 8, so it

becomes 8 plus x.

All right.

Not too bad.

That was C.

Let's do E.

We have 4 times m plus 7 minus
6 times 4 minus m.

Let's do the distributive
property.

4 times m is 4m plus
4 times 7 is 28.

And then we could
do it two ways.

Let's do it this way first.
So we could have minus

6 times 4 is 24.

6 times negative
m is minus 6m.

And notice, I could have just
said, times negative 6, and

have a plus here, but I'm
doing it in two steps.

I'm doing the 6 first, and then
I'll do the negative 1.

And so this is going to be
4m plus 28, and then you

distribute the negative sign.

You can view this as a negative
1 times all of this.

So negative 1 times
24 is minus 24.

Negative 1 times minus
6m is plus 6m.

Now you add the m terms.
4m plus 6m is 10m.

And then add the constant terms.
28 minus 24, that is

equal to plus 4.

Let's go down here.

Use the distributive property
to simplify

the following fractions.

So I'll do every other
one again.

So the first one is, a
is 8x plus 12 over 4.

So the reason why they're
saying the distributive

property, you're essentially
saying, let's divide this

whole thing by 4.

And to divide the whole thing by
4, you have to divide each

of the things by 4.

You could even view this as,
this is the same thing as

multiplying 1/4 times
8x plus 12.

These two things
are equivalent.

Here you're dividing each
by 4, here you're

multiplying each by 4.

If you did it this way, this is
the same thing as 8x over 4

plus 12 over 4.

You're kind of doing a adding
fractions problem in reverse.

And then this 8 divided
by 4 is going to be,

this'll be 2x plus 3.

That's one way to do it.

Or you could do it this way.

1/4 times 8x is 2x, plus
1/4 times 12 is 3.

Either way, we got
the same answer.

C.

We have 11x plus 12 over 2.

Just like here.

We could say, this is the same
thing as 11-- We could write

it as 11 over 2x, if we like.

Or 11x over 2, either way.

Plus 12 over 2 plus 6.

And let's just do one more.

E.

This looks interesting.

We have a negative out in front,
and then we have a 6z

minus 2 over 3.

So one way we can view this,
this is the same thing, this

is equal to, negative 1/3
times 6z minus 2.

These two things
are equivalent.

Right?

This is a negative 1/3.

You could imagine a
1 right out here.

Right?

Negative 1/3 times 6z minus 2.

And then you just do the
distributive property.

Negative 1/3 times 6z is
going to be minus 2z.

And then negative 1/3 times
negative 2, negatives cancel

out, you get plus 2/3.

And you are done.