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# Equivalent forms of expressions

## Video transcript

Which expressions are equivalent
to 2 times the quantity 4f plus 2g? Mark all that apply. So the first one
here is 8f plus 4g. So could I manipulate
this somehow, in a valid way, so
I get 8f plus 4g? Well, the most obvious thing
is I just distribute the 2. So 2 times this
whole quantity is going to be 2 times 4f
plus 2 times 2g, which is equal to 8f plus 4g. So this expression is indeed
equivalent to that expression right over there. Now let's see what
they tried to do. 2f times this thing
right over here. Is that equivalent
to that over there? Well, it doesn't look
like it's going to be. And you could even try to
distribute this right over here. You're going to get--
if we distribute the 2f, you'll get 2f times 4 is 8f. And 2f times 2g
is plus 4fg, which is very different
than 8f plus 4g. This is 8f plus 4fg. So this one is not an
equivalent expression. And here that you
have 8f plus 2g. Well, 8f plus 2g,
we already know, is different than 8f
plus 4g, And 8f plus 4g is an equivalent expression. And these two things
aren't equivalent, so we can cross that out. Now they have a 4
times 2f plus g. Well, what happens if we
were to factor a 4 out of 8f plus 4g, which
we already know is equivalent to our
original expression? So if you try to factor
out a 4 right over here, so you divide 8
by 4, you get 2f. And you divide 4g
by 4, you get g. And you can't just
only divide by 4. Then you then have to
multiply by 4 in order to not change the actual
value of the expressions. So all we did is we divided
by 4 and then multiplied by 4, which doesn't
change the actual value. Or you could think of it
as we undistributed a 4. We factored out a 4. So 4 times 2f plus 4g
is indeed the same thing as 8f plus 4g, which
is, we already know, is the same thing as
our original expression. And you can distribute
this to verify that. So this is also a
valid expression. Let's do a couple more of these. Fill in the blank to
produce an expression equivalent to mu
plus mu plus mu. Well, I have 3 mus
right over here. So this is literally just
going to be 3 mu, 3 times the variable mu. That's all that's going
on right over there. Let's do a couple more of this. That was pretty fast. Which expressions are equivalent
to 6l plus 5m minus 3n? Mark all that apply. So let's look at this first one. If I were to distribute the 3,
I would get 6l minus 3n plus 5. And that and that
are equivalent. If you just change the
order-- oh, a plus 5m. Let me be very careful here. So if I were to
distribute the 3, 6l minus 3n plus 5m-- and this
expression and this expression are going to be
equivalent if you just swap the negative 3m and the 5m,
which you could completely do. Addition is commutative. It doesn't matter which
order you actually add in. So this is legitimate. Now let's see. 3n plus 6l. So already, something
goes shady here. Here you have a minus 3n, or you
could view it as a negative 3n. Here you have a
positive 3n, and they don't fix it anywhere else. So this does not seem like
a legitimate expression. And they also have
a negative 5m, while it was a positive
5m right over there. So that's definitely
not the case. So here you have
5m, a positive 5m. Well, you have a positive
5m right over there. Then you have plus 6l minus 3n. Putting these parentheses,
these are essentially reassociating what
operation I would do first. But you could actually
remove the parentheses here, and it won't change the value. You could think about it as
distributing the positive sign or distributing the
positive 1 here. It would just become 5m plus
6l minus 3n, which is just a reordering of this
right over here. But it's completely legitimate. So let me write that
right over there. And then they have 5l. And if you distribute
this, this would be 3m. So they mixed everything up. This is 6l. This is a 5l. So this one right
over here is no good. This is a lot of fun. Let's keep going. If we take the
expression 2 times a plus 2b-- in parentheses--
and ignore the parentheses, we can write another
expression, 2a plus 2b, if you were to do that. Is 2a plus 2b equivalent to 2
times a plus 2b in parentheses? So when they're saying all
of this thing over here, they're saying,
hey, look, I'm just this irresponsible
mathematician. And I like to just
ignore parentheses without thinking about it fully. And if I did, I would
just get 2a plus 2b. Can I do this? And you might
already be imagining, based on the tone of my
voice, what the answer is. And to think about it, you just
have to realize, well, look. You're multiplying 2
not just times the a. You're multiplying 2 times
the entire quantity a plus 2b. You have to distribute that 2. This is going to be equal to 2
times a plus 2 times 2b, which is equal to 2a plus 4b. So you can't just
ignore that there. 2a plus 4b is very
different than 2a plus 2b. So are these two
things equivalent? No. No, this is seriously
irresponsible mathematics.