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.