Introduction to solving an equation with variables on both sides

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.810
Let's try to solve a
more involved equation.

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So, let's say that we have 2x
plus 3, 2x plus 3 is equal

00:00:15.190 --> 00:00:23.040
to is equal to 5x minus 2.

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So this might look a
little daunting at first.

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We have x's on both
sides of the equation.

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We're adding and
subtracting numbers.

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How do you solve it?

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And we'll do it in a
couple of different ways.

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The the important thing
to remember is, we just

00:00:33.950 --> 00:00:35.260
want to isolate an x.

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Once you've isolated an x,
you have x equals something.

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Or x equals something.

00:00:39.900 --> 00:00:41.640
You're done, you've
solved the equation.

00:00:41.640 --> 00:00:44.050
You can actually go back and
check whether that works, So

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what we're going to do is just
do a bunch of operations on

00:00:46.400 --> 00:00:49.375
both sides of the equation,
to eventually isolate the x.

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But while we do those, I
actually want to visualize

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what's occurring.

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Because I don't want you just
say, oh what are the rules or

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the steps of solving equations.

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And I forgot whether this is a
allowed or that isn't allowed.

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If you visualize what's
happening, it'll actually be

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common sense what's allowed.

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So let's visualize it.

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So we have 2x right here
on the left-hand side.

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So that's literally,
that's x plus x.

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And then you have plus 3.

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Plus 3, I'll do it like this.

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So that's equal to plus
1, plus 1, plus 1.

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That's the same thing as 3.

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I could've drawn 3
circles here as well.

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Let do the same color.

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Plus 3.

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And then that is
equal to 5 x's.

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Do that in blue.

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That is equal to 5 x's.

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So, 1, 2, 3, 5, 6.

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And I want to make it clear.

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You would never actually
have to do it this way when

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you're solving the problem.

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You would just have to
do the algebraic steps.

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But I'm doing this for you so
you can actually visualize

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what this equation is saying.

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the left-hand side is these
two orange x's plus 3.

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The right-hand side
is 5x minus 2.

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So minus 2, we could write
as -- so let me do this

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in a different color,
I'll do it in pink.

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So, minus 2, I'll do as
minus 1 and minus 1.

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Now, we want to isolate
the x's on the same

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side of the equation.

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So, how could we do that?

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Well, there's two
ways of doing it.

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We could subtract these
two x's from both

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sides of the equation.

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And that would be
pretty reasonable.

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Because then you'd have
5 x's minus the 2 x's.

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You'd have a positive number of
x's on the right-hand side.

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Or, you could actually
subtract 5x from both sides.

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And that's what's
neat about algebra.

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As long as you do legitimate
operations, you will eventually

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get the right answer.

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So let's just start off
subtracting 2x from both

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sides of the equation.

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And what I mean there, I mean
we're going to remove 2 x's

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from the left-hand side.

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And if we were to move 2 x's on
the left-hand side, we have to

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remove 2 x's the
right-hand side.

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Just like that.

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So what does that give us?

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We're subtracting 2 x's.

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2 x's from the left.

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And we're also going to
subtract 2 x's from the right.

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Now, what does our left-hand
side simplify to?

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We have 2x plus 3 minus 2x.

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The 2 x's cancel out.

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So you're just left with --
you're just left with the 3.

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And you see that over here.

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We took 2 of these x's away.

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We're just left with the
plus 1, plus 1, plus 1.

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And then on the right-hand
side, 5x minus 2x.

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We have it over here.

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We have 5 x's minus 2 x's.

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You only have 1, 2,
3, x's left over.

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3 is equal to 3x.

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And then you have
your minus 2 there.

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You have your minus 2.

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So, normally if you were to do
the problem, you would just

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have to write what we have
here on the left-hand side.

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So what can we do next?

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Remember, we want to
isolate the x's.

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Well, we have all of our x's on
the right-hand side right here.

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If we could get rid of this
negative 2, off of the

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right-hand side, then
the x's will be alone.

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They'll be isolated.

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So how can we get rid of
this negative 2, if we

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visualize it over here.

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This negative 1,
this negative 1.

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Well, we could add 2 to both
sides of this equation.

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Think about what happens there.

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So, if we add 2, so I'm
going to do it like this.

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Plus 1, plus 1.

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So you could literally see.

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We're adding 2.

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And then we're going to add
2 to the left-hand side.

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1 plus, 1 plus.

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What happens?

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Let me do it over here as well.

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So we're going to add 2.

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We're going to add 2.

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So what happens to
the left-hand side?

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3 plus 2 is going
to be equal to 5.

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And that is going to be
equal to 3x minus 2 plus 2.

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These guys cancel out.

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And you're just left with 3x.

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And we see it over here.

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We have the left-hand side is 1
plus 1 plus 1 plus 1 plus 1.

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We have 5 1's, or 5.

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And the right-hand side,
we have the 3 x's,

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right over there.

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And then we have the
negative 1, negative 1.

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Plus 1, plus 1, negative
1, these cancel out.

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They get us to 0.

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They cancel out.

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So we're just left with
5 is equal to 3x.

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So we have 1, 2, 3,
4, 5 is equal to 3x.

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Let me clear everything that
we've removed, so it looks

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a little bit cleaner.

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These are all of the things
that we've removed.

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Let me clear that out.

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And then let me clear
that out, like that.

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Edit.

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Clear.

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So now we are just left
with 1, 2, 3, 4, 5.

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Actually, let me
move this over.

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So I could just move this
over right over here.

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We now have 1, 2, 3, 4, 5.

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These are the two that we
added here, is equal to 3x.

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These guys canceled out.

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That's why we have
nothing there.

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Now, to solve this, we
just divide both sides

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of this equation by 3.

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And this is going to
be a little hard to

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visualize over here.

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But if we divide over here both
sides by 3, what do we get?

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We divide the left by 3.

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We divide the right by 3.

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The whole reason why we divided
by 3 is because the x was

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being multiplied by 3.

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3 is the coefficient on the x.

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Fancy word, it literally
just means the number

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multiplying the variable.

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The number we're solving, the
variable we're solving for.

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So these 3's cancel out.

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The right-hand side of
the equation is just x.

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The left-hand side is 5/3.

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So 5/3, we could say
is is equal to 5/3.

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And this is different than
everything we've seen so far.

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I now have the x on the
right-hand side, the value

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on the left-hand side.

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That's completely fine.

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This is the exact same thing as
saying 5/3 is is equal to x is

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the same thing as saying
x's equal to 5/3.

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Completely equivalent.

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Completely equivalent.

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We sometimes get more used
to this one, but this is

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completely the same thing.

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Now, if we wanted to write this
as a mixed number, if we want

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to write this as a mixed
number, 3 goes into 5 one

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time with remainder 2.

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So it's going to be 1 2/3.

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So it's going to be 1 2/3.

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So we could also write
that x is equal to 1 2/3.

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And I'll leave it up for you
to actually substitute back

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into this original equation.

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And see that it works out.

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Now, to visualize it over here,
you know, how did he get 1

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2/3, let's think about it.

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Instead of doing 1, I'm
going to do circles.

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I am going to do circles.

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Actually, even better,
I'm going to do squares.

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So I'm going to have 5 squares
on the left-hand side.

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I'll do it in this same
yellow color right here.

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So I have 1, 2, 3, 4, 5.

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And that is going to be
equal to the 3 x's.

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x plus x plus x.

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Now, we're dividing both
sides of the equation by 3.

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We're dividing both sides
of the equation by 3.

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Actually, that's where
we did it up here, we

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divided both sides by 3.

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So how do you do that
right-hand side's pretty

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straightforward.

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You want to divide these
3 x's into 3 groups.

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That's 1, 2, 3 groups.

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1, 2, 3.

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Now, how do you
divide 5 into 3?

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And they have to be thought
through even groups.

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And the answer tells us.

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Each group is going
to be 1 2/3.

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So, 1 2/3.

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So it's going to be 2/3
of this, the next one.

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And then we're going
to have 1 2/3.

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So this is 1/3.

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We're going to need another.

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Another 1, so this is 1 1/3.

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We're going to need 1 more
1/3, so this is going

00:08:05.770 --> 00:08:07.870
to be right here.

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And then we're left
with 2/3 and 1.

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So we've broken it
up into 3 groups.

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This right here.

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Let me make it clear.

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Let me make it clear, this
right here is 1 2/3.

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1 2/3.

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And then this right
here, this 1/3.

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That's another 1/3, so
that's 2/3, and then

00:08:26.140 --> 00:08:27.010
that's 1 right there.

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So that's 1 2/3.

00:08:28.970 --> 00:08:32.480
And then finally this
is 2/3 and this is

00:08:32.480 --> 00:08:34.890
1, so this is 1 2/3.

00:08:34.890 --> 00:08:40.030
So when you divide both
sides by 3 you get 1 2/3.

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Each section, each bucket, is
1 2/3 on the left-hand side.

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On the left-hand side, or 5/3.

00:08:46.080 --> 00:08:48.120
And on the right-hand
side we just have an x.

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So it still works.

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A little bit harder to
visualize with fractions.


Solving a more complicated equation

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.390
Let's try some slightly more
complicated equations.

00:00:03.390 --> 00:00:13.340
Let's say we have 3 times
x plus 5-- I want to make

00:00:13.340 --> 00:00:24.655
sure I get all the colors
nice-- is equal to 17.

00:00:27.430 --> 00:00:29.790
So what's different about this
than what we saw in the last

00:00:29.790 --> 00:00:32.290
video is, all of a sudden,
now we have this plus 5.

00:00:32.290 --> 00:00:35.250
If it was just 3x is equal to
17, you could divide both

00:00:35.250 --> 00:00:37.570
sides by 3, and you'd
get your answer.

00:00:37.570 --> 00:00:41.080
But now this 5 seems to mess
things up a little bit.

00:00:41.080 --> 00:00:43.610
Now, before we even solve
it, let's think about

00:00:43.610 --> 00:00:44.170
what it's saying.

00:00:44.170 --> 00:00:47.500
Let's solve it kind of in a
tangible way, then we'll solve

00:00:47.500 --> 00:00:49.800
it using operations that
hopefully will make

00:00:49.800 --> 00:00:50.650
sense after that.

00:00:50.650 --> 00:00:54.440
So 3 times x literally means--
so let me write it over here.

00:00:54.440 --> 00:00:55.870
So we have 3 times x.

00:00:55.870 --> 00:00:59.960
So we literally have an
x plus an x plus an x.

00:00:59.960 --> 00:01:01.630
That right there is a 3x.

00:01:01.630 --> 00:01:04.170
And then that's plus 5, and
I'm actually going to write

00:01:04.170 --> 00:01:05.770
it out as five objects.

00:01:05.770 --> 00:01:12.130
So plus 1, 2, 3, 4, 5.

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This right here, this 3x
plus 5, is equal to 17.

00:01:19.010 --> 00:01:20.845
So let me write the equal sign.

00:01:23.520 --> 00:01:25.810
Now, let me draw
17 objects here.

00:01:25.810 --> 00:01:38.020
So 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17.

00:01:38.020 --> 00:01:40.910
Now, these two things are
equal, so anything you do

00:01:40.910 --> 00:01:42.440
to this side, you have
to do to that side.

00:01:42.440 --> 00:01:44.460
If we were to get rid of one
object here, you'd want to get

00:01:44.460 --> 00:01:46.730
rid of one object there in
order for the equality

00:01:46.730 --> 00:01:48.730
to still be true.

00:01:48.730 --> 00:01:52.530
Now, what can we do to both
sides of this equation so we

00:01:52.530 --> 00:01:54.690
can get it in the form that
we're used to, where we

00:01:54.690 --> 00:01:57.040
only have a 3x on the
left-hand side, where

00:01:57.040 --> 00:01:58.840
we don't have this 5?

00:01:58.840 --> 00:02:00.810
Well, ideally, we would
just get rid of these

00:02:00.810 --> 00:02:02.210
five objects here.

00:02:02.210 --> 00:02:04.460
You would literally get rid
of these five objects:

00:02:04.460 --> 00:02:06.710
1, 2, 3, 4, 5.

00:02:06.710 --> 00:02:09.300
But, like I said, if the
original thing was equal to the

00:02:09.300 --> 00:02:12.410
original thing on the right, if
we get rid of five objects from

00:02:12.410 --> 00:02:14.680
the left-hand side, we have
to get rid of five objects

00:02:14.680 --> 00:02:15.900
from the right-hand side.

00:02:15.900 --> 00:02:20.600
So we have to do it here,
too: 1, 2, 3, 4, 5.

00:02:20.600 --> 00:02:24.030
Now, what is a symbolic way
of representing taking

00:02:24.030 --> 00:02:25.800
away five things?

00:02:25.800 --> 00:02:29.550
Well, you're subtracting 5 from
both sides of this equation.

00:02:29.550 --> 00:02:32.100
So that's what we're doing here
when we took away 5 from the

00:02:32.100 --> 00:02:33.410
left and from the right.

00:02:33.410 --> 00:02:35.660
So we're subtracting
5 from the left.

00:02:35.660 --> 00:02:36.820
That's what we did here.

00:02:36.820 --> 00:02:39.800
And we're also subtracting
5 from the right.

00:02:39.800 --> 00:02:41.390
Do that right over there.

00:02:41.390 --> 00:02:43.790
Now, what does the left-hand
side of the equation

00:02:43.790 --> 00:02:45.220
now become?

00:02:45.220 --> 00:02:47.300
The left-hand side,
you have 5 minus 5.

00:02:47.300 --> 00:02:48.170
These cancel out.

00:02:48.170 --> 00:02:49.800
You're just left with the 3x.

00:02:52.680 --> 00:02:54.530
It's a different
shade of green.

00:02:54.530 --> 00:02:58.930
You are just left with the 3x.

00:02:58.930 --> 00:03:01.260
The 5 and the negative
5 canceled out.

00:03:01.260 --> 00:03:02.300
And you see that here.

00:03:02.300 --> 00:03:04.070
When you got rid of these
five objects, we were

00:03:04.070 --> 00:03:06.410
just left with the 3x's.

00:03:06.410 --> 00:03:09.340
This right here is the 3x.

00:03:09.340 --> 00:03:11.490
And the whole reason why we
subtracted 5 is because we

00:03:11.490 --> 00:03:13.780
wanted this 5 to go away.

00:03:13.780 --> 00:03:16.790
Now, what does the right-hand
side of the equation look like?

00:03:16.790 --> 00:03:19.450
So it's 3x is going to be-- let
me write the equality sign

00:03:19.450 --> 00:03:22.210
right under it-- is equal to--
or you could either just

00:03:22.210 --> 00:03:23.060
do it mathematically.

00:03:23.060 --> 00:03:24.980
Say, OK, 17 minus 5.

00:03:24.980 --> 00:03:28.310
17 minus 5 is 12.

00:03:28.310 --> 00:03:30.120
Or you could just
count over here.

00:03:30.120 --> 00:03:30.910
I had 17 things.

00:03:30.910 --> 00:03:31.900
I took away 5.

00:03:31.900 --> 00:03:37.800
I have 12 left: 1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12.

00:03:37.800 --> 00:03:39.090
That's what's subtraction is.

00:03:39.090 --> 00:03:41.180
It's just taking
away five things.

00:03:41.180 --> 00:03:43.820
So now we have it in a pretty
straightforward form.

00:03:43.820 --> 00:03:46.350
3x is equal to 12.

00:03:46.350 --> 00:03:48.450
All we have to do is
divide both sides of

00:03:48.450 --> 00:03:50.330
this equation by 3.

00:03:50.330 --> 00:03:53.930
So we're just left with an
x on the left-hand side.

00:03:53.930 --> 00:03:56.840
So we divide by-- let me pick
a nicer color than that.

00:03:56.840 --> 00:04:00.490
Let me do this pink color.

00:04:00.490 --> 00:04:02.340
So you divide the
left-hand side by 3, the

00:04:02.340 --> 00:04:03.540
right-hand side by 3.

00:04:03.540 --> 00:04:05.260
And remember what
that's equivalent to.

00:04:05.260 --> 00:04:08.070
The left-hand side, none of
this stuff exists anymore,

00:04:08.070 --> 00:04:09.570
so we should ignore it.

00:04:09.570 --> 00:04:11.120
None of this stuff
exists anymore.

00:04:11.120 --> 00:04:13.270
In fact, let me clear it
out, just so that we don't

00:04:13.270 --> 00:04:14.190
even have to look at it.

00:04:14.190 --> 00:04:17.730
We subtracted it, so
let me clear it.

00:04:17.730 --> 00:04:20.350
Let me clear it over here.

00:04:20.350 --> 00:04:21.740
Let me clear it over here.

00:04:21.740 --> 00:04:24.300
And so now we are dividing
both sides by 3.

00:04:24.300 --> 00:04:25.640
Divide the left-hand side by 3.

00:04:25.640 --> 00:04:27.890
It's 1, 2, 3.

00:04:27.890 --> 00:04:30.580
So three groups, each of
them have an x in it.

00:04:30.580 --> 00:04:35.670
If you divide this right-hand
side by 3, you have 1, 2 and 3.

00:04:35.670 --> 00:04:38.560
So it's three groups of four.

00:04:38.560 --> 00:04:40.550
So when you do it
mathematically here,

00:04:40.550 --> 00:04:41.680
the 3's cancel out.

00:04:41.680 --> 00:04:43.975
3 times something divided by
3 is just the something.

00:04:43.975 --> 00:04:48.220
So you're left with x is
equal to, and then 12

00:04:48.220 --> 00:04:50.103
divided by 3 is 4.

00:04:53.100 --> 00:04:55.150
You get x is equal to 4,
and you get that exact

00:04:55.150 --> 00:04:56.070
same thing over here.

00:04:56.070 --> 00:04:59.140
When you divided 3x into
groups of three, each of

00:04:59.140 --> 00:05:01.730
the groups had an x in it.

00:05:01.730 --> 00:05:04.610
When you divided 12 into groups
of three, each of the groups

00:05:04.610 --> 00:05:09.230
have a 4 in it, so x
must be equal to 4.

00:05:09.230 --> 00:05:12.390
x is equal to 4.

00:05:12.390 --> 00:05:15.110
Let's do another one, and this
time I won't draw it all out

00:05:15.110 --> 00:05:17.610
like this, but hopefully,
you'll see that the same type

00:05:17.610 --> 00:05:20.750
of processes are involved.

00:05:20.750 --> 00:05:24.380
Let's say I have-- let me
scroll down a little bit.

00:05:24.380 --> 00:05:29.060
Let's say I have 7x.

00:05:29.060 --> 00:05:36.880
So 7x-- and I'll do a slightly
more complicated one this time.

00:05:36.880 --> 00:05:48.330
7x minus 2 is equal to-- I'll
make the numbers not work

00:05:48.330 --> 00:05:54.360
out nice and clean-- is
equal to negative 10.

00:05:54.360 --> 00:05:57.400
Now, this all of a sudden
becomes a lot more-- you know,

00:05:57.400 --> 00:05:58.570
we have a negative sign.

00:05:58.570 --> 00:06:00.310
We have a negative over
here, but we're going to

00:06:00.310 --> 00:06:02.300
do the exact same thing.

00:06:02.300 --> 00:06:04.510
The first thing we want to do
if we want to get the left-hand

00:06:04.510 --> 00:06:07.450
side simplified to just 7x
is we want to get rid

00:06:07.450 --> 00:06:08.950
of this negative 2.

00:06:08.950 --> 00:06:12.170
And what can we add or subtract
to both sides of the equation

00:06:12.170 --> 00:06:14.200
to get rid of this negative 2?

00:06:14.200 --> 00:06:18.050
Well, if we add 2 to the
left-hand side, these two

00:06:18.050 --> 00:06:19.470
guys will cancel out.

00:06:19.470 --> 00:06:21.420
But remember, this
is equal to that.

00:06:21.420 --> 00:06:23.950
If we want the equality to
still hold, if we add 2 to the

00:06:23.950 --> 00:06:28.070
left-hand side, we also have to
do it to the right-hand side.

00:06:28.070 --> 00:06:31.990
So what is the new left-hand
side going to be equal to?

00:06:31.990 --> 00:06:36.950
So we have 7x, negative
2 plus 2 is just 0.

00:06:36.950 --> 00:06:39.090
I could write plus 0, or I
could just write nothing there,

00:06:39.090 --> 00:06:40.130
and I'll just write nothing.

00:06:40.130 --> 00:06:44.920
So we get 7x is equal to-- now,
what's negative 10 plus 2?

00:06:44.920 --> 00:06:46.670
And this is a little bit
of review of adding and

00:06:46.670 --> 00:06:48.970
subtracting negative numbers.

00:06:48.970 --> 00:06:49.810
Remember.

00:06:49.810 --> 00:06:51.840
I'll draw the number
line here for you.

00:06:51.840 --> 00:06:56.090
If I draw the number line--
so this is 9, this is 1.

00:06:56.090 --> 00:06:58.360
We could keep going in
the positive direction.

00:06:58.360 --> 00:06:59.550
Negative 10 is out here.

00:06:59.550 --> 00:07:03.850
Negative 10, negative 9,
negative 8, negative 7.

00:07:03.850 --> 00:07:05.860
There's a bunch
of numbers here.

00:07:05.860 --> 00:07:07.040
You know, dot, dot, dot.

00:07:07.040 --> 00:07:09.350
I don't have space to draw them
all, but we're starting at

00:07:09.350 --> 00:07:14.350
negative 10, and we're adding 2
to it, so we're moving in the

00:07:14.350 --> 00:07:15.780
positive direction on
the number lines.

00:07:15.780 --> 00:07:17.960
So we're going 1, 2.

00:07:17.960 --> 00:07:19.520
So it's negative 8.

00:07:19.520 --> 00:07:20.540
Don't get confused.

00:07:20.540 --> 00:07:23.130
Don't say, OK, 10 plus 2
is 12, so negative 10

00:07:23.130 --> 00:07:24.070
plus 2 is negative 12.

00:07:24.070 --> 00:07:24.800
No!

00:07:24.800 --> 00:07:27.260
Negative 10 minus 2 would be
negative 12 because you'd

00:07:27.260 --> 00:07:28.440
be going more negative.

00:07:28.440 --> 00:07:30.200
Here, we have a negative
number, but we're

00:07:30.200 --> 00:07:30.960
going to the right.

00:07:30.960 --> 00:07:32.640
We're going in the positive
direction, so this

00:07:32.640 --> 00:07:35.020
is negative 8.

00:07:35.020 --> 00:07:38.010
So we have 7x is
equal to negative 8.

00:07:38.010 --> 00:07:40.990
So now you might be saying,
well, how do I do this

00:07:40.990 --> 00:07:42.760
type of a problem?

00:07:42.760 --> 00:07:44.170
You know, I have a
negative number here.

00:07:44.170 --> 00:07:45.425
You do it the exact same way.

00:07:45.425 --> 00:07:49.410
If we want to just have an x on
the left-hand side, we have to

00:07:49.410 --> 00:07:53.850
divide the left-hand side by 7,
so that the 7x divided by 7,

00:07:53.850 --> 00:07:56.610
just the 7's cancel out,
you're left with x.

00:07:56.610 --> 00:07:58.220
So let's do that.

00:07:58.220 --> 00:08:00.880
If you divide by 7, those
cancel out, but you can't just

00:08:00.880 --> 00:08:02.130
do it to the left-hand side.

00:08:02.130 --> 00:08:03.880
Anything you do to the left,
you have to do to the right

00:08:03.880 --> 00:08:06.950
in order for the equality
to still hold true.

00:08:06.950 --> 00:08:10.360
So let's divide the
right by 7 as well.

00:08:10.360 --> 00:08:15.270
And we are left with
just an x is equal to

00:08:15.270 --> 00:08:16.760
negative 8 divided by 7.

00:08:16.760 --> 00:08:17.560
We could work it out.

00:08:17.560 --> 00:08:19.810
It'll be some type of a
decimal, if you were to use a

00:08:19.810 --> 00:08:21.810
calculator, or you could just
leave it in fraction form.

00:08:21.810 --> 00:08:26.200
Negative 8 divided by
7 is negative 8/7.

00:08:26.200 --> 00:08:29.560
Negative 8/7, or if you want to
write it as a mixed number, x

00:08:29.560 --> 00:08:33.500
is equal to 7 goes into 8 one
time and has a remainder of 1,

00:08:33.500 --> 00:08:36.560
so it's negative 1 and 1/7.

00:08:36.560 --> 00:08:39.690
Either one would be acceptable.


How to solve equations of the form ax = b

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:12.390
Let's say we have the equation
7 times x is equal to 14.

00:00:12.390 --> 00:00:15.970
Now before even trying to solve
this equation, what I want to

00:00:15.970 --> 00:00:20.080
do is think a little bit about
what this actually means.

00:00:20.080 --> 00:00:26.870
7x equals 14, this is the exact
same thing as saying 7 times x

00:00:26.870 --> 00:00:35.000
-- let me write it this way --
7 times x -- we'll do the x in

00:00:35.000 --> 00:00:41.270
orange again -- 7 times
x is equal to 14.

00:00:41.270 --> 00:00:42.860
Now you might be able to
do this in your head.

00:00:42.860 --> 00:00:45.490
You could literally go
through the 7 times table.

00:00:45.490 --> 00:00:48.960
You say well 7 times 1 is equal
to 7, so that won't work.

00:00:48.960 --> 00:00:54.350
7 times 2 is equal to
14, so 2 works here.

00:00:54.350 --> 00:00:55.920
So you would immediately
be able to solve it.

00:00:55.920 --> 00:00:58.510
You would immediately, just
by trying different numbers

00:00:58.510 --> 00:01:01.220
out, say hey, that's
going to be a 2.

00:01:01.220 --> 00:01:03.200
But what we're going to do in
this video is to think about

00:01:03.200 --> 00:01:04.780
how to solve this
systematically.

00:01:04.780 --> 00:01:07.100
Because what we're going to
find is as these equations get

00:01:07.100 --> 00:01:09.680
more and more complicated,
you're not going to be able to

00:01:09.680 --> 00:01:11.670
just think about it and
do it in your head.

00:01:11.670 --> 00:01:14.590
So it's really important that
one, you understand how to

00:01:14.590 --> 00:01:16.510
manipulate these equations,
but even more important to

00:01:16.510 --> 00:01:18.280
understand what they
actually represent.

00:01:18.280 --> 00:01:22.090
This literally just says 7
times x is equal to 14.

00:01:22.090 --> 00:01:24.530
In algebra we don't
write the times there.

00:01:24.530 --> 00:01:27.620
When you write two numbers next
to each other or a number next

00:01:27.620 --> 00:01:30.280
to a variable like this, it
just means that you

00:01:30.280 --> 00:01:31.100
are multiplying.

00:01:31.100 --> 00:01:33.920
It's just a shorthand,
a shorthand notation.

00:01:33.920 --> 00:01:37.040
And in general we don't use the
multiplication sign because

00:01:37.040 --> 00:01:40.970
it's confusing, because x is
the most common variable

00:01:40.970 --> 00:01:41.840
used in algebra.

00:01:41.840 --> 00:01:49.330
And if I were to write 7 times
x is equal to 14, if I write my

00:01:49.330 --> 00:01:52.140
times sign or my x a little
bit strange, it might look

00:01:52.140 --> 00:01:54.150
like xx or times times.

00:01:54.150 --> 00:01:56.410
So in general when you're
dealing with equations,

00:01:56.410 --> 00:01:58.910
especially when one of the
variables is an x, you

00:01:58.910 --> 00:02:01.350
wouldn't use the traditional
multiplication sign.

00:02:01.350 --> 00:02:05.020
You might use something like
this -- you might use dot to

00:02:05.020 --> 00:02:06.130
represent multiplication.

00:02:06.130 --> 00:02:10.040
So you might have 7
times is equal to 14.

00:02:10.040 --> 00:02:11.720
But this is still
a little unusual.

00:02:11.720 --> 00:02:13.750
If you have something
multiplying by a variable

00:02:13.750 --> 00:02:15.080
you'll just write 7x.

00:02:15.080 --> 00:02:18.210
That literally means 7 times x.

00:02:18.210 --> 00:02:22.750
Now, to understand how you can
manipulate this equation to

00:02:22.750 --> 00:02:25.040
solve it, let's visualize this.

00:02:25.040 --> 00:02:27.710
So 7 times x, what is that?

00:02:27.710 --> 00:02:29.690
That's the same thing -- so I'm
just going to re-write this

00:02:29.690 --> 00:02:32.720
equation, but I'm going to
re-write it in visual form.

00:02:32.720 --> 00:02:34.880
So 7 times x.

00:02:34.880 --> 00:02:37.740
So that literally means x
added to itself 7 times.

00:02:37.740 --> 00:02:40.230
That's the definition
of multiplication.

00:02:40.230 --> 00:02:47.900
So it's literally x plus x plus
x plus x plus x -- let's see,

00:02:47.900 --> 00:02:51.830
that's 5 x's -- plus x plus x.

00:02:51.830 --> 00:02:54.660
So that right there
is literally 7 x's.

00:02:54.660 --> 00:02:56.300
This is 7x right there.

00:02:56.300 --> 00:02:57.270
Let me re-write it down.

00:02:57.270 --> 00:03:03.030
This right here is 7x.

00:03:03.030 --> 00:03:07.610
Now this equation tells us
that 7x is equal to 14.

00:03:07.610 --> 00:03:10.800
So just saying that
this is equal to 14.

00:03:10.800 --> 00:03:13.300
Let me draw 14 objects here.

00:03:13.300 --> 00:03:19.640
So let's say I have 1,
2, 3, 4, 5, 6, 7, 8,

00:03:19.640 --> 00:03:23.640
9, 10, 11, 12, 13, 14.

00:03:23.640 --> 00:03:26.790
So literally we're saying
7x is equal to 14 things.

00:03:26.790 --> 00:03:29.070
These are equivalent
statements.

00:03:29.070 --> 00:03:32.280
Now the reason why I drew
it out this way is so that

00:03:32.280 --> 00:03:34.950
you really understand what
we're going to do when we

00:03:34.950 --> 00:03:37.110
divide both sides by 7.

00:03:37.110 --> 00:03:39.820
So let me erase
this right here.

00:03:39.820 --> 00:03:44.175
So the standard step whenever
-- I didn't want to do that,

00:03:44.175 --> 00:03:48.720
let me do this, let me
draw that last circle.

00:03:48.720 --> 00:03:52.360
So in general, whenever you
simplify an equation down to a

00:03:52.360 --> 00:03:55.080
-- a coefficient is just the
number multiplying

00:03:55.080 --> 00:03:55.750
the variable.

00:03:55.750 --> 00:03:58.465
So some number multiplying the
variable or we could call that

00:03:58.465 --> 00:04:00.990
the coefficient times a
variable equal to

00:04:00.990 --> 00:04:01.990
something else.

00:04:01.990 --> 00:04:05.210
What you want to do is just
divide both sides by 7 in

00:04:05.210 --> 00:04:08.240
this case, or divide both
sides by the coefficient.

00:04:08.240 --> 00:04:12.130
So if you divide both sides
by 7, what do you get?

00:04:12.130 --> 00:04:16.150
7 times something divided
by 7 is just going to be

00:04:16.150 --> 00:04:17.690
that original something.

00:04:17.690 --> 00:04:22.350
7's cancel out and 14
divided by 7 is 2.

00:04:22.350 --> 00:04:26.180
So your solution is going
to be x is equal to 2.

00:04:26.180 --> 00:04:29.200
But just to make it very
tangible in your head, what's

00:04:29.200 --> 00:04:32.570
going on here is when we're
dividing both sides of the

00:04:32.570 --> 00:04:36.260
equation by 7, we're literally
dividing both sides by 7.

00:04:36.260 --> 00:04:37.250
This is an equation.

00:04:37.250 --> 00:04:39.850
It's saying that this
is equal to that.

00:04:39.850 --> 00:04:43.230
Anything I do to the left hand
side I have to do to the right.

00:04:43.230 --> 00:04:46.130
If they start off being equal,
I can't just do an operation

00:04:46.130 --> 00:04:48.970
to one side and have
it still be equal.

00:04:48.970 --> 00:04:50.100
They were the same thing.

00:04:50.100 --> 00:04:53.420
So if I divide the left hand
side by 7, so let me divide

00:04:53.420 --> 00:04:55.150
it into seven groups.

00:04:55.150 --> 00:04:58.920
So there are seven x's here,
so that's one, two, three,

00:04:58.920 --> 00:05:01.690
four, five, six, seven.

00:05:01.690 --> 00:05:04.780
So it's one, two, three, four,
five, six, seven groups.

00:05:04.780 --> 00:05:07.240
Now if I divide that into
seven groups, I'll also want

00:05:07.240 --> 00:05:11.350
to divide the right hand
side into seven groups.

00:05:11.350 --> 00:05:16.530
One, two, three, four,
five, six, seven.

00:05:16.530 --> 00:05:20.370
So if this whole thing is equal
to this whole thing, then each

00:05:20.370 --> 00:05:25.660
of these little chunks that we
broke into, these seven chunks,

00:05:25.660 --> 00:05:27.860
are going to be equivalent.

00:05:27.860 --> 00:05:32.070
So this chunk you could say
is equal to that chunk.

00:05:32.070 --> 00:05:34.530
This chunk is equal to
this chunk -- they're

00:05:34.530 --> 00:05:35.490
all equivalent chunks.

00:05:35.490 --> 00:05:37.360
There are seven chunks
here, seven chunks here.

00:05:37.360 --> 00:05:41.390
So each x must be equal
to two of these objects.

00:05:41.390 --> 00:05:46.240
So we get x is equal to, in
this case -- in this case

00:05:46.240 --> 00:05:48.830
we had the objects drawn
out where there's two of

00:05:48.830 --> 00:05:50.680
them. x is equal to 2.

00:05:50.680 --> 00:05:53.210
Now, let's just do a couple
more examples here just so it

00:05:53.210 --> 00:05:55.565
really gets in your mind that
we're dealing with an equation,

00:05:55.565 --> 00:05:58.320
and any operation that you do
on one side of the equation

00:05:58.320 --> 00:06:00.540
you should do to the other.

00:06:00.540 --> 00:06:04.040
So let me scroll
down a little bit.

00:06:04.040 --> 00:06:13.550
So let's say I have I say
I have 3x is equal to 15.

00:06:13.550 --> 00:06:15.640
Now once again, you might be
able to do is in your head.

00:06:15.640 --> 00:06:17.780
You're saying this is
saying 3 times some

00:06:17.780 --> 00:06:19.490
number is equal to 15.

00:06:19.490 --> 00:06:22.380
You could go through your 3
times tables and figure it out.

00:06:22.380 --> 00:06:25.150
But if you just wanted to do
this systematically, and it

00:06:25.150 --> 00:06:27.680
is good to understand it
systematically, say OK, this

00:06:27.680 --> 00:06:30.500
thing on the left is equal
to this thing on the right.

00:06:30.500 --> 00:06:31.995
What do I have to do to
this thing on the left

00:06:31.995 --> 00:06:33.640
to have just an x there?

00:06:33.640 --> 00:06:36.690
Well to have just an x there,
I want to divide it by 3.

00:06:36.690 --> 00:06:39.930
And my whole motivation for
doing that is that 3 times

00:06:39.930 --> 00:06:43.620
something divided by 3, the 3's
will cancel out and I'm just

00:06:43.620 --> 00:06:45.120
going to be left with an x.

00:06:45.120 --> 00:06:48.190
Now, 3x was equal to 15.

00:06:48.190 --> 00:06:52.240
If I'm dividing the left side
by 3, in order for the equality

00:06:52.240 --> 00:06:57.230
to still hold, I also have to
divide the right side by 3.

00:06:57.230 --> 00:06:58.560
Now what does that give us?

00:06:58.560 --> 00:07:01.070
Well the left hand side, we're
just going to be left with

00:07:01.070 --> 00:07:04.130
an x, so it's just
going to be an x.

00:07:04.130 --> 00:07:08.040
And then the right hand side,
what is 15 divided by 3?

00:07:08.040 --> 00:07:10.760
Well it is just 5.

00:07:10.760 --> 00:07:13.570
Now you could also done this
equation in a slightly

00:07:13.570 --> 00:07:16.140
different way, although they
are really equivalent.

00:07:16.140 --> 00:07:21.160
If I start with 3x is equal to
15, you might say hey, Sal,

00:07:21.160 --> 00:07:25.260
instead of dividing by 3, I
could also get rid of this 3, I

00:07:25.260 --> 00:07:28.110
could just be left with an x if
I multiply both sides of

00:07:28.110 --> 00:07:30.670
this equation by 1/3.

00:07:30.670 --> 00:07:34.710
So if I multiply both sides
of this equation by 1/3

00:07:34.710 --> 00:07:35.830
that should also work.

00:07:35.830 --> 00:07:38.980
You say look, 1/3 of 3 is 1.

00:07:38.980 --> 00:07:41.710
When you just multiply this
part right here, 1/3 times

00:07:41.710 --> 00:07:46.170
3, that is just 1, 1x.

00:07:46.170 --> 00:07:52.130
1x is equal to 15 times
1/3 third is equal to 5.

00:07:52.130 --> 00:07:56.140
And 1 times x is the same thing
as just x, so this is the same

00:07:56.140 --> 00:07:58.400
thing as x is equal to 5.

00:07:58.400 --> 00:08:01.660
And these are actually
equivalent ways of doing it.

00:08:01.660 --> 00:08:05.180
If you divide both sides by
3, that is equivalent to

00:08:05.180 --> 00:08:10.340
multiplying both sides
of the equation by 1/3.

00:08:10.340 --> 00:08:12.160
Now let's do one more and I'm
going to make it a little

00:08:12.160 --> 00:08:15.270
bit more complicated.

00:08:15.270 --> 00:08:16.840
And I'm going to change the
variable a little bit.

00:08:16.840 --> 00:08:36.200
So let's say I have 2y
plus 4y is equal to 18.

00:08:36.200 --> 00:08:37.956
Now all of a sudden it's
a little harder to

00:08:37.956 --> 00:08:38.540
do it in your head.

00:08:38.540 --> 00:08:42.420
We're saying 2 times something
plus 4 times that same

00:08:42.420 --> 00:08:45.260
something is going
to be equal to 18.

00:08:45.260 --> 00:08:47.400
So it's harder to think
about what number that is.

00:08:47.400 --> 00:08:48.110
You could try them.

00:08:48.110 --> 00:08:51.910
Say if y was 1, it'd be 2
times 1 plus 4 times 1,

00:08:51.910 --> 00:08:52.710
well that doesn't work.

00:08:52.710 --> 00:08:54.390
But let's think about how
to do it systematically.

00:08:54.390 --> 00:08:56.250
You could keep guessing and
you might eventually get

00:08:56.250 --> 00:08:58.840
the answer, but how do you
do this systematically.

00:08:58.840 --> 00:08:59.830
Let's visualize it.

00:08:59.830 --> 00:09:04.540
So if I have two y's,
what does that mean?

00:09:04.540 --> 00:09:08.800
It literally means I have two
y's added to each other.

00:09:08.800 --> 00:09:11.990
So it's literally y plus y.

00:09:11.990 --> 00:09:15.300
And then to that I'm
adding four y's.

00:09:15.300 --> 00:09:18.350
To that I'm heading four y's,
which are literally four

00:09:18.350 --> 00:09:20.090
y's added to each other.

00:09:20.090 --> 00:09:25.220
So it's y plus y plus y plus y.

00:09:25.220 --> 00:09:28.630
And that has got to
be equal to 18.

00:09:28.630 --> 00:09:34.530
So that is equal to 18.

00:09:34.530 --> 00:09:38.840
Now, how many y's do I have
here on the left hand side?

00:09:38.840 --> 00:09:41.520
How many y's do I have?

00:09:41.520 --> 00:09:45.590
I have one, two, three,
four, five, six y's.

00:09:45.590 --> 00:09:49.070
So you could simplify this
as 6y is equal to 18.

00:09:49.070 --> 00:09:50.930
And if you think about it
it makes complete sense.

00:09:50.930 --> 00:09:57.350
So this thing right here,
the 2y plus the 4y is 6y.

00:09:57.350 --> 00:10:00.320
So 2y plus 4y is 6y,
which makes sense.

00:10:00.320 --> 00:10:02.840
If I have 2 apples plus
4 apples, I'm going

00:10:02.840 --> 00:10:03.850
to have 6 apples.

00:10:03.850 --> 00:10:07.810
If I have 2 y's plus 4 y's
I'm going to have 6 y's.

00:10:07.810 --> 00:10:09.615
Now that's going to
be equal to 18.

00:10:12.270 --> 00:10:15.250
And now, hopefully, we
understand how to do this.

00:10:15.250 --> 00:10:18.490
If I have 6 times something is
equal to 18, if I divide both

00:10:18.490 --> 00:10:22.710
sides of this equation by 6,
I'll solve for the something.

00:10:22.710 --> 00:10:29.920
So divide the left hand
side by 6, and divide the

00:10:29.920 --> 00:10:31.870
right hand side by 6.

00:10:31.870 --> 00:10:39.100
And we are left with
y is equal to 3.

00:10:39.100 --> 00:10:40.160
And you could try it out.

00:10:40.160 --> 00:10:41.750
That's what's cool
about an equation.

00:10:41.750 --> 00:10:44.290
You can always check to see
if you got the right answer.

00:10:44.290 --> 00:10:45.220
Let's see if that works.

00:10:45.220 --> 00:10:52.120
2 times 3 plus 4 times
3 is equal to what?

00:10:52.120 --> 00:10:56.640
2 times 3, this
right here is 6.

00:10:56.640 --> 00:10:59.580
And then 4 times 3 is 12.

00:10:59.580 --> 00:11:03.410
6 plus 12 is, indeed,
equal to 18.

00:11:03.410 --> 00:11:05.640
So it works out.


Converting Farenheit to Celsius

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.500
A thermometer in a science lab
displays the temperature in

00:00:03.500 --> 00:00:05.510
both Celsius and Fahrenheit.

00:00:05.510 --> 00:00:09.785
If the mercury in the
thermometer rises to 56

00:00:09.785 --> 00:00:13.480
degrees Fahrenheit, what is
the corresponding Celsius

00:00:13.480 --> 00:00:14.290
temperature?

00:00:14.290 --> 00:00:16.560
So right from the get go,
there's a formula that if you

00:00:16.560 --> 00:00:18.400
get have the Celsius
temperature, it'll give you

00:00:18.400 --> 00:00:20.210
the Fahrenheit, if you have the
Fahrenheit, it'll give you

00:00:20.210 --> 00:00:21.160
the Celsius.

00:00:21.160 --> 00:00:23.040
Now I have a bad memory,
so I always have to

00:00:23.040 --> 00:00:23.920
pseudo-derive it.

00:00:23.920 --> 00:00:25.830
And maybe in future videos,
we'll totally derive it.

00:00:25.830 --> 00:00:28.270
Once we get a little bit more
algebra, if you just know the

00:00:28.270 --> 00:00:30.920
boiling and freezing points in
both scales, you should be

00:00:30.920 --> 00:00:32.960
able to figure out the
relationship between the two.

00:00:32.960 --> 00:00:35.870
Now the way I always remember
it is that there's a

00:00:35.870 --> 00:00:38.570
relationship of 5/9 or
9/5 between the two.

00:00:38.570 --> 00:00:40.950
And then you have to
add or subtract 32.

00:00:40.950 --> 00:00:45.490
And the way I think about it is,
the Fahrenheit number is

00:00:45.490 --> 00:00:46.960
always going to be larger.

00:00:46.960 --> 00:00:50.340
So if you start with-- or at
least while we are in our

00:00:50.340 --> 00:00:53.620
normal, everyday living scales
when water is in its liquid

00:00:53.620 --> 00:00:54.760
form and all that.

00:00:54.760 --> 00:00:57.020
So if you start with a Celsius,
I say, oh I have to

00:00:57.020 --> 00:00:57.740
make it bigger.

00:00:57.740 --> 00:01:00.860
So I have to say, OK, I'm going
to multiply that by 9/5.

00:01:00.860 --> 00:01:02.900
That will make it bigger.

00:01:02.900 --> 00:01:04.959
And then I'm going
to add 32 to it.

00:01:04.959 --> 00:01:07.450
And the reason why I know to add
32 is because when Celsius

00:01:07.450 --> 00:01:10.940
is 0, 0 times 9/5 is still 0.

00:01:10.940 --> 00:01:13.950
And we know when Celsius is 0,
Fahrenheit is going to be 32.

00:01:13.950 --> 00:01:17.060
So I know I have to
add 32 to it.

00:01:17.060 --> 00:01:22.070
So, at least my gut is telling
me, that the formula is 9/5

00:01:22.070 --> 00:01:24.070
times the Celsius degrees,
plus 32.

00:01:24.070 --> 00:01:26.100
And you can verify that this
works for the boiling point.

00:01:26.100 --> 00:01:27.360
This definitely works for
the freezing point.

00:01:27.360 --> 00:01:30.740
When Celsius is 0, this is going
to be 0, you add 32,

00:01:30.740 --> 00:01:31.930
that's going to be Fahrenheit.

00:01:31.930 --> 00:01:38.650
When Celsius is 100, 9/5 times
100 is 180, plus 32 is 212.

00:01:38.650 --> 00:01:39.900
So this works.

00:01:43.150 --> 00:01:45.450
They've given us the Fahrenheit
temperature, and we

00:01:45.450 --> 00:01:47.580
need to get the Celsius
temperature.

00:01:47.580 --> 00:01:51.190
Now I could just put the 56 in
here, and solve for C, and

00:01:51.190 --> 00:01:52.660
that will completely work.

00:01:52.660 --> 00:01:56.110
Or I could solve for C using
this, so I have the equation

00:01:56.110 --> 00:01:57.730
for Celsius in terms
of Fahrenheit.

00:01:57.730 --> 00:01:58.650
Either way.

00:01:58.650 --> 00:02:00.020
So let's do it the
second way, so we

00:02:00.020 --> 00:02:01.240
have the second equation.

00:02:01.240 --> 00:02:11.510
So if we subtract 32 from both
sides of this equation, we get

00:02:11.510 --> 00:02:18.370
Fahrenheit minus 32 is equal
to 9/5 times Celsius.

00:02:18.370 --> 00:02:20.870
Now to solve for Celsius, we
can multiply both sides of

00:02:20.870 --> 00:02:22.120
this equation times 5/9.

00:02:29.850 --> 00:02:33.690
And so we will get 9 cancels
with 9, 5 cancels with 5.

00:02:33.690 --> 00:02:35.930
That was the whole point behind
multiplying by the

00:02:35.930 --> 00:02:37.180
reciprocal.

00:02:39.550 --> 00:02:42.710
Remember, this 5/9 is going to
be times this whole left side.

00:02:42.710 --> 00:02:44.780
We're left with Celsius.

00:02:44.780 --> 00:02:51.770
Celsius is equal to 5/9 times
Fahrenheit, minus 32.

00:02:51.770 --> 00:02:54.740
So we just rearranged this
equation up here to get this

00:02:54.740 --> 00:02:57.280
equation down here.

00:02:57.280 --> 00:02:59.450
I didn't derive the equation in
this video, I just kind of

00:02:59.450 --> 00:03:01.660
pseudo-remembered it, and made
sure I got the equation right.

00:03:01.660 --> 00:03:04.670
But you can derive it if
you just remember, it's

00:03:04.670 --> 00:03:05.730
essentially, if you know
how to find the

00:03:05.730 --> 00:03:07.850
equation of a line.

00:03:07.850 --> 00:03:11.170
You say, well, when Celsius-- if
you use the x-coordinate as

00:03:11.170 --> 00:03:21.550
Celsius-- when Celsius is
0, Fahrenheit is 32.

00:03:21.550 --> 00:03:26.490
And when Celsius is 100,
Fahrenheit is 212, and find

00:03:26.490 --> 00:03:28.280
the equation of the line between
these two points.

00:03:28.280 --> 00:03:30.140
Don't worry about it if
you don't know that.

00:03:30.140 --> 00:03:32.580
All you need to know for this
problem is this right here.

00:03:32.580 --> 00:03:34.490
But if you know how to solve the
equation for the line, you

00:03:34.490 --> 00:03:36.660
would come up with this answer
right here, and you wouldn't

00:03:36.660 --> 00:03:38.950
have to memorize it.

00:03:38.950 --> 00:03:43.610
Now that we've actually gotten
this far, let's just put in 56

00:03:43.610 --> 00:03:45.670
degrees for the Fahrenheit
temperature, and figure out

00:03:45.670 --> 00:03:47.720
what the Celsius
temperature is.

00:03:47.720 --> 00:03:52.100
So if we put 56 in there, we're
going to have 5/9 times

00:03:52.100 --> 00:03:59.680
56 minus 32 is equal to the
Celsius temperature.

00:03:59.680 --> 00:04:03.500
So 56 minus 32-- let me
continue it up here.

00:04:09.000 --> 00:04:14.810
56 minus 30 is 26,
minus 2 is 24.

00:04:14.810 --> 00:04:16.529
So this is going to be 24.

00:04:16.529 --> 00:04:20.990
So it's going to be
5/9 times 24.

00:04:20.990 --> 00:04:24.400
This right here is 24.

00:04:24.400 --> 00:04:27.280
5/9 times 24 is going
to be equal to C.

00:04:27.280 --> 00:04:31.720
And let's see if we can
simplify this at all.

00:04:31.720 --> 00:04:36.110
So this is the same thing
as 5 times 24 over 9.

00:04:36.110 --> 00:04:37.820
24 is divisible by 3.

00:04:37.820 --> 00:04:39.030
You divide it by 3, you get 8.

00:04:39.030 --> 00:04:41.600
9 divided by 3 is 3.

00:04:41.600 --> 00:04:47.890
So 5 times 8 is equal to 40/3
Celsius, degrees Celsius.

00:04:47.890 --> 00:04:50.870
Now if we wanted to write this
as a mixed number, 3 goes into

00:04:50.870 --> 00:04:54.660
40 thirteen times.

00:04:54.660 --> 00:04:56.160
And you have a remainder of 1.

00:04:56.160 --> 00:04:57.535
So it's 13 and 1/3.

00:04:57.535 --> 00:04:58.860
Now we could even check that.

00:04:58.860 --> 00:05:02.510
3 times 13 is 39,
plus 1 is 40.

00:05:02.510 --> 00:05:04.560
And if this wasn't obvious to
you, you could actually just

00:05:04.560 --> 00:05:07.350
take 3 into 40.

00:05:07.350 --> 00:05:08.640
3 goes into 4 one time.

00:05:08.640 --> 00:05:10.350
1 times 3 is 3.

00:05:10.350 --> 00:05:11.070
Subtract.

00:05:11.070 --> 00:05:12.200
You get 1.

00:05:12.200 --> 00:05:13.290
Bring down that 0.

00:05:13.290 --> 00:05:15.400
3 goes into 10 three times.

00:05:15.400 --> 00:05:17.060
3 times 3 is 9.

00:05:17.060 --> 00:05:17.720
You subtract.

00:05:17.720 --> 00:05:18.800
You get a remainder of 1.

00:05:18.800 --> 00:05:21.200
So it's 3 and 1/3.

00:05:21.200 --> 00:05:25.460
So this is 13 and 1/3
degrees Celsius.

00:05:25.460 --> 00:05:29.890
So if it's 56 degrees outside in
Fahrenheit, it's 13 and 1/3

00:05:29.890 --> 00:05:32.240
degrees in Celsius.


Converting within the metric system

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:06.190
Convert 51.8 decimeters
to kilometers.

00:00:06.190 --> 00:00:08.740
So let's just think about what
decimeters means in terms of

00:00:08.740 --> 00:00:11.440
meters, and then we can think
about what kilometers mean in

00:00:11.440 --> 00:00:12.770
terms of meters, and
then convert them.

00:00:12.770 --> 00:00:21.210
So we're going to start
with 51.8 decimeters.

00:00:21.210 --> 00:00:23.600
Now, as a starting point, just
to keep things simple, we're

00:00:23.600 --> 00:00:26.360
going to convert them into
meters, because this prefix

00:00:26.360 --> 00:00:28.710
tells us a lot about how
many meters this is.

00:00:28.710 --> 00:00:31.390
So if we want to convert it
into meters, our units are

00:00:31.390 --> 00:00:35.640
going to have to be meters
per decimeter.

00:00:41.010 --> 00:00:44.260
And we need this because this
decimeter and that decimeter

00:00:44.260 --> 00:00:46.820
are going to cancel out, and
we're going to be just left

00:00:46.820 --> 00:00:47.700
with meters.

00:00:47.700 --> 00:00:50.830
Now 1 meter is how
many decimeters?

00:00:50.830 --> 00:00:57.130
Well, the prefix deci
means 1/10.

00:00:57.130 --> 00:01:00.880
So 1 decimeter is
1/10 of a meter.

00:01:00.880 --> 00:01:03.410
Or another way of thinking
about it is there are 10

00:01:03.410 --> 00:01:05.630
decimeters for every meter.

00:01:05.630 --> 00:01:07.110
This is a smaller unit.

00:01:07.110 --> 00:01:12.100
10 of these make up 1 of these,
or 1/10 of a meter per

00:01:12.100 --> 00:01:14.540
decimeter, which is exactly
what that is telling you.

00:01:14.540 --> 00:01:16.550
1/10 of a meter.

00:01:16.550 --> 00:01:19.850
Now if you multiply these two
things, what do we get?

00:01:19.850 --> 00:01:25.140
Well, the decimeters cancel out,
and we divide 51.8 by 10.

00:01:25.140 --> 00:01:28.800
Multiplying by 1/10 is the same
thing as dividing by 10.

00:01:28.800 --> 00:01:32.300
So this is going to be equal
to 51.8 divided by 10.

00:01:32.300 --> 00:01:33.840
The decimal's just going
to move to the left.

00:01:33.840 --> 00:01:35.840
It's going to be 5.18.

00:01:35.840 --> 00:01:40.970
So this is going to
be 5.18 meters.

00:01:40.970 --> 00:01:42.360
And you should always
do a reality check.

00:01:42.360 --> 00:01:43.490
Did it make sense?

00:01:43.490 --> 00:01:46.740
Does it make sense that
51 decimeters

00:01:46.740 --> 00:01:48.940
are equal to 5 meters?

00:01:48.940 --> 00:01:49.650
Well, sure.

00:01:49.650 --> 00:01:50.490
It should go down.

00:01:50.490 --> 00:01:53.520
We're going from a smaller unit
to a larger unit, so we

00:01:53.520 --> 00:01:57.420
should have fewer of that larger
unit to make the same

00:01:57.420 --> 00:01:58.370
distance in this case.

00:01:58.370 --> 00:02:00.260
A meter is a unit of length.

00:02:00.260 --> 00:02:07.290
Now, we're at 5.18 meters, and
we want to now convert it to

00:02:07.290 --> 00:02:08.540
kilometers.

00:02:10.580 --> 00:02:15.030
So we're going to want meters
in the denominator and

00:02:15.030 --> 00:02:19.100
kilometers in the numerator.

00:02:19.100 --> 00:02:19.960
Why?

00:02:19.960 --> 00:02:23.450
Because this meter and that
meter will cancel out.

00:02:23.450 --> 00:02:27.570
Now, how many meters make
up a kilometer?

00:02:27.570 --> 00:02:29.900
The kilo tells us 1,000.

00:02:29.900 --> 00:02:32.010
That is 1,000 meters.

00:02:32.010 --> 00:02:36.770
So 1,000 meters are equal
to 1 kilometer.

00:02:36.770 --> 00:02:39.170
So if we were to multiply this,
the meters cancel out

00:02:39.170 --> 00:02:40.910
and we'll just be left
with kilometers.

00:02:40.910 --> 00:02:44.590
And multiplying something by
1/1,000 is the exact same

00:02:44.590 --> 00:02:47.840
thing as dividing by 1,000,
and that makes sense.

00:02:47.840 --> 00:02:51.190
We should get a smaller number
here because we're going from

00:02:51.190 --> 00:02:53.020
a smaller unit to
a larger unit.

00:02:53.020 --> 00:02:55.670
So we're going to need
a much smaller amount

00:02:55.670 --> 00:02:57.190
of the larger unit.

00:02:57.190 --> 00:03:05.940
So this is going to be equal
to 5.18/1,000 kilometers.

00:03:05.940 --> 00:03:15.890
Now, the easy way to do this,
if we take 5.18/10 is equal

00:03:15.890 --> 00:03:17.880
to-- we just move the decimal
one to the left.

00:03:17.880 --> 00:03:22.080
It's going to be
equal to 0.518.

00:03:22.080 --> 00:03:26.950
If we take 5.18 and divide it
by 100, we're going to take

00:03:26.950 --> 00:03:28.130
the decimal two to the left.

00:03:28.130 --> 00:03:31.100
So one, two, we're going to
have to add a zero here.

00:03:31.100 --> 00:03:36.090
So it's going to be 0.0518.

00:03:36.090 --> 00:03:44.356
And if we take 5.18 and we
divide by 1,000, we're going

00:03:44.356 --> 00:03:47.230
to move the decimal
three to the left.

00:03:47.230 --> 00:03:48.200
So we'll just do it.

00:03:48.200 --> 00:03:52.610
So if we start with 5.18-- the
decimal started here-- we're

00:03:52.610 --> 00:03:55.470
going to go one, two,
so that's one

00:03:55.470 --> 00:03:57.470
zero, and then three.

00:03:57.470 --> 00:03:59.380
And that's where the decimal
is going to sit there.

00:03:59.380 --> 00:04:06.170
So this is Or another way, this
is a little over 5/1,000.

00:04:06.170 --> 00:04:08.030
This right here is 5/1,000.

00:04:08.030 --> 00:04:09.620
It's a little bit
more than that.

00:04:09.620 --> 00:04:22.930
So our final answer: 51.8
decimeters is equal to 0.00518

00:04:22.930 --> 00:04:24.220
kilometers.

00:04:24.220 --> 00:04:25.800
And always do a reality check.

00:04:25.800 --> 00:04:27.310
Does that make sense?

00:04:27.310 --> 00:04:28.070
Well, sure.

00:04:28.070 --> 00:04:30.800
We're going from a really small
unit here, a bunch of a

00:04:30.800 --> 00:04:33.950
small unit, to a very small
amount of a large unit.

00:04:33.950 --> 00:04:34.940
It makes sense.

00:04:34.940 --> 00:04:38.960
It makes sense that this number
goes down while the

00:04:38.960 --> 00:04:41.790
actual unit becomes larger.


U.S. customary and metric units

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.300
We're asked to sort the
following units of measurement

00:00:03.300 --> 00:00:06.860
into two categories:
U.S. customary

00:00:06.860 --> 00:00:10.260
units and metric units.

00:00:10.260 --> 00:00:12.430
So these are just two different
systems. You'll get

00:00:12.430 --> 00:00:13.840
more and more familiar
with them.

00:00:13.840 --> 00:00:17.110
Then indicate whether each unit
measures length, weight,

00:00:17.110 --> 00:00:18.390
mass, or volume.

00:00:18.390 --> 00:00:20.970
Let's do the first. Let's see
which of these are U.S.

00:00:20.970 --> 00:00:23.590
customary unit versus
metric units.

00:00:23.590 --> 00:00:27.900
So the liter is a metric unit.

00:00:27.900 --> 00:00:30.020
You would use it in
the metric system.

00:00:30.020 --> 00:00:33.390
A gallon is a U.S.
customary unit.

00:00:33.390 --> 00:00:35.700
We've been dealing with that.

00:00:35.700 --> 00:00:39.090
If you fill your gasoline in
Europe, you're going to be

00:00:39.090 --> 00:00:40.410
filling it in terms of liters.

00:00:40.410 --> 00:00:41.730
In the U.S., you're going
to be filling

00:00:41.730 --> 00:00:43.570
it in terms of gallons.

00:00:43.570 --> 00:00:45.190
And we're going to talk about
whether they're units of

00:00:45.190 --> 00:00:46.970
volume and whatnot
in a little bit.

00:00:46.970 --> 00:00:48.985
Decigram, that is
metric system.

00:00:53.530 --> 00:00:56.750
In general, whenever you see
these prefixes, deci, centi,

00:00:56.750 --> 00:00:59.000
kilo, you're dealing with
the metric system.

00:00:59.000 --> 00:01:00.840
No one ever talks about
a kilopound.

00:01:00.840 --> 00:01:03.010
I guess you could, but no one
really talks about it.

00:01:03.010 --> 00:01:04.379
Same thing, millimeter.

00:01:04.379 --> 00:01:05.519
This is metric system.

00:01:05.519 --> 00:01:07.190
A gram is metric system.

00:01:07.190 --> 00:01:08.700
Meter is metric system.

00:01:08.700 --> 00:01:12.700
The foot is a U.S.
customary unit.

00:01:12.700 --> 00:01:14.750
We'll talk about whether it's
distance or any of that in a

00:01:14.750 --> 00:01:15.450
little bit.

00:01:15.450 --> 00:01:17.900
Kilogram, once again,
it is metric units.

00:01:20.630 --> 00:01:22.750
In case you haven't gotten what
I'm doing here, blue for

00:01:22.750 --> 00:01:27.540
metric, red for U.S. customary
units, or I guess magenta.

00:01:27.540 --> 00:01:31.720
Centiliter, that is metric.

00:01:31.720 --> 00:01:34.180
Centimeter, meters are metric.

00:01:34.180 --> 00:01:36.880
And notice we have the
prefix in both cases.

00:01:36.880 --> 00:01:39.110
Centi means 1/100.

00:01:39.110 --> 00:01:42.840
Cup, that is U.S.
customary units.

00:01:42.840 --> 00:01:44.800
I have to do that
in the magenta.

00:01:44.800 --> 00:01:47.140
Cup, U.S. customary units.

00:01:47.140 --> 00:01:49.830
Meter, that is the
metric system.

00:01:52.360 --> 00:01:55.530
Pound, U.S. customary units.

00:01:55.530 --> 00:01:57.220
It's getting a little tedious.

00:01:57.220 --> 00:02:01.250
Inch, same thing, that's what
we use in the U.S. Ounce, we

00:02:01.250 --> 00:02:05.600
use that in the U.S. And then
the yard, we also use that in

00:02:05.600 --> 00:02:06.110
the U.S.

00:02:06.110 --> 00:02:07.190
Now we've divided them up.

00:02:07.190 --> 00:02:11.120
All the magenta ones are used
in the U.S. All of the blue

00:02:11.120 --> 00:02:13.230
ones are used really in the
rest of the world, and

00:02:13.230 --> 00:02:15.790
actually some places in
the U.S. as well.

00:02:15.790 --> 00:02:18.260
I think a lot of the world is
frustrated that the U.S., that

00:02:18.260 --> 00:02:20.450
we're not all converted to this
because the metric system

00:02:20.450 --> 00:02:21.760
is actually a little
bit more logical.

00:02:21.760 --> 00:02:23.870
It's easy to just figure out
what it's saying, and we'll

00:02:23.870 --> 00:02:25.970
deal with that in more
detail in the future.

00:02:25.970 --> 00:02:29.090
Now the next thing we to
figure out is whether

00:02:29.090 --> 00:02:33.290
something is a measure of
length, weight/mass-- and

00:02:33.290 --> 00:02:34.640
they're not exactly
the same thing.

00:02:34.640 --> 00:02:38.580
Mass is how much of a substance
you have. Weight is

00:02:38.580 --> 00:02:41.060
how much force with
which gravity is

00:02:41.060 --> 00:02:42.590
pulling on that mass.

00:02:42.590 --> 00:02:44.690
And it would change depending
on what planet you're on.

00:02:44.690 --> 00:02:47.870
But on Earth, they tend to be
used interchangeably, so we'll

00:02:47.870 --> 00:02:49.930
use it roughly interchangeably
here.

00:02:49.930 --> 00:02:51.680
And then you have volume,
or how much space

00:02:51.680 --> 00:02:52.820
something takes up.

00:02:52.820 --> 00:02:54.380
So this is distance.

00:02:54.380 --> 00:02:58.000
This is moving in
one dimension.

00:02:58.000 --> 00:02:59.790
Mass is how much
stuff there is.

00:02:59.790 --> 00:03:02.540
Weight is how much the force
that stuff is pulled on, on a

00:03:02.540 --> 00:03:05.470
planet, by gravity, or I
guess a star anywhere.

00:03:05.470 --> 00:03:10.030
And volume is how much space
does that stuff take up.

00:03:10.030 --> 00:03:11.000
Now let's think about it.

00:03:11.000 --> 00:03:13.580
Liter is volume.

00:03:13.580 --> 00:03:15.380
This right here is volume.

00:03:15.380 --> 00:03:17.390
How much space do you take up.

00:03:17.390 --> 00:03:19.860
Gallon is also volume.

00:03:19.860 --> 00:03:23.390
That's in the U.S. And in
Europe, or in the metric

00:03:23.390 --> 00:03:24.830
system, it would be a liter.

00:03:24.830 --> 00:03:27.420
That's a gram.

00:03:27.420 --> 00:03:29.745
Gram is a unit of mass.

00:03:32.760 --> 00:03:36.110
So decigram just means
1/10 of a gram.

00:03:36.110 --> 00:03:36.810
Millimeter.

00:03:36.810 --> 00:03:38.610
Meter is a unit.

00:03:38.610 --> 00:03:43.890
Meter right here, that is the
unit of distance or of length.

00:03:43.890 --> 00:03:48.500
Millimeter, milli means
1/1,000 of a meter.

00:03:48.500 --> 00:03:52.570
Foot, that is also
a unit of length.

00:03:52.570 --> 00:03:56.250
Kilogram, that just
means 1,000 grams.

00:03:56.250 --> 00:03:57.320
Kilo means a thousand.

00:03:57.320 --> 00:04:00.800
Gram, we already said,
is a unit of mass.

00:04:00.800 --> 00:04:04.900
Centiliter, that means
1/100 of a liter.

00:04:04.900 --> 00:04:08.640
Liter, we already figured out,
is a unit of volume.

00:04:08.640 --> 00:04:10.330
Centimeter, we already
figured out.

00:04:10.330 --> 00:04:12.700
Meter is a unit of length.

00:04:12.700 --> 00:04:15.650
Centimeter means 1/100
of a meter.

00:04:15.650 --> 00:04:20.630
So this is a unit of length.

00:04:20.630 --> 00:04:23.370
Cup, we've seen multiple
times already.

00:04:23.370 --> 00:04:27.110
It is a unit of volume,
how much space does

00:04:27.110 --> 00:04:28.240
something take up.

00:04:28.240 --> 00:04:30.390
Meter, that is length.

00:04:30.390 --> 00:04:32.210
We've seen it multiple
times already.

00:04:32.210 --> 00:04:34.820
Pound, that is actually
a unit of weight.

00:04:38.720 --> 00:04:42.490
An inch is a unit of length.

00:04:42.490 --> 00:04:43.950
We're all familiar with it.

00:04:43.950 --> 00:04:48.020
An ounce-- you have to be
careful here-- if someone just

00:04:48.020 --> 00:04:50.830
has an ounce, that is
1/16 of a pound.

00:04:50.830 --> 00:04:53.330
It as a unit of weight.

00:04:53.330 --> 00:04:56.460
If it was written fluid ounce,
then we'd be talking about

00:04:56.460 --> 00:04:59.790
1/16 of a pint, and then it
would be a unit of volume.

00:04:59.790 --> 00:05:01.300
But since it's just ounce,
it's a unit of

00:05:01.300 --> 00:05:03.380
weight, 1/16 of a pound.

00:05:03.380 --> 00:05:07.690
And then finally, a yard
is a unit of length.

00:05:07.690 --> 00:05:09.460
And we are done.


Comparing Celsius and Fahrenheit temperature scales

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:02.840
Look at the two thermometers
below.

00:00:02.840 --> 00:00:06.450
Identify which is Celsius and
which is Fahrenheit, and then

00:00:06.450 --> 00:00:10.600
label the boiling and freezing
points of water on each.

00:00:10.600 --> 00:00:12.490
Now, the Celsius scale
is what's used in

00:00:12.490 --> 00:00:13.740
the most of the world.

00:00:22.830 --> 00:00:24.990
And the easy way to tell that
you're dealing with the

00:00:24.990 --> 00:00:31.810
Celsius scale is on the Celsius
scale, 0 degrees is

00:00:31.810 --> 00:00:37.960
freezing of water at standard
temperature and pressure, and

00:00:37.960 --> 00:00:44.300
100 degrees is the boiling point
of water at standard

00:00:44.300 --> 00:00:45.570
temperature and pressure.

00:00:45.570 --> 00:00:50.180
Now, on the Fahrenheit scale,
which is used mainly in the

00:00:50.180 --> 00:01:02.360
United States, the freezing
point of water is 32 degrees,

00:01:02.360 --> 00:01:09.300
and boiling of water
is 212 degrees.

00:01:09.300 --> 00:01:11.670
As you could tell, Celsius,
the whole scale came from

00:01:11.670 --> 00:01:15.160
using freezing as 0 of regular
water at standard temperature

00:01:15.160 --> 00:01:17.990
and pressure and setting
100 to be boiling.

00:01:17.990 --> 00:01:20.120
On some level, it makes a little
bit more logical sense,

00:01:20.120 --> 00:01:22.790
but at least here in the U.S.,
we still use Fahrenheit.

00:01:22.790 --> 00:01:24.450
Now let's figure out which of
these are Fahrenheit and which

00:01:24.450 --> 00:01:24.990
are Celsius.

00:01:24.990 --> 00:01:27.460
Now remember, regardless of
which thermometer you're

00:01:27.460 --> 00:01:30.120
using, water will always
actually boil at the exact

00:01:30.120 --> 00:01:30.850
same temperature.

00:01:30.850 --> 00:01:35.070
So Fahrenheit, 32 degrees, this
has to be the same thing

00:01:35.070 --> 00:01:37.050
as Celsius 0 degrees.

00:01:37.050 --> 00:01:38.410
So let's see what happens.

00:01:38.410 --> 00:01:42.370
So when this temperature right
here is 0, this one over here,

00:01:42.370 --> 00:01:44.650
it looks like it's negative
something.

00:01:44.650 --> 00:01:46.700
So this one right here doesn't
look like Celsius.

00:01:46.700 --> 00:01:49.610
Here, if we say this is Celsius,
this looks pretty

00:01:49.610 --> 00:01:53.530
close to 32 on this one.

00:01:53.530 --> 00:01:55.660
Let me do that in
a darker color.

00:01:55.660 --> 00:02:00.780
So this one right here looks
like Celsius, and this one

00:02:00.780 --> 00:02:04.300
right here looks like
Fahrenheit.

00:02:04.300 --> 00:02:07.550
And the way I was able to tell
is that the 0 degrees Celsius

00:02:07.550 --> 00:02:10.759
needs to be the same thing
as 32 degrees Fahrenheit.

00:02:10.759 --> 00:02:15.940
In both cases, this is
where water freezes,

00:02:15.940 --> 00:02:17.355
the freezing point.

00:02:17.355 --> 00:02:19.000
That is water freezing.

00:02:19.000 --> 00:02:19.900
And let's make sure
we're right.

00:02:19.900 --> 00:02:22.330
So if this is the Celsius scale,
this is where water

00:02:22.330 --> 00:02:25.370
will boil, 100 degrees Celsius,
and that looks like

00:02:25.370 --> 00:02:28.060
it is right about 212
on the other scale.

00:02:28.060 --> 00:02:33.210
So right there is where water
is boiling at standard

00:02:33.210 --> 00:02:34.280
temperature and pressure.

00:02:34.280 --> 00:02:37.070
So this thing on the right,
right here, I guess I'll

00:02:37.070 --> 00:02:41.890
circle it in orange,
that is Celsius.

00:02:41.890 --> 00:02:45.960
And then the one on the left,
I'll do it in magenta, the one

00:02:45.960 --> 00:02:49.370
on the left is Fahrenheit.


Applying the metric system

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:06.640
The perimeter of a rectangular
fence measures 0.72

00:00:06.640 --> 00:00:08.270
kilometers.

00:00:08.270 --> 00:00:12.670
The length of the fenced
area is 160 meters.

00:00:12.670 --> 00:00:14.550
What is its width?

00:00:14.550 --> 00:00:16.280
So we have a rectangular
fence.

00:00:18.980 --> 00:00:19.950
Let me draw it.

00:00:19.950 --> 00:00:21.710
So that is a rectangle.

00:00:21.710 --> 00:00:23.880
You can imagine we're
looking from above.

00:00:23.880 --> 00:00:26.540
This line is the top
of the fence.

00:00:26.540 --> 00:00:29.470
And if you take its perimeter,
the perimeter is the distance

00:00:29.470 --> 00:00:30.520
around the fence.

00:00:30.520 --> 00:00:32.820
If you take this distance plus
this distance plus that

00:00:32.820 --> 00:00:36.960
distance plus that distance,
it's going to be 0.72

00:00:36.960 --> 00:00:37.920
kilometers.

00:00:37.920 --> 00:00:40.350
That's the total distance
of all of the sides.

00:00:40.350 --> 00:00:43.510
Now, the length of the fenced
area is 160 meters.

00:00:43.510 --> 00:00:44.640
Let's call this the length.

00:00:44.640 --> 00:00:52.120
So this distance right
here is 160 meters.

00:00:52.120 --> 00:00:54.960
Since it's a rectangle, that
distance over there is also

00:00:54.960 --> 00:00:58.510
going to be 160 meters, and we
need to figure out its width.

00:00:58.510 --> 00:01:00.560
They want us to figure
out the width.

00:01:00.560 --> 00:01:04.650
The width is this distance right
over here, which is also

00:01:04.650 --> 00:01:09.120
going to be this distance
right over there.

00:01:09.120 --> 00:01:11.050
Now, what is the perimeter
of the fence?

00:01:11.050 --> 00:01:14.810
The perimeter is the sum of this
length plus this length

00:01:14.810 --> 00:01:17.270
plus that length plus
that length.

00:01:17.270 --> 00:01:23.600
So the perimeter-- let me write
it in orange-- is going

00:01:23.600 --> 00:01:34.260
to be equal to w, the width,
plus 160 meters, plus the

00:01:34.260 --> 00:01:44.650
width, plus 160 meters.

00:01:44.650 --> 00:01:46.990
And let's assume
w is in meters.

00:01:50.470 --> 00:01:51.100
So we could add it all up.

00:01:51.100 --> 00:01:53.010
So if you were to add all of
these up, you'd get a certain

00:01:53.010 --> 00:01:53.800
number of meters.

00:01:53.800 --> 00:01:56.690
If you were to add all of these
up, you have a w plus a

00:01:56.690 --> 00:02:07.100
w, so you'd be 2w, plus-- 160
plus 160 is 320-- so plus 320.

00:02:07.100 --> 00:02:09.550
And this whole thing
is in meters.

00:02:09.550 --> 00:02:12.120
We're assuming that
w is in meters.

00:02:12.120 --> 00:02:14.610
Now, they also told us that the
perimeter of the fence is

00:02:14.610 --> 00:02:17.200
0.72 kilometers.

00:02:17.200 --> 00:02:21.240
So the perimeter-- let me do it
in that same orange color.

00:02:21.240 --> 00:02:31.400
The perimeter is also equal to
0.72 kilometers, and we can

00:02:31.400 --> 00:02:34.580
abbreviate that with km.

00:02:34.580 --> 00:02:37.560
Now, if we wanted to solve an
equation, if we wanted to set

00:02:37.560 --> 00:02:41.530
this thing equal to the
perimeter they gave us, we

00:02:41.530 --> 00:02:43.260
have to make sure that the
units are the same.

00:02:43.260 --> 00:02:44.390
Here it's in meters.

00:02:44.390 --> 00:02:46.430
Here it's in kilometers.

00:02:46.430 --> 00:02:50.470
So let's convert this 0.72
kilometers to meters.

00:02:50.470 --> 00:02:53.530
And the way to do that, we want
the kilometers in the

00:02:53.530 --> 00:02:56.260
denominator so it cancels out
with the kilometers, and you

00:02:56.260 --> 00:02:58.330
want meters in the numerator.

00:02:58.330 --> 00:03:00.900
Now, how many meters
equal a kilometer?

00:03:00.900 --> 00:03:04.720
Well, it's 1,000 meters
for every 1 kilometer.

00:03:04.720 --> 00:03:06.890
1 kilometer is 1,000 meters.

00:03:06.890 --> 00:03:09.200
Now, if you multiply these
two things, the

00:03:09.200 --> 00:03:10.870
kilometers cancel out.

00:03:10.870 --> 00:03:13.160
You're just going to be
left with the meters.

00:03:13.160 --> 00:03:17.820
And then when you multiply 0.72
times 1,000, you could

00:03:17.820 --> 00:03:20.490
just first say, OK, let's
take 72 times 1,000.

00:03:20.490 --> 00:03:24.050
72 times 1,000 would
be 72 with three

00:03:24.050 --> 00:03:26.110
zeroes: one, two, three.

00:03:26.110 --> 00:03:27.860
And then out of the two numbers
that you're taking the

00:03:27.860 --> 00:03:30.070
product of, we have two
numbers behind the

00:03:30.070 --> 00:03:31.620
decimal: one, two.

00:03:31.620 --> 00:03:34.360
So you put two numbers behind
the decimal, one, two, right

00:03:34.360 --> 00:03:35.350
over there.

00:03:35.350 --> 00:03:40.380
So 0.72 times 1,000 is 720.

00:03:40.380 --> 00:03:41.230
And that makes sense.

00:03:41.230 --> 00:03:45.520
0.72, it's not quite 1, but
it's getting close to 1,

00:03:45.520 --> 00:03:49.720
almost 3/4 of a whole, and this
is almost 3/4 of 1,000.

00:03:49.720 --> 00:03:50.740
So it makes sense.

00:03:50.740 --> 00:03:54.510
So the perimeter, this
information right here, 0.72

00:03:54.510 --> 00:03:58.400
kilometers, is the same
thing as 720 meters.

00:03:58.400 --> 00:04:03.910
So if you take 2 times the width
in meters, and add that

00:04:03.910 --> 00:04:09.700
to the combined lengths-- it's
160 and 160, that's 320, and

00:04:09.700 --> 00:04:16.019
all of this is in meters-- that
is going to be equal to

00:04:16.019 --> 00:04:22.790
the total perimeter, which
is 720 meters.

00:04:22.790 --> 00:04:25.990
Now that we have the units
right, we can solve for w, and

00:04:25.990 --> 00:04:30.370
w will tell us this
width in meters.

00:04:30.370 --> 00:04:35.780
So 2w plus 320 is
equal to 720.

00:04:35.780 --> 00:04:38.790
We can subtract 320 from both
sides of this equation.

00:04:38.790 --> 00:04:41.620
Remember, an equation is saying,
that is equal to that.

00:04:41.620 --> 00:04:44.120
If we want them both always to
be equal, whatever we do to

00:04:44.120 --> 00:04:46.670
the left-hand side, we have
to do the right-hand side.

00:04:46.670 --> 00:04:48.915
So we subtract 320
from both sides.

00:04:54.220 --> 00:04:56.190
The left-hand side,
that cancels out.

00:04:56.190 --> 00:04:58.100
That's why we subtracted 320.

00:04:58.100 --> 00:05:03.000
And you just have left 2 times
the width is equal to 720

00:05:03.000 --> 00:05:05.550
minus 320, which is 400.

00:05:05.550 --> 00:05:07.585
Now you can divide both
sides of this by 2.

00:05:11.120 --> 00:05:13.030
And then you're left with
the w is equal to

00:05:13.030 --> 00:05:16.820
400/2, which is 200.

00:05:16.820 --> 00:05:20.790
And we said that w was
in meters, so the

00:05:20.790 --> 00:05:27.350
width is 200 meters.

00:05:27.350 --> 00:05:28.610
Let's see if it makes sense.

00:05:28.610 --> 00:05:33.680
200 plus 200 is 400,
plus 320 is 720.

00:05:33.680 --> 00:05:38.200
So the total perimeter is 720
meters, which is the exact

00:05:38.200 --> 00:05:42.020
same thing as 0.72 kilometers.


Conversion between metric units

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:01.490
We're asked how many

00:00:01.490 --> 00:00:04.410
centiliters are in one dekaliter?

00:00:04.410 --> 00:00:06.710
So the first thing we want to
do is just think about how

00:00:06.710 --> 00:00:09.270
much is a centiliter relative to
a liter, and how much is a

00:00:09.270 --> 00:00:11.120
dekaliter relative to a liter?

00:00:11.120 --> 00:00:12.710
And I'll write the
prefixes down.

00:00:12.710 --> 00:00:15.173
And really, you should have
these memorized because you're

00:00:15.173 --> 00:00:17.250
going to see these prefixes
over and over again for

00:00:17.250 --> 00:00:18.880
different types of units.

00:00:18.880 --> 00:00:22.890
So the prefix, kilo, sometimes
[? ki-lo, ?]

00:00:22.890 --> 00:00:27.260
this means 1,000.

00:00:27.260 --> 00:00:33.670
If you see hecto,
hecto means 100.

00:00:33.670 --> 00:00:38.100
Deka means 10.

00:00:38.100 --> 00:00:41.220
If you have nothing, then
that just means 1.

00:00:41.220 --> 00:00:42.670
Let's put that there.

00:00:42.670 --> 00:00:47.940
Then if you have deci,
this means 1/10.

00:00:47.940 --> 00:00:54.810
If you have centi,
this means 1/100.

00:00:54.810 --> 00:00:59.570
If you have milli, this
means 1/1,000.

00:00:59.570 --> 00:01:01.220
So let's go back to what we
have. We have centiliters.

00:01:04.819 --> 00:01:06.640
Let me write this in
a different color.

00:01:06.640 --> 00:01:22.350
If you have a centiliter, this
is equal to 1/100 of a liter,

00:01:22.350 --> 00:01:26.430
Or you could say 1 liter for
every 100 centiliters, so you

00:01:26.430 --> 00:01:34.310
could also write it like this:
1 liter for every 100, or per

00:01:34.310 --> 00:01:42.060
every 100, centiliters.

00:01:42.060 --> 00:01:44.940
So we got the centi, now let's
think about the dekaliter.

00:01:44.940 --> 00:01:46.820
So the deka is right
over here.

00:01:46.820 --> 00:01:54.880
So a dekaliter means
10 liters.

00:01:54.880 --> 00:02:02.300
Or another way to say it is for
every 10 liters, you will

00:02:02.300 --> 00:02:05.930
have 1 dekaliter.

00:02:05.930 --> 00:02:08.090
Now, before I actually
work out the problem,

00:02:08.090 --> 00:02:10.240
what's going on here?

00:02:10.240 --> 00:02:13.380
We're going from a smaller
unit to a larger unit, so

00:02:13.380 --> 00:02:16.440
there are going to be many of
the smaller units in one of

00:02:16.440 --> 00:02:17.380
the larger ones.

00:02:17.380 --> 00:02:19.190
And we can do it
multiple ways.

00:02:19.190 --> 00:02:23.740
So we want to essentially
convert 1 dekaliter into

00:02:23.740 --> 00:02:24.600
centiliters.

00:02:24.600 --> 00:02:26.590
Now, we could just do it by
looking at this chart, or we

00:02:26.590 --> 00:02:29.930
could do it with the dimensional
analysis, making

00:02:29.930 --> 00:02:31.020
sure the dimensions work out.

00:02:31.020 --> 00:02:32.240
Let's do it the first way.

00:02:32.240 --> 00:02:37.680
So if you have one dekaliter,
how many liters is that?

00:02:42.540 --> 00:02:47.690
1 dekaliter over here would be
the same thing as 10 liters.

00:02:47.690 --> 00:02:48.860
That's liters.

00:02:48.860 --> 00:02:51.290
We're assuming that our
unit is liters here.

00:02:51.290 --> 00:02:54.380
And then 10 liters is going
to be how many deciliters?

00:02:54.380 --> 00:02:56.990
It's going to be 100
deciliters, right?

00:02:56.990 --> 00:02:59.370
Because each of these is
10 deciliters, and

00:02:59.370 --> 00:03:00.750
you have 10 of them.

00:03:00.750 --> 00:03:02.790
So every time you go down,
you're going to be multiplying

00:03:02.790 --> 00:03:04.510
by a factor of 10.

00:03:04.510 --> 00:03:07.560
100 deciliters is how
many centiliters?

00:03:07.560 --> 00:03:11.000
Well, 100 deciliters, each of
them is going to be worth 10

00:03:11.000 --> 00:03:15.650
centiliters, so that's going
to be 1,000 centiliters.

00:03:15.650 --> 00:03:18.920
So we have our answer:
1 dekaliter is 1,000

00:03:18.920 --> 00:03:20.170
centiliters.

00:03:23.600 --> 00:03:27.690
Now, the other way to do it is
you could convert a dekaliter

00:03:27.690 --> 00:03:30.480
to liters, and then convert
a liter to centiliters.

00:03:30.480 --> 00:03:37.780
So if we have one dekaliter--
and whenever you do unit

00:03:37.780 --> 00:03:40.550
things, just make sure
that it makes sense.

00:03:40.550 --> 00:03:43.380
Sometimes, people, instead of
multiplying, they would

00:03:43.380 --> 00:03:45.710
divide, and then they'd get,
oh, 1 dekaliter is equal to

00:03:45.710 --> 00:03:47.440
1/1,000 of a centiliter.

00:03:47.440 --> 00:03:48.300
And they say, no, no, no, no.

00:03:48.300 --> 00:03:51.130
A dekaliter is a much larger
unit that a centiliter.

00:03:51.130 --> 00:03:53.720
So 1 dekaliter has to be
a bunch of centiliters.

00:03:53.720 --> 00:03:55.970
This should be a large number,
so you should always do that

00:03:55.970 --> 00:03:59.020
reality check whenever you're
dealing with units.

00:03:59.020 --> 00:04:01.540
Now, let's do it the dimensional
analysis way.

00:04:01.540 --> 00:04:03.640
We're starting with
one dekaliter.

00:04:03.640 --> 00:04:05.250
We want to convert
it to liters.

00:04:05.250 --> 00:04:07.800
So if you're converting it to
liters, you want the dekaliter

00:04:07.800 --> 00:04:13.120
in the denominator and you want
liters in the numerator.

00:04:13.120 --> 00:04:16.060
Now, how many liters
are 1 dekaliter?

00:04:16.060 --> 00:04:20.620
Well, you could say 10 liters
is equal to 1 dekaliter.

00:04:24.950 --> 00:04:27.300
So 1 dekaliter is equal
to-- these cancel out.

00:04:27.300 --> 00:04:31.270
1 times 10 is 10 liters.

00:04:31.270 --> 00:04:33.570
Now, if we wanted to convert
this to centiliters, we're

00:04:33.570 --> 00:04:36.850
going to want the liters in the
denominator, and you want

00:04:36.850 --> 00:04:40.855
the centiliters in
the numerator.

00:04:40.855 --> 00:04:41.230
Now,

00:04:41.230 --> 00:04:43.900
how many centiliters are
there per liter?

00:04:43.900 --> 00:04:46.135
How many centiliters?

00:04:46.135 --> 00:04:49.020
Well, 1 liter is 100
centiliters.

00:04:49.020 --> 00:04:51.790
Centiliter is 1/100
of a liter.

00:04:51.790 --> 00:04:54.280
Notice, this and this are
the inverse statements.

00:04:54.280 --> 00:04:55.540
They're saying same the
exact same thing.

00:04:55.540 --> 00:04:56.760
1 liter per 100 centiliters.

00:04:56.760 --> 00:04:59.370
Here, writing 1 liter
per 100 centiliters.

00:04:59.370 --> 00:05:01.450
We've just flipped it, but
they're giving the same

00:05:01.450 --> 00:05:02.400
information.

00:05:02.400 --> 00:05:04.400
And the reason why we flipped
it is so that the liters

00:05:04.400 --> 00:05:07.860
cancel out, and then we're just
left with 10 times 100 is

00:05:07.860 --> 00:05:13.880
1,000 centiliters.

00:05:13.880 --> 00:05:15.850
And we are done!


Unit conversion: arithmetic calculations on units of volume

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:07.830
16 fluid ounces plus 3 pints
plus 4 quarts is equal to how

00:00:07.830 --> 00:00:09.640
many gallons?

00:00:09.640 --> 00:00:12.140
Now the one unfamiliar thing
here that we haven't seen

00:00:12.140 --> 00:00:14.940
before is this 16
fluid ounces.

00:00:14.940 --> 00:00:18.090
We've seen ounces before,
but not fluid ounces.

00:00:18.090 --> 00:00:21.280
And to make things confusing,
they're actually very

00:00:21.280 --> 00:00:23.970
different things.

00:00:23.970 --> 00:00:28.030
Fluid ounces are a measure
of a volume.

00:00:28.030 --> 00:00:32.420
Fluid ounces measure volume,
just like cups and pints and

00:00:32.420 --> 00:00:35.570
quarts and gallons do.

00:00:35.570 --> 00:00:40.200
While ounces measure weight.

00:00:40.200 --> 00:00:42.250
And this might make it
kind of confusing,

00:00:42.250 --> 00:00:43.690
or maybe less confusing.

00:00:43.690 --> 00:00:47.310
But it also happens that a
fluid ounce of water, at

00:00:47.310 --> 00:00:51.360
standard temperature and
pressure, is going to weigh a

00:00:51.360 --> 00:00:54.680
little bit more than
an actual ounce.

00:00:54.680 --> 00:00:56.820
So that's why they're
kind of close.

00:00:56.820 --> 00:01:00.330
A fluid ounce of water is not
that different in weight than

00:01:00.330 --> 00:01:01.880
an ounce, but they're two

00:01:01.880 --> 00:01:03.100
fundamentally different things.

00:01:03.100 --> 00:01:04.060
This is volume.

00:01:04.060 --> 00:01:05.310
This is weight.

00:01:05.310 --> 00:01:07.630
Now a fluid ounce, there's
many ways you

00:01:07.630 --> 00:01:08.280
could think of this.

00:01:08.280 --> 00:01:11.680
One is that you have 8-- let me
write it here-- you have 8

00:01:11.680 --> 00:01:18.130
fluid ounces per cup.

00:01:18.130 --> 00:01:24.420
Or, since there are 2 cups per
pint, you also have 16 fluid

00:01:24.420 --> 00:01:28.810
ounces per pint.

00:01:28.810 --> 00:01:29.940
You could view it either way.

00:01:29.940 --> 00:01:32.960
Or you could say that you have
32 fluid ounces per quart.

00:01:32.960 --> 00:01:35.340
Or you can multiply that times
4, and that's how many fluid

00:01:35.340 --> 00:01:37.060
ounces you have per gallon.

00:01:37.060 --> 00:01:39.230
Ounces, which is a unit
of weight, you have

00:01:39.230 --> 00:01:44.430
16 ounces per pound.

00:01:44.430 --> 00:01:47.460
And since I told you that a
fluid ounce is a little bit

00:01:47.460 --> 00:01:50.850
more that an ounce, you can kind
of figure out that, hey,

00:01:50.850 --> 00:01:53.780
a pint of water, at standard
temperature and pressure and

00:01:53.780 --> 00:01:57.790
all of that, will weigh
a little bit more.

00:01:57.790 --> 00:02:00.060
Almost a pound, or actually a
little bit more than a pound,

00:02:00.060 --> 00:02:01.200
but roughly equal to a pound.

00:02:01.200 --> 00:02:03.340
So that's something that
might make things

00:02:03.340 --> 00:02:04.640
simpler, or more difficult.

00:02:04.640 --> 00:02:06.630
Now let's actually tackle
the problem.

00:02:06.630 --> 00:02:08.820
So we have 16 fluid ounces.

00:02:08.820 --> 00:02:12.890
I just told you that there are
16 fluid ounces per pint.

00:02:12.890 --> 00:02:15.780
So 16 fluid ounces, let
me write over here.

00:02:15.780 --> 00:02:27.280
So 16 fluid ounces times-- now
we want the fluid ounces to

00:02:27.280 --> 00:02:30.280
cancel out, so we want fluid
ounces in the denominator

00:02:30.280 --> 00:02:34.100
times fluid ounces.

00:02:34.100 --> 00:02:36.420
Now how many fluid ounces
are there per pint?

00:02:36.420 --> 00:02:38.670
We want a pint in the numerator
so that this and

00:02:38.670 --> 00:02:39.830
this cancel out.

00:02:39.830 --> 00:02:44.810
We just saw it, there are 16
fluid ounces per pint, or that

00:02:44.810 --> 00:02:48.010
1/16 of a pint per
fluid ounce.

00:02:48.010 --> 00:02:48.910
Either way.

00:02:48.910 --> 00:02:51.360
But we see this and this
will cancel out.

00:02:51.360 --> 00:02:55.350
And so 16 over 16, because 16
times 1, divided by 16, 1

00:02:55.350 --> 00:02:58.060
equal to 1 pint.

00:02:58.060 --> 00:02:58.640
And we know that.

00:02:58.640 --> 00:03:00.230
16 fluid ounces equal a pint.

00:03:00.230 --> 00:03:05.340
If we have 16 fluid ounces right
over here, this is 1

00:03:05.340 --> 00:03:07.150
pint right over there.

00:03:07.150 --> 00:03:10.760
So we have 1 pint plus another
3 pints, so if we were to add

00:03:10.760 --> 00:03:15.690
all of this part right here
together, 1 pint plus 3 pints

00:03:15.690 --> 00:03:16.940
is 4 pints.

00:03:19.540 --> 00:03:21.553
Now we have 4 pints, let's
convert to quarts.

00:03:24.170 --> 00:03:26.980
Now we know that there are
2 pints per quart.

00:03:26.980 --> 00:03:29.980
Or we can view it this way, we
want the pints to cancel out,

00:03:29.980 --> 00:03:38.550
or 1 quart is 2 pints, or
1/2 quart per pint.

00:03:38.550 --> 00:03:40.700
And we wrote it this way instead
of 2 over 1 because we

00:03:40.700 --> 00:03:43.015
wanted the quarts in the
numerator, and the pints in

00:03:43.015 --> 00:03:45.780
the denominator, so that that,
and that, cancel out.

00:03:45.780 --> 00:03:46.580
And it makes sense.

00:03:46.580 --> 00:03:48.650
If you're converting a smaller
unit to a larger unit you're

00:03:48.650 --> 00:03:49.700
going to have fewer of them.

00:03:49.700 --> 00:03:51.680
So you're essentially
dividing by 2.

00:03:51.680 --> 00:03:58.200
So 4 times 1/2, or 4 divided
by 2, is equal to 2 quarts.

00:03:58.200 --> 00:04:01.560
So this whole piece right here,
these 4 pints, are equal

00:04:01.560 --> 00:04:04.200
to 2 quarts.

00:04:04.200 --> 00:04:05.060
That's 2 quarts.

00:04:05.060 --> 00:04:07.750
And you could visualize it.

00:04:07.750 --> 00:04:09.990
2 of these combined make 1
quart, so 4 of them is going

00:04:09.990 --> 00:04:11.010
to be 2 quarts.

00:04:11.010 --> 00:04:13.570
So you have 2 quarts here,
plus another 4 quarts.

00:04:13.570 --> 00:04:18.519
So this entire quantity of 2
quarts plus this 4 quarts are

00:04:18.519 --> 00:04:22.070
going to be 6 quarts.

00:04:22.070 --> 00:04:24.080
So the entire quantity
is 6 quarts.

00:04:24.080 --> 00:04:25.850
And we have to convert
that to gallons.

00:04:25.850 --> 00:04:26.660
So let me write it right here.

00:04:26.660 --> 00:04:31.970
We have 6 quarts and we want
to convert to gallons.

00:04:31.970 --> 00:04:36.320
So we want gallons and the
numerator, and we want quarts

00:04:36.320 --> 00:04:40.210
in the denominator, so that it
cancels out with this quarts

00:04:40.210 --> 00:04:41.730
when we multiply them.

00:04:41.730 --> 00:04:43.380
And how many quarts are
there in a gallon?

00:04:43.380 --> 00:04:47.840
Or 1 gallon is 4 quarts.

00:04:47.840 --> 00:04:50.400
Or 1/4 of a gallon per quart.

00:04:50.400 --> 00:04:52.100
Now we can multiply.

00:04:52.100 --> 00:04:53.430
The quarts cancel out.

00:04:53.430 --> 00:04:59.190
You have 6 times 1/4, which
is 6/4 gallons.

00:04:59.190 --> 00:05:00.460
And now we can simplify this.

00:05:00.460 --> 00:05:03.250
We can divide the numerator
and the denominator by 2.

00:05:03.250 --> 00:05:07.540
This is the same thing
as 3/2 gallons.

00:05:07.540 --> 00:05:09.950
So 3/2 would be an acceptable
answer right here.

00:05:09.950 --> 00:05:11.280
This would be 3/2 gallons.

00:05:11.280 --> 00:05:14.620
If you want to write it as a
mixed number, 3 divided by 2

00:05:14.620 --> 00:05:17.190
is 1 with the remainder 1.

00:05:17.190 --> 00:05:19.360
So it's 1 and 1/2 gallons.

00:05:19.360 --> 00:05:22.610
Either one would be an
acceptable answer.