Scale and Indirect Measurement

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:05.000
Let's do some scale and indirect
measurement problems.

00:00:05.000 --> 00:00:09.060
So they say use the scale
diagram to the right to

00:00:09.060 --> 00:00:11.180
determine-- I think this is
the scale diagram they're

00:00:11.180 --> 00:00:13.540
talking-- it's really
above the question.

00:00:13.540 --> 00:00:15.770
And they want to determine--
let's see first, the length of

00:00:15.770 --> 00:00:16.460
the helicopter.

00:00:16.460 --> 00:00:19.080
When they say the scale diagram,
you notice this

00:00:19.080 --> 00:00:20.380
helicopter looks pretty small.

00:00:20.380 --> 00:00:23.080
I don't know how big your screen
is, but looks maybe,

00:00:23.080 --> 00:00:25.680
one or two inches, depending
on what resolution you're

00:00:25.680 --> 00:00:27.090
watching this video on.

00:00:27.090 --> 00:00:28.250
But they're saying this
helicopter is

00:00:28.250 --> 00:00:29.170
much bigger than that.

00:00:29.170 --> 00:00:31.420
They're saying just this little
distance right here--

00:00:31.420 --> 00:00:33.720
just that little distance, that
little white box there--

00:00:33.720 --> 00:00:37.195
that's actually a foot on
the real helicopter.

00:00:37.195 --> 00:00:40.380
So what they're able to do is to
draw a small version of the

00:00:40.380 --> 00:00:42.860
helicopter and essentially give
us a ruler and tell us

00:00:42.860 --> 00:00:45.920
that that little distance
is equivalent to a foot.

00:00:45.920 --> 00:00:49.200
And this whole distance right
here, that whole distance

00:00:49.200 --> 00:00:53.895
right there, is equivalent to
1, 2, 3, 4, 5, 6, 7 feet.

00:00:53.895 --> 00:00:54.780
Did I get that right?

00:00:54.780 --> 00:00:58.060
1, 2, 3, 4, 5, 6, 7.

00:00:58.060 --> 00:01:01.660
So this thing is equal
to 7 feet.

00:01:01.660 --> 00:01:03.850
So given that, let's try to
answer their question.

00:01:03.850 --> 00:01:06.760
So the first one, the length
of the helicopter.

00:01:06.760 --> 00:01:12.860
So that's from cabin to tail, is
that distance right there.

00:01:12.860 --> 00:01:15.480
Let's see how many of
these things it is.

00:01:15.480 --> 00:01:17.970
So if I take that right there,
let me see if I can copy and

00:01:17.970 --> 00:01:19.650
paste that.

00:01:19.650 --> 00:01:24.880
So if I-- let's see, if I copy
it and then I paste it,

00:01:24.880 --> 00:01:27.160
there's one right there.

00:01:27.160 --> 00:01:28.870
One.

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Edit, paste.

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Two right there.

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Edit, paste.

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And then we get not
quite three.

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Almost three, but not quite.

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We get about that far.

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So it looks like about-- so
this right here is-- this

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right here is 7 feet.

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That right there is
another 7 feet.

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And I know that, because
that is 7 feet

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according to the scale.

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And then this looks like
we got about that far.

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So it looks like this is
maybe another 5 feet,

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if I were to estimate.

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Actually, if we went to go all
the way to the end of the

00:02:04.540 --> 00:02:06.210
tail-- if we went to the end
of here-- it actually looks

00:02:06.210 --> 00:02:08.300
like another 7 feet.

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We actually complete
three 7 feets.

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So I would say that part a,
I would go it's 27 feet.

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Sorry, not 27.

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21 feet.

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7 feet times 3 is equal to 21.

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I'd go part one.

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Because if you go all the way
to this tippy, tippy of the

00:02:26.650 --> 00:02:30.370
rear rotor right there, then
you're going to get 21.

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And you're going to get three
of these lengths.

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Let's do part b.

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The height of the helicopter.

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Floor to rotors.

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So from here-- from
floor to rotors.

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Well, I'll go from down
here to the rotors.

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They might be talking about
this floor, but who knows?

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So what is this distance
right here in feet?

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What is that distance?

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We could just eyeball
a little bit.

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They didn't give us
a vertical scale.

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We could just-- this scale looks
like about that high.

00:03:01.990 --> 00:03:05.020
So that would be about 7 feet.

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Let me copy and paste that.

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So let me copy it.

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Let me paste it.

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So that would be 7
feet right there.

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Then if I were do another one,
we're getting about another

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1/2 of 7 feet.

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So that'd be another
3 and 1/2 feet.

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So I would say that this
is 7 right there.

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And then we get about
3 and 1/2.

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So 7 plus 3 and 1/2
is 10.5 feet.

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The length of one main rotor--
and I think the main rotor--

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they're talking about this
length right there.

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That length right there.

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That is the length of
one main rotor.

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Or actually, this length
right here.

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That is the length of
one main rotor.

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And let's compare it to
this scale over here.

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So let's compare it.

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That's 7 feet.

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So let's copy it.

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And then let's paste it.

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I lost it.

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So let me copy it.

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Let me copy and then paste.

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So that's one length.

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So we want one rear rotor.

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So that's 7 feet right there.

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7 feet right there.

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And we can do another one.

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The important thing is to
get the idea of it.

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You don't have to get the
exact measurement.

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We're not about to go out there
and build a helicopter.

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So what is this length?

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We have 7 feet.

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And then we have 1, 2, 3,
about another 4 feet.

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So I would go with part c, I'd
go with 11 feet for one rotor

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if we're getting to see
this actually head on.

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Maybe it's a little bit longer
if it's at an angle.

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But that gives us our
general idea.

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Actually it looks a little
bit longer than that.

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But the idea is as long as you
understand how I'm using the

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scale is the main thing.

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The width of the cabin.

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So here-- that's the cabin
width right there.

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This distance.

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That distance right there
is the cabin width.

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And if we compare it to this
right here, we already said

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that this right here
is 7 feet roughly.

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Edit, copy, edit, paste.

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So that is 7 feet.

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That looks like about the
width of the cabin.

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It's a good estimate.

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So the width of the cabin,
I would say, is 7 feet.

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And then find the diameter
of the rear rotor system.

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The rear rotor system
is this right here.

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That is the rear rotor system.

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So if we were to copy and paste
our 7 foot lengths--

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actually, let me copy and
paste the actual ruler.

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So this is our actual ruler.

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So how long is the rear
rotor system?

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Let's look at our ruler.

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Looks like 1, 2, 3, 4 feet
is my best estimate.

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So I'd say the diameter of the
rear rotor system is 4 feet.

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So you might have already known
how to do this, but now

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you have a good sense.

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Look, this is telling us that
this little distance on the

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picture is actually equivalent
to a foot.

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And you just compare it to the
actual picture, and you know

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how big the actual
helicopter is.

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That's all that's
going on there.

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Let's do another scale and
indirect measurement problem.

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So it says on a Sunday morning,
the shadow of the

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Empire State Building
is 600 feet long.

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At the same time, the shadow
of a yard stick, which is 3

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feet long, is 1 foot
5 1/4 inches.

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How high is the Empire
State Building?

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So we could say height of Empire
to length of shadow.

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So length of the shadow of
Empire should be equal to

00:06:59.210 --> 00:07:05.690
taking the length of a yard
stick, so height of yard

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stick-- I'm assuming that
they're pointing up the yard

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stick like a skyscraper-- height
of yard stick divided

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by length of shadow
of yard stick.

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Now this problem looks a little
bit hairier because we

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have feet here, and then we
have 1 foot 5 1/4 inches.

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And you do everything
in inches.

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And you do everything in feet.

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And the interesting
thing is here.

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You can look at the units.

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As long as we calculate this in
feet and we calculate this

00:07:37.900 --> 00:07:41.580
in feet down here, these units
are going to cancel out.

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If you have x feet divided by
600 feet, in this case, the

00:07:46.590 --> 00:07:47.490
feet are going to cancel out.

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You'll have a unit-less
quantity.

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So it's really just going
to be a pure ratio.

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Same thing here.

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You can do it in feet.

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Or you can do it in inches.

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And you could actually do this
in feet and do this side in

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inches, because you get
feet divide by feet.

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The feet cancel out.

00:08:00.920 --> 00:08:02.090
The inches cancel out.

00:08:02.090 --> 00:08:03.435
So they stay unit-less.

00:08:03.435 --> 00:08:06.060
So it actually becomes a lot
easier when you can keep the

00:08:06.060 --> 00:08:08.160
right side in inches and
you can keep the

00:08:08.160 --> 00:08:09.790
left side in feet.

00:08:09.790 --> 00:08:12.790
So let's say the height of the
Empire State Building is x.

00:08:12.790 --> 00:08:18.250
So we could say x feet over
the length of the shadow.

00:08:18.250 --> 00:08:20.110
The shadow of the Empire State
Building is 600 feet.

00:08:20.110 --> 00:08:26.590
Over 600 feet is equal to the
length of a yard stick

00:08:26.590 --> 00:08:27.220
standing up.

00:08:27.220 --> 00:08:28.500
Or the height of a yard stick.

00:08:28.500 --> 00:08:29.220
Let's do it in inches.

00:08:29.220 --> 00:08:31.980
3 feet long is 36 inches.

00:08:31.980 --> 00:08:33.080
36 inches.

00:08:33.080 --> 00:08:35.270
And I want to show you that
the units work out.

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36 inches divided by the
length of its shadow.

00:08:38.750 --> 00:08:41.460
1 foot 5 1/4 inches.

00:08:41.460 --> 00:08:42.030
What's that?

00:08:42.030 --> 00:08:43.840
One foot is 12 inches.

00:08:43.840 --> 00:08:47.760
12 plus 5 1/4 is 17 1/4.

00:08:47.760 --> 00:08:52.780
Or 17.25 inches.

00:08:52.780 --> 00:08:55.000
And notice the units
cancel out.

00:08:55.000 --> 00:08:56.700
Feet divided by feet.

00:08:56.700 --> 00:08:57.970
Inches divided by inches.

00:08:57.970 --> 00:08:59.900
So they just become
pure ratios.

00:08:59.900 --> 00:09:05.270
x/600 is equal to 36/17.25.

00:09:05.270 --> 00:09:07.630
The reason why I could use feet
here and inches here is

00:09:07.630 --> 00:09:08.940
because the units cancel out.

00:09:08.940 --> 00:09:11.270
I want to make that very,
very, very clear.

00:09:11.270 --> 00:09:15.140
Now we can just cross multiply,
which is essentially

00:09:15.140 --> 00:09:19.200
just multiplying both sides of
the equation by 600 and 17.25.

00:09:19.200 --> 00:09:35.940
So 17.25 times x is going to
be equal to 36 times 600.

00:09:35.940 --> 00:09:43.200
And then we can divide
both sides by 17.25.

00:09:43.200 --> 00:09:49.600
So x is equal to 36 times
600 over 17.25.

00:09:49.600 --> 00:09:53.010
And let's get the
calculator out.

00:09:53.010 --> 00:09:58.350
So x is equal to 36 times
600 divided by

00:09:58.350 --> 00:10:03.690
17.25 is equal to 1,252.2.

00:10:03.690 --> 00:10:05.400
Let's just round it.

00:10:05.400 --> 00:10:13.220
So x is equal to 1,252.2 feet,
which is almost 1/4 of a mile.

00:10:13.220 --> 00:10:15.300
So the Empire State Building
is pretty high.

00:10:15.300 --> 00:10:18.120
But this is pretty neat, because
this is something that

00:10:18.120 --> 00:10:19.820
you could actually do.

00:10:19.820 --> 00:10:24.230
You could actually measure the
shadow of a skyscraper and

00:10:24.230 --> 00:10:26.250
then you could actually
measure the

00:10:26.250 --> 00:10:28.900
shadow of a yard stick.

00:10:28.900 --> 00:10:32.180
And then doing that, you can
actually figure out the height

00:10:32.180 --> 00:10:34.330
of a skyscraper.

00:10:34.330 --> 00:10:36.920
Which is pretty neat, because
if you didn't know how to do

00:10:36.920 --> 00:10:38.910
this math, you'd have to get
into a helicopter, and drop a

00:10:38.910 --> 00:10:40.940
huge-- I mean it'd be a much
harder thing to do.

00:10:40.940 --> 00:10:43.620
So hopefully, you found that
pretty interesting.


Ratio and Proportion

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:02.570
Let's start with a warm
up ratio problem.

00:00:02.570 --> 00:00:06.280
Then we can tackle some harder
word problems. So I have the

00:00:06.280 --> 00:00:15.750
ratio 13/6 is equal to 5/x.

00:00:15.750 --> 00:00:18.740
I don't like having this x in
the denominator, so let's

00:00:18.740 --> 00:00:22.420
multiply both sides of
this equation by x.

00:00:22.420 --> 00:00:27.150
So if I multiply both sides by
x, what's going to happen?

00:00:27.150 --> 00:00:29.810
On the right hand side, this
x cancels out with that x.

00:00:29.810 --> 00:00:34.300
And then the left hand side
going to become 13 over 6x is

00:00:34.300 --> 00:00:36.910
equal to-- you're just going
to have a 5 there.

00:00:36.910 --> 00:00:39.650
And then to solve for x, you
just multiply both sides by

00:00:39.650 --> 00:00:42.010
the inverse of 13/6.

00:00:42.010 --> 00:00:46.670
6/13.

00:00:46.670 --> 00:00:48.320
These, obviously, cancel out.

00:00:48.320 --> 00:00:49.940
That's why I multiplied
it by the inverse.

00:00:49.940 --> 00:00:55.320
And you get x is equal to 5
times 6, which is 30/13.

00:00:55.320 --> 00:00:58.220
Now one way that you might see
this done-- it's kind of

00:00:58.220 --> 00:01:01.470
skipping a step-- is called
cross multiplying.

00:01:01.470 --> 00:01:03.150
You look at a ratio like this,
and you immediately say the

00:01:03.150 --> 00:01:06.430
numerator on this side times the
denominator on that side

00:01:06.430 --> 00:01:08.690
is equal to the numerator
on this side times the

00:01:08.690 --> 00:01:09.520
denominator on that side.

00:01:09.520 --> 00:01:10.710
Let me write that out.

00:01:10.710 --> 00:01:14.380
So you might sometimes see
people immediately go to-- let

00:01:14.380 --> 00:01:17.680
me just rewrite the problem
actually-- So that original

00:01:17.680 --> 00:01:21.750
problem was 13/6 is
equal to 5/x.

00:01:21.750 --> 00:01:23.700
You might sometimes immediately
see someone go to

00:01:23.700 --> 00:01:29.950
13 times x is equal
to 5 times 6.

00:01:29.950 --> 00:01:31.550
And it might look like magic.

00:01:31.550 --> 00:01:32.470
How does that work out?

00:01:32.470 --> 00:01:33.520
Why does that make sense?

00:01:33.520 --> 00:01:36.610
And really, all they're doing
to get to this point is they

00:01:36.610 --> 00:01:40.430
are simultaneously multiplying
both sides of the equation by

00:01:40.430 --> 00:01:41.620
both denominators.

00:01:41.620 --> 00:01:42.550
Let me show you what I mean.

00:01:42.550 --> 00:01:46.150
If I multiply both sides of this
equation by 6 and an x,

00:01:46.150 --> 00:01:47.245
what's going to happen?

00:01:47.245 --> 00:01:52.490
If I multiply it by 6x times
both sides of this equation--

00:01:52.490 --> 00:01:53.700
And where did I get the 6?

00:01:53.700 --> 00:01:54.190
From here.

00:01:54.190 --> 00:01:55.030
Where did I get the x?

00:01:55.030 --> 00:01:55.460
From there.

00:01:55.460 --> 00:01:56.850
Both denominators.

00:01:56.850 --> 00:01:58.040
What's going to happen?

00:01:58.040 --> 00:02:00.470
On this side of the equation,
the 6 is going to cancel out

00:02:00.470 --> 00:02:01.890
with this denominator.

00:02:01.890 --> 00:02:04.580
And on the right hand side of
the equation, the x is going

00:02:04.580 --> 00:02:05.600
to cancel with this
denominator.

00:02:05.600 --> 00:02:08.870
So you're going to be left
with 13 times x is

00:02:08.870 --> 00:02:11.210
equal to 5 times 6.

00:02:11.210 --> 00:02:12.290
So nothing fancy there.

00:02:12.290 --> 00:02:15.570
You're just multiplying by the
denominators of both sides of

00:02:15.570 --> 00:02:16.000
the equation.

00:02:16.000 --> 00:02:18.060
And it looks like you're
cross multiplying.

00:02:18.060 --> 00:02:21.090
13x is equal to 5 times 6.

00:02:21.090 --> 00:02:23.600
And then from here, of course,
you divide both sides by 13.

00:02:23.600 --> 00:02:28.950
You get x is equal to 30/13.

00:02:28.950 --> 00:02:32.910
Now that we're all warmed up,
let's tackle some actual word

00:02:32.910 --> 00:02:35.740
problems.

00:02:35.740 --> 00:02:37.990
So we have the highest
mountain in

00:02:37.990 --> 00:02:39.800
Canada is Mount Yukon.

00:02:39.800 --> 00:02:45.990
It is 298/67 the size
of then Ben Nevis.

00:02:45.990 --> 00:02:55.470
Let's Y for Yukon is equal to
298/67 the size of-- let's say

00:02:55.470 --> 00:02:57.630
N for Nevis.

00:02:57.630 --> 00:03:00.990
That's what this in
green tells us.

00:03:00.990 --> 00:03:02.800
The highest peak in Scotland.

00:03:02.800 --> 00:03:04.840
Mount Elbert in Colorado
is the highest peak

00:03:04.840 --> 00:03:05.680
in the Rocky Mountains.

00:03:05.680 --> 00:03:08.790
Mount Elbert-- so we have this
other information here-- Mount

00:03:08.790 --> 00:03:12.450
Elbert is 220/67 the height
of Ben Nevis.

00:03:12.450 --> 00:03:14.850
So let's say, E for Elbert.

00:03:14.850 --> 00:03:21.720
E is equal to 22/67
times Nevis.

00:03:21.720 --> 00:03:24.290
Times the same Ben Nevis,
right there.

00:03:24.290 --> 00:03:25.420
And they're telling us more.

00:03:25.420 --> 00:03:30.230
And, it is 44/48 the
size of Mont Blanc.

00:03:30.230 --> 00:03:37.950
So Elbert is equal to 44/48
the size of Mont Blanc.

00:03:37.950 --> 00:03:40.890
Let's write B for Mount Blanc.

00:03:40.890 --> 00:03:44.300
They also tell us Mont Blanc
is 4,800 meters high.

00:03:44.300 --> 00:03:46.090
Mont Blanc is 4,800.

00:03:46.090 --> 00:03:46.700
meters high.

00:03:46.700 --> 00:03:50.860
So B is equal to 4,800.

00:03:50.860 --> 00:03:53.120
And they ask us, how high
is Mount Yukon?

00:03:53.120 --> 00:03:54.900
So we have to figure out Y.

00:03:54.900 --> 00:03:57.190
So let's see if we can work
backwards, and figure out all

00:03:57.190 --> 00:03:59.230
the variables in between.

00:03:59.230 --> 00:04:00.810
So let's start with this
information here.

00:04:00.810 --> 00:04:02.560
B is equal to 4,800.

00:04:02.560 --> 00:04:05.140
E is equal to 44/48 times B.

00:04:05.140 --> 00:04:14.490
So E-- so Elbert-- is equal to
44/48 times Mont Blanc, which

00:04:14.490 --> 00:04:17.279
is 4,800 meters.

00:04:17.279 --> 00:04:23.690
Now if you divide that by 48--
4,800 divided by 48 is 100.

00:04:23.690 --> 00:04:28.210
So Elbert is 44 times
100 meters high.

00:04:28.210 --> 00:04:33.870
So it's equal to 4,400 meters.

00:04:33.870 --> 00:04:34.730
Fair enough.

00:04:34.730 --> 00:04:35.960
Now we can use this
information and

00:04:35.960 --> 00:04:37.650
substitute it over here.

00:04:37.650 --> 00:04:42.360
We get Elbert, which is 4,400
meters high, is equal to

00:04:42.360 --> 00:04:47.510
220/67 times Ben Nevis.

00:04:47.510 --> 00:04:48.850
N for Nevis.

00:04:48.850 --> 00:04:51.340
To solve for Nevis, we multiply
both sides by the

00:04:51.340 --> 00:04:53.440
inverse of this coefficient
right here.

00:04:53.440 --> 00:04:57.740
So we multiply both
sides by 67/220.

00:04:57.740 --> 00:05:05.750
So times 67/220.

00:05:05.750 --> 00:05:07.810
The 67 cancels with that 67.

00:05:07.810 --> 00:05:09.920
That 220 cancels
with that 220.

00:05:09.920 --> 00:05:13.870
And then you get-- let's see,
if I take 4,400 divided by

00:05:13.870 --> 00:05:17.480
220-- 440 divided by 220 is 2.

00:05:17.480 --> 00:05:20.350
So this is going to be 20.

00:05:20.350 --> 00:05:24.640
So 4,400 divided by
220 is just 20.

00:05:24.640 --> 00:05:28.360
So you get Nevis is equal
to-- I'll swap sides.

00:05:28.360 --> 00:05:33.770
So Ben Nevis is equal to
67 times 20 meters.

00:05:33.770 --> 00:05:35.930
And now that's what?

00:05:35.930 --> 00:05:39.420
1,340 meters.

00:05:39.420 --> 00:05:40.620
Is that is right?

00:05:40.620 --> 00:05:42.250
Well, lets just leave it like
that, because we could--

00:05:42.250 --> 00:05:44.570
actually it looks like that's
67-- I'm going to leave Nevis

00:05:44.570 --> 00:05:46.530
as 67 times 20 meters.

00:05:46.530 --> 00:05:48.800
And substitute it right there.

00:05:48.800 --> 00:05:51.940
So Yukon-- I'll just go down
here, because I have more real

00:05:51.940 --> 00:05:58.050
estate there-- Yukon is equal
to 298 over 67 times the

00:05:58.050 --> 00:05:59.280
height of Nevis.

00:05:59.280 --> 00:06:01.120
Nevis is 67 times 20.

00:06:01.120 --> 00:06:04.340
So times 67 times 20.

00:06:04.340 --> 00:06:08.180
Well I can divide 67 by
67, and I get Yukon is

00:06:08.180 --> 00:06:11.430
298 times 20 meters.

00:06:11.430 --> 00:06:15.380
So Yukon is equal
to 298 times 20.

00:06:15.380 --> 00:06:16.430
And what is that equal to?

00:06:16.430 --> 00:06:21.580
That is equal to-- Let's
see that's 2 times 298

00:06:21.580 --> 00:06:26.510
is going to be 396.

00:06:26.510 --> 00:06:30.380
Oh sorry, it's going
to be 596.

00:06:30.380 --> 00:06:34.290
This is almost 300, so it
should be close to 600.

00:06:34.290 --> 00:06:35.820
This is 2 less than 300,
so this should

00:06:35.820 --> 00:06:38.730
be 4 less than 300.

00:06:38.730 --> 00:06:40.340
And then, I have a 0 here.

00:06:40.340 --> 00:06:45.200
So it's going to be
5,960 meters.

00:06:45.200 --> 00:06:46.650
And we are done.

00:06:46.650 --> 00:06:50.920
Let's do one more of these
word problems. All right.

00:06:50.920 --> 00:06:55.390
At a large high school, it is
estimated that 2 out of every

00:06:55.390 --> 00:06:59.630
3 students have a cell phone.

00:06:59.630 --> 00:07:05.790
And 1 in 5 of all students have
a cellphone that is one

00:07:05.790 --> 00:07:08.140
year old or less.

00:07:08.140 --> 00:07:08.550
All right.

00:07:08.550 --> 00:07:09.940
So let's think about it.

00:07:09.940 --> 00:07:16.915
Let's say that x is equal to the
total number of students.

00:07:20.930 --> 00:07:24.670
This first line, 2 out of 3
students have a cell phone, so

00:07:24.670 --> 00:07:30.640
we could say that 2/3
x have cell phone.

00:07:33.340 --> 00:07:35.630
That's what that green
statement tells us.

00:07:35.630 --> 00:07:37.810
And then that purple statement--
1 in 5 of all

00:07:37.810 --> 00:07:41.080
students have a cellphone that
is one year old or less.

00:07:41.080 --> 00:07:48.136
So 1/5 x have less than
1/5 year cell phone.

00:07:50.970 --> 00:07:54.720
So they want to know, out of
the students who own a cell

00:07:54.720 --> 00:07:57.950
phone-- so it's out of this--
that's our denominator.

00:07:57.950 --> 00:07:59.340
So let me write that down.

00:07:59.340 --> 00:08:00.440
That is our denominator.

00:08:00.440 --> 00:08:03.420
So out of the students who
have a cellphone-- that's

00:08:03.420 --> 00:08:06.770
right there-- they want to know
what proportion owns a

00:08:06.770 --> 00:08:10.160
phone that is more than
one year old.

00:08:10.160 --> 00:08:13.280
So how many students have a cell
phone that is more than

00:08:13.280 --> 00:08:14.540
one year old?

00:08:14.540 --> 00:08:17.090
Well, we could take the total
number that have a cellphone,

00:08:17.090 --> 00:08:19.500
which is 2/3 x.

00:08:19.500 --> 00:08:21.620
2/3 of all the students
have a cell phone.

00:08:21.620 --> 00:08:24.430
We can subtract out all of the
students that have a new

00:08:24.430 --> 00:08:27.560
cellphone-- a cell phone that
is less than one year.

00:08:27.560 --> 00:08:29.420
Remember they're saying more
than one year here.

00:08:29.420 --> 00:08:32.630
So we want to subtract out all
the students with the new

00:08:32.630 --> 00:08:36.919
cellphone, minus 1/5x, and you
will then have the proportion

00:08:36.919 --> 00:08:39.700
of students who have
this right here.

00:08:39.700 --> 00:08:41.590
This is right here.

00:08:41.590 --> 00:08:49.600
This is, have greater than
1/5 year cell phone.

00:08:49.600 --> 00:08:52.470
They have a phone, but it's
more than 1/5 years old.

00:08:52.470 --> 00:08:54.230
This is all of them that
have a cellphone.

00:08:54.230 --> 00:08:55.780
We subtract out the new ones.

00:08:55.780 --> 00:08:58.470
So this is, essentially, all
of the people who have an

00:08:58.470 --> 00:09:00.510
older than one-year
old cell phone.

00:09:00.510 --> 00:09:03.300
So to solve this, we just
subtract the fractions.

00:09:03.300 --> 00:09:07.140
So this is just going to be,
let's see, 2/3 is the same

00:09:07.140 --> 00:09:10.080
thing as 10/15.

00:09:10.080 --> 00:09:13.170
That's 2/3 minus 1/5.

00:09:13.170 --> 00:09:16.960
The same thing as 3/15 x.

00:09:16.960 --> 00:09:21.900
Which is equal to 10
minus 3 is 7/15 x.

00:09:21.900 --> 00:09:23.830
Is the total proportion of
students-- that's this

00:09:23.830 --> 00:09:26.450
orange-- what proportion owns
a phone that is more

00:09:26.450 --> 00:09:27.160
than one year old?

00:09:27.160 --> 00:09:28.590
It's 7/15 x.

00:09:28.590 --> 00:09:30.090
That's an actual number.

00:09:30.090 --> 00:09:32.810
So if you want to know, out of
the students who own a cell

00:09:32.810 --> 00:09:35.230
phone-- so out of the students
who own a cell phone, right

00:09:35.230 --> 00:09:38.560
there-- 2/3x, what proportion
owns a phone that it is more

00:09:38.560 --> 00:09:39.800
than one year old?

00:09:39.800 --> 00:09:41.870
This is the number that own a
cellphone that is more than

00:09:41.870 --> 00:09:43.700
one year old.

00:09:43.700 --> 00:09:46.820
And this whole value is the
proportion out of the students

00:09:46.820 --> 00:09:48.330
who have a cell phone.

00:09:48.330 --> 00:09:51.270
Lucky for us, the
x's cancel out.

00:09:51.270 --> 00:09:56.300
And we are left with this is
equal to 7/15 times the

00:09:56.300 --> 00:09:57.950
inverse of the denominator.

00:09:57.950 --> 00:09:58.980
You divide by 2/3.

00:09:58.980 --> 00:10:03.000
That's the same thing as
multiplying by 3/2.

00:10:03.000 --> 00:10:03.940
And what does this equal to?

00:10:03.940 --> 00:10:05.250
Divide by 3.

00:10:09.750 --> 00:10:12.740
We are left with 7/10.

00:10:12.740 --> 00:10:16.840
So of the students who own a
cell phone, 7 out of 10 of the

00:10:16.840 --> 00:10:20.080
students who own a cell phone,
own a cell phone that is more

00:10:20.080 --> 00:10:21.730
than one year old.

00:10:21.730 --> 00:10:24.110
And we are done.


Ex 3: Distributive property to simplify

WEBVTT
Kind: captions
Language: en

00:00:01.420 --> 00:00:04.470
Like the last video, I want
to start with two warm up

00:00:04.470 --> 00:00:06.210
problems. And then we'll
do an actual word.

00:00:06.210 --> 00:00:08.050
And you're going to see these
are going to be a little bit

00:00:08.050 --> 00:00:10.340
more involved than the equations
in the last video.

00:00:10.340 --> 00:00:13.190
But we're still going to
be doing the exact same

00:00:13.190 --> 00:00:15.630
operations, or what we could
consider legitimate

00:00:15.630 --> 00:00:17.130
operations, to get our answer.

00:00:17.130 --> 00:00:21.880
So here we have 3 times x
minus 1 is equal to 2

00:00:21.880 --> 00:00:23.590
times x plus 3.

00:00:23.590 --> 00:00:25.880
So let's see what
we can do here.

00:00:25.880 --> 00:00:28.880
The first thing I like to do--
and there's no definite right

00:00:28.880 --> 00:00:30.730
way to do it-- there's several
ways you could do these

00:00:30.730 --> 00:00:32.439
problems-- but I like
to distribute out

00:00:32.439 --> 00:00:34.000
the 3 and the 2.

00:00:34.000 --> 00:00:39.240
So 3 times x minus 1, that's the
same thing as 3x minus 3--

00:00:39.240 --> 00:00:42.440
I just distributed the
3-- is equal to--

00:00:42.440 --> 00:00:43.810
distribute out the 2.

00:00:43.810 --> 00:00:48.570
2 times x, plus 2 times
3, which is 6.

00:00:48.570 --> 00:00:54.100
Now what I like to do is get
all of my constant terms on

00:00:54.100 --> 00:00:57.220
the same side of the equation
and all my variable terms on

00:00:57.220 --> 00:00:58.390
the same side of the equation.

00:00:58.390 --> 00:01:01.660
So let's see if we can get
rid of this 2x term on

00:01:01.660 --> 00:01:02.580
the right hand side.

00:01:02.580 --> 00:01:04.220
So let's subtract 2x.

00:01:04.220 --> 00:01:06.120
I'm going to do a slightly
different notation this time

00:01:06.120 --> 00:01:08.290
because you might see
it done this way.

00:01:08.290 --> 00:01:10.880
Or you might find it easier
to visualize it this way.

00:01:10.880 --> 00:01:11.410
It doesn't matter.

00:01:11.410 --> 00:01:13.360
It's the same thing we did
in the last video.

00:01:13.360 --> 00:01:16.160
But I want to subtract 2x from
this side of the equation.

00:01:16.160 --> 00:01:18.960
But if I subtract 2x from the
side of the equation, I also

00:01:18.960 --> 00:01:23.610
have to subtract 2x from that
side of the equation.

00:01:23.610 --> 00:01:25.930
So then when we subtract 2x
from both sides of the

00:01:25.930 --> 00:01:27.470
equation, what do we get?

00:01:27.470 --> 00:01:29.810
Here we get 3x minus 2x.

00:01:29.810 --> 00:01:32.900
That's just 1x, or x minus 3.

00:01:36.690 --> 00:01:40.390
2x minus 2x is no x's, or 0.

00:01:40.390 --> 00:01:42.130
Then you just have the 6.

00:01:42.130 --> 00:01:45.620
So we get x minus
3 is equal to 6.

00:01:45.620 --> 00:01:47.980
That was by getting rid of the
2x from the right hand side;

00:01:47.980 --> 00:01:50.890
subtracting it from both
sides of this equation.

00:01:50.890 --> 00:01:54.140
Now we have this negative
3 on the left hand side.

00:01:54.140 --> 00:01:54.920
I don't want it there.

00:01:54.920 --> 00:01:56.410
I just want an x there.

00:01:56.410 --> 00:01:58.640
So to get rid of that
we can add 3 to both

00:01:58.640 --> 00:01:59.890
sides of this equation.

00:02:03.040 --> 00:02:05.910
You could imagine this is being
adding this equation to

00:02:05.910 --> 00:02:07.980
the equation 3 is equal to 3.

00:02:07.980 --> 00:02:09.570
3 is, obviously, equal to 3.

00:02:09.570 --> 00:02:13.250
Negative 2x is, obviously,
equal to negative 2x.

00:02:13.250 --> 00:02:14.070
You could do it either way.

00:02:14.070 --> 00:02:17.410
But if you add 3 to both sides
of this equation the left hand

00:02:17.410 --> 00:02:21.170
side of the equation becomes
just an x, because these two

00:02:21.170 --> 00:02:26.710
guys cancel out. x equal,
and 6 plus 3 is 9.

00:02:26.710 --> 00:02:27.650
And we are done.

00:02:27.650 --> 00:02:29.520
And we can even check
our answer.

00:02:29.520 --> 00:02:33.950
3 times 9 minus 1 is what?

00:02:33.950 --> 00:02:35.890
This is 3 times 8.

00:02:35.890 --> 00:02:37.450
This is 24.

00:02:37.450 --> 00:02:39.400
So that's what the left
hand side equals.

00:02:39.400 --> 00:02:41.220
What does the right
hand side equal?

00:02:41.220 --> 00:02:44.890
That is 2 times 9 plus 3.

00:02:44.890 --> 00:02:49.390
That's the 2 times 12, which
is also equals 24.

00:02:49.390 --> 00:02:53.790
So it all works out.
x is equals to 9.

00:02:53.790 --> 00:02:55.430
Next problem.

00:02:55.430 --> 00:03:01.830
z over 16 is equal to 2 times 3z
plus 1, all of that over 9.

00:03:01.830 --> 00:03:04.210
So it's a hairy looking
problem.

00:03:04.210 --> 00:03:09.820
Let's multiply both sides
of this equation by 9.

00:03:09.820 --> 00:03:12.520
So if you multiply both sides
of this equation by

00:03:12.520 --> 00:03:13.510
9, what do we get?

00:03:13.510 --> 00:03:20.490
We get 9 over 16z is equal to--
this 9 and that 9 will

00:03:20.490 --> 00:03:26.250
cancel out-- 2 times
3z plus 1.

00:03:26.250 --> 00:03:28.240
Now let's distribute this 2.

00:03:28.240 --> 00:03:38.540
So we get 9 over 16z is equal
to 2 times 3z is 6z plus 2.

00:03:38.540 --> 00:03:41.900
Now let's get all of z's on the
same side of the equation.

00:03:41.900 --> 00:03:46.070
So let's subtract 6z from both
sides of the equation.

00:03:46.070 --> 00:03:50.210
So let's subtract minus 6z
there, and that, of course,

00:03:50.210 --> 00:03:52.370
equals minus 6z there.

00:03:52.370 --> 00:03:54.390
And what do we get?

00:03:54.390 --> 00:04:02.250
On the left hand side,
we get 9/16 minus 6.

00:04:02.250 --> 00:04:05.640
What is 6 if I have 16
as a denominator?

00:04:05.640 --> 00:04:09.030
6 is equal to what over 16?

00:04:09.030 --> 00:04:09.956
Let me think about it.

00:04:09.956 --> 00:04:11.890
60 plus 36.

00:04:11.890 --> 00:04:14.980
It's equal to 96/16.

00:04:14.980 --> 00:04:20.140
So it's 9 minus 96 over 16z.

00:04:20.140 --> 00:04:23.240
This is just 6-- I just rewrote
minus 6 here-- is

00:04:23.240 --> 00:04:25.360
equal to-- these
two cancel out.

00:04:25.360 --> 00:04:27.985
That's why I subtracted
6z in the first place.

00:04:27.985 --> 00:04:29.940
So it's going to equal 2.

00:04:29.940 --> 00:04:32.710
So what does this equal
right over here?

00:04:32.710 --> 00:04:34.100
Let me do it in orange.

00:04:34.100 --> 00:04:38.320
9 minus 96.

00:04:38.320 --> 00:04:38.820
Well let's see.

00:04:38.820 --> 00:04:43.710
The difference between 9 and
96 is going to be 87.

00:04:43.710 --> 00:04:45.650
And, of course, we're
subtracting the larger from

00:04:45.650 --> 00:04:46.130
the smaller.

00:04:46.130 --> 00:04:53.460
So it's going to be negative
87 over 16z is equal to 2.

00:04:53.460 --> 00:04:54.460
And we're almost there.

00:04:54.460 --> 00:04:57.540
We just have to multiply both
sides of this equation by the

00:04:57.540 --> 00:04:59.730
inverse of this coefficient.

00:04:59.730 --> 00:05:10.630
So multiply both sides of this
equation by negative 16/87.

00:05:10.630 --> 00:05:11.750
These cancel out.

00:05:11.750 --> 00:05:14.420
87, 87, 16 and 16.

00:05:14.420 --> 00:05:15.660
The negative signs plus.

00:05:15.660 --> 00:05:18.990
You're just left with z is equal
to 2 times this thing.

00:05:18.990 --> 00:05:25.830
So 2 times negative 16
is negative 32/87.

00:05:25.830 --> 00:05:27.160
And we are done.

00:05:27.160 --> 00:05:29.970
That's not a pretty looking
answer, but

00:05:29.970 --> 00:05:30.620
that's what we got.

00:05:30.620 --> 00:05:33.460
And you can try it with
multiple methods.

00:05:33.460 --> 00:05:35.850
Maybe you can multiply both of
the equation by 16 first.

00:05:35.850 --> 00:05:38.970
Maybe you can distribute out the
2 9's first. All sorts of

00:05:38.970 --> 00:05:39.550
things you can do.

00:05:39.550 --> 00:05:43.970
And you could also verify that
this is, indeed, the answer.

00:05:43.970 --> 00:05:47.240
Let's do a word problem now
that we're warmed up.

00:05:47.240 --> 00:05:48.030
All right.

00:05:48.030 --> 00:05:52.360
It says Manoj and Tomar are
arguing about how a number

00:05:52.360 --> 00:05:54.200
trick they heard goes.

00:05:54.200 --> 00:05:56.880
Tomar tells Andrew to
think of a number.

00:05:56.880 --> 00:06:00.770
Let's say that x is the number
that Andrew thinks of.

00:06:00.770 --> 00:06:01.750
Think of a number.

00:06:01.750 --> 00:06:03.010
Multiply it by 5.

00:06:03.010 --> 00:06:05.730
So this is what Tomar
is saying.

00:06:05.730 --> 00:06:10.105
ƒ is telling Andrew to think of
a number, multiply it by 5,

00:06:10.105 --> 00:06:11.130
so let me do that.

00:06:11.130 --> 00:06:17.435
So multiply it by 5, and
subtract 3 from the result.

00:06:20.150 --> 00:06:21.325
We'll do that in a
different color.

00:06:21.325 --> 00:06:23.280
We'll do it in blue.

00:06:23.280 --> 00:06:26.670
Subtract 3 from the result.

00:06:26.670 --> 00:06:29.030
So subtract 3 from the result.

00:06:29.030 --> 00:06:30.740
Then Manoj comes along.

00:06:30.740 --> 00:06:33.490
So here's Manoj.

00:06:33.490 --> 00:06:36.010
Manoj comes along and tells
Andrew once again

00:06:36.010 --> 00:06:37.820
to think of a number.

00:06:37.820 --> 00:06:39.150
So once again, think
of a number.

00:06:39.150 --> 00:06:40.600
We'll call that x.

00:06:40.600 --> 00:06:43.250
So they both are telling him
to think of a number.

00:06:43.250 --> 00:06:44.780
Add five.

00:06:44.780 --> 00:06:47.960
So now he's saying add
five to the number.

00:06:47.960 --> 00:06:49.370
So you add 5 to number.

00:06:49.370 --> 00:06:55.290
And then multiply
the result by 3.

00:06:55.290 --> 00:06:58.990
So multiply the result by 3.

00:06:58.990 --> 00:07:01.580
Andrew says that which ever way
he does the trick, he gets

00:07:01.580 --> 00:07:02.730
the same answer.

00:07:02.730 --> 00:07:04.100
What was Andrew's number?

00:07:04.100 --> 00:07:07.080
So regardless of whether he does
5 times the number, and

00:07:07.080 --> 00:07:10.170
then subtracts 3, or whether he
adds 5 and then multiples

00:07:10.170 --> 00:07:12.950
the whole answer by 3, he
gets the same number.

00:07:12.950 --> 00:07:16.470
So that must mean that these
two things are equal.

00:07:16.470 --> 00:07:18.070
He gets the same answer.

00:07:18.070 --> 00:07:19.630
That means that these
two are equals.

00:07:19.630 --> 00:07:22.540
So let's solve this equation.

00:07:22.540 --> 00:07:30.380
We get 5x minus 3 is equal
to 3 times x plus 5.

00:07:30.380 --> 00:07:33.870
A good place for me-- I like to
distribute out this 3, so

00:07:33.870 --> 00:07:37.620
that is equal to 3x plus 15.

00:07:37.620 --> 00:07:39.820
Always have to remember to
multiply the 3 times all of

00:07:39.820 --> 00:07:41.330
the terms in parentheses.

00:07:41.330 --> 00:07:43.020
This is a 15.

00:07:43.020 --> 00:07:47.460
And, of course, 5x minus
3 is equal to that.

00:07:47.460 --> 00:07:50.040
Now, I think you know how
I like to operate.

00:07:50.040 --> 00:07:53.090
I like to get all of my
ex-coeffients on one side.

00:07:53.090 --> 00:07:54.590
So let's get them all on
the left hand side.

00:07:54.590 --> 00:07:56.770
Which means, let's get rid of
them on the right hand side.

00:07:56.770 --> 00:07:59.120
So let's subtract 3x
from both sides.

00:07:59.120 --> 00:08:04.420
So minus 3x minus 3x and
then what do we get?

00:08:04.420 --> 00:08:08.410
Our equation becomes
5x minus 3x is 2x.

00:08:08.410 --> 00:08:12.420
You still have a minus 3 is
equal to-- these cancel out--

00:08:12.420 --> 00:08:14.380
equal to 15.

00:08:14.380 --> 00:08:17.170
Equal to that yellow
15 right there.

00:08:17.170 --> 00:08:19.300
Now we want to get rid
of this negative 3 on

00:08:19.300 --> 00:08:20.170
the left hand side.

00:08:20.170 --> 00:08:22.110
And the best way to do it
is to add 3 to both

00:08:22.110 --> 00:08:24.170
sides of the equation.

00:08:24.170 --> 00:08:28.620
So add 3 to both sides
of this equation.

00:08:28.620 --> 00:08:29.770
So what do you get?

00:08:29.770 --> 00:08:35.539
You now get 2x-- and these
cancel out-- is equal to 18.

00:08:35.539 --> 00:08:38.144
And then you divide both
sides of this by 2.

00:08:40.980 --> 00:08:44.159
And you get x is equal
to 18 over 2.

00:08:44.159 --> 00:08:45.340
Or 9.

00:08:45.340 --> 00:08:48.560
So in either situation, Andrew
was thinking of the number 9.

00:08:48.560 --> 00:08:51.430
Let's see if it works according
to both Tomar and

00:08:51.430 --> 00:08:52.720
Manoj's method.

00:08:52.720 --> 00:08:58.190
If I were to take 5 times-- if
this is 9-- you get 5 times 9

00:08:58.190 --> 00:09:01.350
is 45 minus 3.

00:09:01.350 --> 00:09:03.270
So you get 42.

00:09:03.270 --> 00:09:04.910
That's when you do
Tomar's method.

00:09:04.910 --> 00:09:08.270
When you do Manoj's method--
this is a 9, right there.

00:09:08.270 --> 00:09:11.110
9 plus 5 is 14.

00:09:11.110 --> 00:09:14.990
So this thing becomes 14, and
it becomes 3 times 14.

00:09:14.990 --> 00:09:18.860
3 times 14 is 30 plus
12, which is 42.

00:09:18.860 --> 00:09:19.790
So Andrew was right.

00:09:19.790 --> 00:09:22.230
When he randomly picked the
number 9, and regardless of

00:09:22.230 --> 00:09:25.030
which method he uses, he
gets the same results.

00:09:25.030 --> 00:09:27.000
He gets 42.


Ex 1: Distributive property to simplify

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.940
In this video, we're going to
a couple of warm-up solving

00:00:03.940 --> 00:00:06.750
equations problems. And you'll
see these require several

00:00:06.750 --> 00:00:08.855
steps, maybe a little bit more
than the ones that we've done

00:00:08.855 --> 00:00:09.710
in the previous video.

00:00:09.710 --> 00:00:12.110
And then we'll do a word problem
that applies our

00:00:12.110 --> 00:00:15.030
equation-solving capabilities.

00:00:15.030 --> 00:00:20.640
So here we have 7 times w plus
20, minus w, is equal to 5.

00:00:20.640 --> 00:00:22.220
Let's see if we can
solve this.

00:00:22.220 --> 00:00:24.870
And like all things, there's
multiple ways to solve these.

00:00:24.870 --> 00:00:28.210
I'll just solve the way that
seems most natural to me.

00:00:28.210 --> 00:00:30.890
So one thing I like to do, is
I like to distribute the

00:00:30.890 --> 00:00:31.610
numbers out.

00:00:31.610 --> 00:00:34.950
Because if I distribute the
numbers out, I get a 7w, and

00:00:34.950 --> 00:00:36.710
then I can subtract
w from there.

00:00:36.710 --> 00:00:39.110
So maybe I can merge the
w terms somehow.

00:00:39.110 --> 00:00:46.620
So this 7w plus 20 I can rewrite
as 7w plus 7 times 20.

00:00:46.620 --> 00:00:48.080
Remember, distributive
properties.

00:00:48.080 --> 00:00:51.140
So plus 140.

00:00:51.140 --> 00:00:55.560
And then we have this minus
w is equal to 5.

00:00:55.560 --> 00:00:57.070
I just rewrote this
part right here.

00:00:57.070 --> 00:00:58.970
I just distributed the 7.

00:00:58.970 --> 00:01:02.430
And now, just like I talked
about, we can merge-- We can

00:01:02.430 --> 00:01:06.030
take 7w, and from that,
we can subtract a w.

00:01:06.030 --> 00:01:09.940
So if you take those two
terms, you get 6w.

00:01:09.940 --> 00:01:12.640
If I have 7 of something, and
I take away 1 of that

00:01:12.640 --> 00:01:14.770
something, I have 6
of that something.

00:01:14.770 --> 00:01:21.530
So I have 6w plus 140
is equal to 5.

00:01:24.410 --> 00:01:26.470
Now, I want get rid
of this 140.

00:01:26.470 --> 00:01:29.070
Because then I'll have 6w is
equaling to something, and

00:01:29.070 --> 00:01:30.920
I'll be able divide by
6, and all of that.

00:01:30.920 --> 00:01:34.770
So to get rid of 140, I can
subtract 140 from both sides

00:01:34.770 --> 00:01:36.030
of the equation.

00:01:36.030 --> 00:01:38.700
And I'll do it in pink.

00:01:38.700 --> 00:01:42.210
Minus 140.

00:01:42.210 --> 00:01:44.510
So I'm just subtracting
140 from both

00:01:44.510 --> 00:01:46.150
sides of this equation.

00:01:46.150 --> 00:01:49.450
If something equals something,
something minus 140 is going

00:01:49.450 --> 00:01:51.390
to equal something minus 140.

00:01:51.390 --> 00:01:53.130
Whatever you do to one side,
you've got to do

00:01:53.130 --> 00:01:54.470
to the other side.

00:01:54.470 --> 00:01:57.680
So the whole point here was for
these two to cancel out.

00:01:57.680 --> 00:02:01.370
You're left with the left side,
which is 6w is equal to

00:02:01.370 --> 00:02:04.040
5 minus 140.

00:02:04.040 --> 00:02:08.160
Well, that's negative 135.

00:02:08.160 --> 00:02:12.120
And now we can divide both sides
of this equation by 6,

00:02:12.120 --> 00:02:15.330
which is equivalent to
multiplying by 1/6, and you

00:02:15.330 --> 00:02:22.820
get w is equal to, let's see,
negative 135 over 6.

00:02:22.820 --> 00:02:26.320
Let's see, is there any place
to simplify this anymore?

00:02:26.320 --> 00:02:26.580
Let's see.

00:02:26.580 --> 00:02:29.850
This isn't divisible by 2,
and it also doesn't look

00:02:29.850 --> 00:02:31.060
divisible by 3.

00:02:31.060 --> 00:02:36.000
So this looks like we are
done with the problem.

00:02:36.000 --> 00:02:37.030
And you can verify.

00:02:37.030 --> 00:02:38.490
Actually let's verify, because
it's kind of a

00:02:38.490 --> 00:02:42.470
strange-looking-- Let's verify
that this is the answer.

00:02:42.470 --> 00:02:49.640
So 7 times negative 135 over
6-- that's our solution

00:02:49.640 --> 00:02:51.700
for w --plus 20.

00:02:51.700 --> 00:02:55.810
Instead of 20, I'm going to
write plus 120 over 6.

00:02:55.810 --> 00:02:59.690
20 is the same thing as
120 over 6, right?

00:02:59.690 --> 00:03:00.940
Minus w.

00:03:03.880 --> 00:03:08.360
So w is negative 135 over 6.

00:03:08.360 --> 00:03:11.970
So subtracting a negative
becomes

00:03:11.970 --> 00:03:13.630
adding a positive, right?

00:03:13.630 --> 00:03:14.880
So let's see what happens.

00:03:14.880 --> 00:03:22.450
This becomes 7 times-- Let's
see, negative 135 plus 120 is

00:03:22.450 --> 00:03:32.930
negative 15 over 6
plus 135 over 6.

00:03:32.930 --> 00:03:34.700
Let's see what we get here.

00:03:34.700 --> 00:03:38.330
So then we get 7 times 15.

00:03:38.330 --> 00:03:39.260
So let me go over here.

00:03:39.260 --> 00:03:40.830
What is 7 times 15?

00:03:40.830 --> 00:03:42.780
It's 70 plus 35.

00:03:42.780 --> 00:03:50.200
So it's negative 105 over 6.

00:03:50.200 --> 00:03:52.630
That is that right there.

00:03:52.630 --> 00:03:58.780
Plus 135 over 6.

00:03:58.780 --> 00:04:00.160
What does that become?

00:04:00.160 --> 00:04:05.090
This becomes 30 over 6,
which is equal to 5.

00:04:05.090 --> 00:04:06.560
Which is exactly what it
needed to be equal to.

00:04:06.560 --> 00:04:08.930
It's equal to 5, so we
got the right answer.

00:04:08.930 --> 00:04:10.630
My spider sense was wrong.

00:04:10.630 --> 00:04:12.480
We did it correctly, even though
we got this strange

00:04:12.480 --> 00:04:13.230
looking answer.

00:04:13.230 --> 00:04:15.900
So now let's do this problem.

00:04:15.900 --> 00:04:18.769
So once again, I like to
distribute out the 9.

00:04:18.769 --> 00:04:20.156
Actually, we don't have
to distribute it out.

00:04:20.156 --> 00:04:21.220
There's multiple ways
to do this.

00:04:21.220 --> 00:04:22.380
Maybe we'll do it both ways.

00:04:22.380 --> 00:04:24.200
So the first way I like to do it
is to distribute out the 9,

00:04:24.200 --> 00:04:26.020
so I don't have to deal
with fractions.

00:04:26.020 --> 00:04:30.040
So you get 9x minus 18. just
distributed the 9.

00:04:30.040 --> 00:04:33.770
Is equal to 3x plus 3.

00:04:33.770 --> 00:04:36.370
Now we want to get the x-terms
together somehow.

00:04:36.370 --> 00:04:37.830
Let's get them together
on the left-hand side.

00:04:37.830 --> 00:04:41.000
So let's get rid of this 3x
on the right-hand side.

00:04:41.000 --> 00:04:43.960
And the best way to get rid of
it is to subtract 3x from the

00:04:43.960 --> 00:04:44.820
right-hand side.

00:04:44.820 --> 00:04:46.100
But if we do it from the
right-hand side, we have to do

00:04:46.100 --> 00:04:48.650
it from the left-hand
side, as well.

00:04:48.650 --> 00:04:51.600
So I'm subtracting 3x
from both sides.

00:04:51.600 --> 00:04:57.280
The left-hand side, 9x minus 3x
is equal to 6x, and then,

00:04:57.280 --> 00:05:02.690
of course, you have your minus
18 is equal to 3x minus 3x.

00:05:02.690 --> 00:05:04.800
That just disappears, those
cancel out, and you just have

00:05:04.800 --> 00:05:06.670
the 3 left over.

00:05:06.670 --> 00:05:08.920
Now there's multiple ways
you could do this.

00:05:08.920 --> 00:05:12.040
I mean, one fun thing-- Well,
let me just do it the most

00:05:12.040 --> 00:05:13.040
traditional way.

00:05:13.040 --> 00:05:20.220
We could add 18 to both sides so
that the 18 disappears from

00:05:20.220 --> 00:05:21.460
the left-hand side.

00:05:21.460 --> 00:05:26.410
So then you are left with 6x--
these two guys cancel out --is

00:05:26.410 --> 00:05:29.800
equal to 3 plus 18,
which is 21.

00:05:29.800 --> 00:05:36.540
Divide both sides by 6, you get
x is equal to 21 over 6,

00:05:36.540 --> 00:05:40.030
or if you divide the numerator
and the denominator by 3, you

00:05:40.030 --> 00:05:44.060
get 7 over 2.

00:05:44.060 --> 00:05:45.970
And you are done.

00:05:45.970 --> 00:05:47.180
Now, I said that there's
multiple

00:05:47.180 --> 00:05:48.380
ways to do this problem.

00:05:48.380 --> 00:05:50.460
Let me do it another
way here in orange.

00:05:50.460 --> 00:05:56.900
So you have 9 times x minus
2 is equal to 3x plus 3.

00:05:56.900 --> 00:05:58.520
Well, I see a 9,
I see some 3's.

00:05:58.520 --> 00:06:02.600
What if I just divide both sides
of this equation by 3?

00:06:02.600 --> 00:06:06.400
So if I divide that side by 3,
and I divide all of these

00:06:06.400 --> 00:06:07.990
terms by 3.

00:06:07.990 --> 00:06:09.200
What do I get?

00:06:09.200 --> 00:06:16.120
This becomes 3 times x minus
2 is equal to x plus 1.

00:06:16.120 --> 00:06:20.240
Maybe at this point, if I want,
I could distribute this.

00:06:20.240 --> 00:06:25.390
So this becomes 3x minus
6 is equal to x plus 1.

00:06:25.390 --> 00:06:28.720
I could subtract x from both
sides of this equation, so I

00:06:28.720 --> 00:06:32.560
get 2x minus 6 is equal to 1.

00:06:32.560 --> 00:06:34.570
Remember, I subtracted that
from both sides, so it

00:06:34.570 --> 00:06:36.210
disappeared on the
right-hand side.

00:06:36.210 --> 00:06:39.210
I could add 6 to both sides
of this equation.

00:06:39.210 --> 00:06:43.350
I get 2x is equal to 1
plus 6, which is 7.

00:06:43.350 --> 00:06:47.060
Divide both sides by 2, you get
x is equal to 7 over 2.

00:06:47.060 --> 00:06:48.500
I went through this a
little bit faster.

00:06:48.500 --> 00:06:51.920
But really, I just wanted to
show you that as long as do

00:06:51.920 --> 00:06:55.650
legitimate operations, you're
going to get the same answer.

00:06:55.650 --> 00:06:58.510
And you could check, verify
that this is indeed the

00:06:58.510 --> 00:07:00.500
correct answer, if you like.

00:07:00.500 --> 00:07:02.150
Now we have a word problem.

00:07:02.150 --> 00:07:04.220
Let's see if we can
tackle this.

00:07:04.220 --> 00:07:06.560
Lydia inherited a
sum of money.

00:07:06.560 --> 00:07:09.460
She split it into
5 equal chunks.

00:07:09.460 --> 00:07:12.850
So let me just say m is the
amount of money she has.

00:07:12.850 --> 00:07:14.910
So she split into
5 equal chunks.

00:07:14.910 --> 00:07:21.670
So let me say, Lydia's money
that she inherited.

00:07:26.790 --> 00:07:28.910
She splits it into
5 equal chunks.

00:07:28.910 --> 00:07:33.350
She invested 3 parts of the
money in a high interest bank.

00:07:33.350 --> 00:07:35.490
So how much did she invest in
the high interest bank?

00:07:45.520 --> 00:07:50.330
She divided it into 5 equal
chunks, and she invested 3 of

00:07:50.330 --> 00:07:53.570
those chunks into a high
interest bank account.

00:07:53.570 --> 00:07:59.400
So she took her money, divided
it into 5 chunks-- So this is

00:07:59.400 --> 00:08:00.320
each of the 5 chunks.

00:08:00.320 --> 00:08:02.770
Then she put 3 of those chunks,
or you could say she

00:08:02.770 --> 00:08:11.450
took 3/5 of her money, and she
put it into the high interest

00:08:11.450 --> 00:08:14.470
bank account, which adds
10% to the value.

00:08:14.470 --> 00:08:16.630
So that's how much she
originally put into high

00:08:16.630 --> 00:08:18.500
interest banking account.

00:08:18.500 --> 00:08:21.830
She placed the rest of her
inheritance plus $500 in the

00:08:21.830 --> 00:08:22.680
stock market.

00:08:22.680 --> 00:08:23.930
So how much was that?

00:08:28.710 --> 00:08:31.900
So she placed the rest
of her inheritance.

00:08:31.900 --> 00:08:34.659
So she put 3/5 of it in the high
interest bank account.

00:08:34.659 --> 00:08:36.140
What's left over?

00:08:36.140 --> 00:08:38.260
What's going to be the 2/5?

00:08:38.260 --> 00:08:41.280
2/5 of her money she is
going to invest in the

00:08:41.280 --> 00:08:42.330
stock market, right?

00:08:42.330 --> 00:08:44.540
You combine 3/5 plus 2/5,
you have all of the

00:08:44.540 --> 00:08:46.070
money that she inherited.

00:08:46.070 --> 00:08:47.720
But she didn't put
just the 2/5.

00:08:47.720 --> 00:08:50.190
She put the rest of her
inheritance, which is the 2/5

00:08:50.190 --> 00:08:57.890
m, plus $500, in the stock
market, but lost 20%.

00:08:57.890 --> 00:09:00.620
So this is how much she put in,
and this is how much she

00:09:00.620 --> 00:09:02.250
ends up with.

00:09:02.250 --> 00:09:11.820
So put in is right there,
and then ends up with.

00:09:11.820 --> 00:09:18.910
So on the checking accounts,
it added 10% of its value.

00:09:18.910 --> 00:09:24.910
So she started with 3/5 of her
money, and it added another

00:09:24.910 --> 00:09:26.180
10% of that.

00:09:26.180 --> 00:09:32.690
So plus, let's say, 0.10
times the amount of

00:09:32.690 --> 00:09:34.010
money she put in.

00:09:34.010 --> 00:09:36.400
Times 3/5 of her money.

00:09:36.400 --> 00:09:37.570
This is how much she
ends up with.

00:09:37.570 --> 00:09:40.940
Her original amount that she put
into the account plus 10%

00:09:40.940 --> 00:09:41.750
of the original amount.

00:09:41.750 --> 00:09:42.740
It grew by 10%.

00:09:42.740 --> 00:09:45.260
It added 10% of the value.

00:09:45.260 --> 00:09:52.580
Now in the stock market, she
started with 2/5 m plus 500,

00:09:52.580 --> 00:09:55.140
but she lost 20%
on that money.

00:09:55.140 --> 00:10:07.760
So she loses 0.20 for 20%,
times 2/5 m plus 500.

00:10:07.760 --> 00:10:09.630
That's how much she loses
on the market.

00:10:09.630 --> 00:10:13.410
She loses 20% of this
amount of money.

00:10:13.410 --> 00:10:16.060
Now at the end it says, if the
two accounts end up with the

00:10:16.060 --> 00:10:18.160
exact same amount of
money in them, how

00:10:18.160 --> 00:10:19.950
much did she inherit?

00:10:19.950 --> 00:10:22.760
So this and this are going to
end up being equal, and we'll

00:10:22.760 --> 00:10:24.130
have to solve for m.

00:10:24.130 --> 00:10:26.100
So let's do that.

00:10:26.100 --> 00:10:32.010
We get 3/5 m plus-- Well, let's
see, this is the same

00:10:32.010 --> 00:10:33.840
thing as 1/10, right?

00:10:33.840 --> 00:10:36.010
Let me write that way.

00:10:36.010 --> 00:10:40.180
So 1/10 times 3/5 is 3/50.

00:10:40.180 --> 00:10:46.890
Plus 3/50 m is equal to-- I just
multiplied the 1/10 times

00:10:46.890 --> 00:10:52.880
3/5 --is equal to 2/5 m-- I
want to do it in that same

00:10:52.880 --> 00:11:06.110
color, in the orange --is
equal to 2/5 m plus 500.

00:11:06.110 --> 00:11:10.520
And then 0.2 is the same
thing as 1/5, right?

00:11:10.520 --> 00:11:16.070
0.2-- let me write it over here
--is equal to 20/100,

00:11:16.070 --> 00:11:17.630
which is equal to 1/5.

00:11:17.630 --> 00:11:20.620
So we can rewrite this
right here as 1/5.

00:11:20.620 --> 00:11:27.200
So 2/5 m plus 500 minus
1/5 times 2/5 plus

00:11:27.200 --> 00:11:32.380
5 2/5 m plus 500.

00:11:32.380 --> 00:11:35.220
So that's a hairy problem, but
well take it step by step and

00:11:35.220 --> 00:11:37.850
see that it's not so bad.

00:11:37.850 --> 00:11:40.600
So right here, let's add
3/5 of something

00:11:40.600 --> 00:11:42.620
plus 3/50 of something.

00:11:42.620 --> 00:11:46.290
Well, 3/5 is the same thing
as 30/50, right?

00:11:46.290 --> 00:11:48.630
If I multiply the numerator
and the denominator by 10.

00:11:48.630 --> 00:11:49.630
And now we can add this.

00:11:49.630 --> 00:11:59.950
30/50 plus 3/50 is 33/50 m is
equal to-- and let's just

00:11:59.950 --> 00:12:03.820
simplify this a little
bit --2/5 m plus 500.

00:12:03.820 --> 00:12:10.360
Distribute the negative 1/5,
so you get negative 2/25 m,

00:12:10.360 --> 00:12:15.630
and then negative 1/5 times
500 is minus 100.

00:12:15.630 --> 00:12:18.040
Let's simplify this even more.

00:12:18.040 --> 00:12:25.090
The left-hand side is still 33
over 50 m is equal to-- And

00:12:25.090 --> 00:12:29.910
now we have we have these
coefficients on our m terms,

00:12:29.910 --> 00:12:30.440
right here.

00:12:30.440 --> 00:12:31.410
Those are our m terms.

00:12:31.410 --> 00:12:38.220
So you could view it as
2/5 minus 2/25 m.

00:12:38.220 --> 00:12:40.370
that takes care of that
term and that term.

00:12:40.370 --> 00:12:42.190
I just factored the m out.

00:12:42.190 --> 00:12:45.190
And then you have the
500 minus 100.

00:12:45.190 --> 00:12:48.860
So that's plus 400.

00:12:48.860 --> 00:12:49.140
Now, let's see.

00:12:49.140 --> 00:12:52.140
2/5, if we multiplied the
numerator and denominator by

00:12:52.140 --> 00:12:56.810
5, this becomes 10/25.

00:12:56.810 --> 00:12:57.170
Right?

00:12:57.170 --> 00:13:06.130
So our whole equation is now
33/50 m is equal to-- what is

00:13:06.130 --> 00:13:07.350
10 minus 2?

00:13:07.350 --> 00:13:13.560
So that's 8/25 m plus 400.

00:13:13.560 --> 00:13:15.080
We're getting close!

00:13:15.080 --> 00:13:15.770
We're getting close.

00:13:15.770 --> 00:13:18.680
Now let's get both
m terms onto the

00:13:18.680 --> 00:13:20.520
left-hand side of the equation.

00:13:20.520 --> 00:13:25.680
So let's subtract 8/25
m from both sides.

00:13:32.600 --> 00:13:35.110
Did that, so that the right-hand
side cancels outs.

00:13:35.110 --> 00:13:38.710
So our right-hand side
is just equal to 400.

00:13:38.710 --> 00:13:44.140
And then our left-hand side
is 33/50 minus 8 over 25.

00:13:44.140 --> 00:13:49.810
So it's equal to 33 over
50 minus 8 over 25.

00:13:49.810 --> 00:13:53.350
That's the same thing as minus
16 over 50, right?

00:13:53.350 --> 00:13:56.190
I just multiply the numerator
and denominator by 2.

00:13:56.190 --> 00:13:58.580
m is equal to 400.

00:13:58.580 --> 00:13:59.535
We're almost there!

00:13:59.535 --> 00:14:01.850
This is a nice, meaty problem.

00:14:01.850 --> 00:14:03.440
Almost there.

00:14:03.440 --> 00:14:11.290
And then 33 minus
16 is 17, right?

00:14:11.290 --> 00:14:17.000
So we're left with 17/50
m is equal to 400.

00:14:17.000 --> 00:14:19.450
And now we can multiply
both sides times

00:14:19.450 --> 00:14:21.300
the inverse of 17/50.

00:14:21.300 --> 00:14:28.220
So 50/17 times 50/17.

00:14:28.220 --> 00:14:32.460
These cancel out, and you get
m is equal to-- and I'll get

00:14:32.460 --> 00:14:34.310
the calculator out for this.

00:14:34.310 --> 00:14:42.860
m is equal to 400 times
50 divided by

00:14:42.860 --> 00:14:55.380
17 is equal to $1,176.47.

00:14:55.380 --> 00:14:58.490
That's how much Lydia
started out with.

00:14:58.490 --> 00:15:00.550
Hopefully you found that fun.


Example: two-step equations

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:03.330
Let's do a few more examples
of solving equations.

00:00:03.330 --> 00:00:05.000
And I think you're going to
see that these equations

00:00:05.000 --> 00:00:07.640
require a few more steps
than the ones we

00:00:07.640 --> 00:00:08.490
did in the last video.

00:00:08.490 --> 00:00:11.420
But the fun thing about these is
that there's more than one

00:00:11.420 --> 00:00:12.060
way to do it.

00:00:12.060 --> 00:00:14.260
But as long as you do legitimate
steps, as long as

00:00:14.260 --> 00:00:16.760
anything you do to the left-hand
side, you also do to

00:00:16.760 --> 00:00:19.760
the right-hand side, you should
move in the correct

00:00:19.760 --> 00:00:22.100
direction, or you shouldn't
get the wrong answer.

00:00:22.100 --> 00:00:23.990
So let's do a couple of these.

00:00:23.990 --> 00:00:28.870
So the first one says-- I'll
rewrite it-- 1.3 times x minus

00:00:28.870 --> 00:00:35.280
0.7 times x is equal to 12.

00:00:35.280 --> 00:00:38.080
Well, here the first thing that
my instinct is to do, is

00:00:38.080 --> 00:00:40.900
to merge these two terms.
Because I have 1.3 of

00:00:40.900 --> 00:00:45.020
something minus 0.7 of
that same something.

00:00:45.020 --> 00:00:46.180
This is the same variable.

00:00:46.180 --> 00:00:50.340
If I have 1.3 apples minus 0.7
apples, well, why don't I

00:00:50.340 --> 00:00:52.840
subtract 0.7 from 1.3?

00:00:52.840 --> 00:01:01.530
And I will get 1.3 minus 0.7
x's, or apples, or whatever

00:01:01.530 --> 00:01:02.610
you want to call them.

00:01:02.610 --> 00:01:04.720
So is equal to 12.

00:01:04.720 --> 00:01:06.800
You could imagine that I did
the reverse distributive

00:01:06.800 --> 00:01:07.430
property out here.

00:01:07.430 --> 00:01:09.080
I factored out an x.

00:01:09.080 --> 00:01:11.610
But the way my head thinks about
it is, I have 1.3 of

00:01:11.610 --> 00:01:15.180
something minus 0.7 of
something, that's going to be

00:01:15.180 --> 00:01:20.240
equal to 1.3 minus 0.7 of those
somethings, that x.

00:01:20.240 --> 00:01:30.570
And of course 1.3 minus 0.7 is
0.6 times x of my somethings

00:01:30.570 --> 00:01:32.750
is equal to 12.

00:01:32.750 --> 00:01:34.970
And now, this looks just like
one of the problems we did in

00:01:34.970 --> 00:01:35.690
the last video.

00:01:35.690 --> 00:01:39.450
We have a coefficient times x is
equal to some other number.

00:01:39.450 --> 00:01:42.140
Well, let's divide both sides
of this equation by that

00:01:42.140 --> 00:01:43.870
coefficient.

00:01:43.870 --> 00:01:45.310
Divide both sides by 0.6.

00:01:50.570 --> 00:01:52.970
So the left-hand side will
just become an x.

00:01:52.970 --> 00:01:56.700
X is equal to-- and what
is 12 divided by 0.6?

00:01:56.700 --> 00:02:02.160
0.6 goes into 12-- let's add
some decimal points here--

00:02:02.160 --> 00:02:06.870
that's the same thing
as 6 going into 120.

00:02:06.870 --> 00:02:08.320
6 goes into 12.

00:02:08.320 --> 00:02:10.630
2 times 2, times 6 is 12.

00:02:10.630 --> 00:02:13.280
Subtract, you get a 0.

00:02:13.280 --> 00:02:15.570
6 goes into 0 0 times.

00:02:15.570 --> 00:02:16.840
So it's going to go 20 times.

00:02:16.840 --> 00:02:20.090
12 divided by 0.6 is 20.

00:02:20.090 --> 00:02:22.070
And we can verify.

00:02:22.070 --> 00:02:23.080
Let's verify this.

00:02:23.080 --> 00:02:24.760
1.3, let's substitute it back.

00:02:24.760 --> 00:02:33.090
1.3 times 20, minus
0.7, times 20.

00:02:33.090 --> 00:02:36.110
Let's verify that that
is equal to 12.

00:02:36.110 --> 00:02:37.760
So I'll take the calculator out,
just so you don't have to

00:02:37.760 --> 00:02:39.590
trust my math.

00:02:39.590 --> 00:02:46.940
So we have 1.3 times
20 is equal to 26.

00:02:46.940 --> 00:02:50.960
So this piece right
here is 26.

00:02:50.960 --> 00:02:54.640
And then 0.7 times 20.

00:02:54.640 --> 00:02:55.920
I don't need a calculator
for that.

00:02:55.920 --> 00:02:57.990
That is 14.

00:02:57.990 --> 00:03:01.440
26 minus 14 is 12.

00:03:01.440 --> 00:03:02.400
So it checks out.

00:03:02.400 --> 00:03:04.590
We got the right answer
for this equation.

00:03:04.590 --> 00:03:07.590
x is equal to 20.

00:03:07.590 --> 00:03:10.210
Let's do this one right here.

00:03:10.210 --> 00:03:12.640
5x minus 3x plus 2
is equal to 1.

00:03:12.640 --> 00:03:13.860
This looks very complicated.

00:03:13.860 --> 00:03:16.420
And whenever something looks
daunting, just do steps that

00:03:16.420 --> 00:03:18.280
look like they're simplifying
the equation.

00:03:18.280 --> 00:03:21.610
And over time, as long as you
do legitimate steps, you

00:03:21.610 --> 00:03:23.520
should be able to make
some progress.

00:03:23.520 --> 00:03:25.530
So the first thing I want to
do, is I want to distribute

00:03:25.530 --> 00:03:27.780
this negative 1 over here.

00:03:27.780 --> 00:03:35.150
So this is the same thing
as 5x minus 3x, minus 2.

00:03:35.150 --> 00:03:35.390
Right?

00:03:35.390 --> 00:03:37.530
I just did the distributive
property on the 3x and the 2.

00:03:37.530 --> 00:03:39.960
This is a negative 1
times 3x plus 2.

00:03:39.960 --> 00:03:43.270
So it's negative 1 times 3x,
plus negative 1 times 2.

00:03:43.270 --> 00:03:45.460
Or negative 3x minus 2.

00:03:45.460 --> 00:03:48.440
And that is going to
be equal to 1.

00:03:48.440 --> 00:03:51.500
Now, I have 5 of something
minus 3

00:03:51.500 --> 00:03:53.720
of that same something.

00:03:53.720 --> 00:03:56.490
So that's going to be equal
to 2 of that something.

00:03:56.490 --> 00:03:59.080
5x minus 3x is 2x.

00:03:59.080 --> 00:04:01.370
5 minus 3 is 2.

00:04:01.370 --> 00:04:06.570
And then I have the minus
2 is equal to 1.

00:04:06.570 --> 00:04:10.470
And now, I like to get into the
form where I have 2x, or I

00:04:10.470 --> 00:04:12.370
have something times x is
equal to something.

00:04:12.370 --> 00:04:14.070
So I want to get rid of
this negative 2 on

00:04:14.070 --> 00:04:15.390
the left-hand side.

00:04:15.390 --> 00:04:16.880
The best way I know how
to do that is to

00:04:16.880 --> 00:04:18.470
add 2 to both sides.

00:04:18.470 --> 00:04:20.209
So add a 2 on the
left-hand side.

00:04:20.209 --> 00:04:21.930
If I do it to the left-hand
side, I've got to do it to the

00:04:21.930 --> 00:04:23.440
right-hand side.

00:04:23.440 --> 00:04:25.870
Plus 2 on the right-hand side.

00:04:25.870 --> 00:04:28.780
These two guys will cancel out,
and you're going to get

00:04:28.780 --> 00:04:35.910
2x is equal to 1 plus
2, is equal to 3.

00:04:35.910 --> 00:04:40.550
And now you can divide both
sides by 2, and you get x is

00:04:40.550 --> 00:04:44.680
equal to 3/2.

00:04:44.680 --> 00:04:47.910
And I'll leave it for you to
verify that this is indeed the

00:04:47.910 --> 00:04:48.900
correct answer.

00:04:48.900 --> 00:04:52.040
Let me draw a little line here
so that our work doesn't get

00:04:52.040 --> 00:04:55.490
messy, although that might have
made it even messier.

00:04:55.490 --> 00:04:57.600
So here we have to
solve for s.

00:04:57.600 --> 00:04:59.850
And look, we have a fraction
and 2 s terms.

00:04:59.850 --> 00:05:01.030
How do we do that?

00:05:01.030 --> 00:05:02.120
Well, just do it the same way.

00:05:02.120 --> 00:05:08.010
We have 1 times s minus-- you
can view this as 3/8 times s

00:05:08.010 --> 00:05:10.500
is equal to 5/6.

00:05:10.500 --> 00:05:13.970
You could view this as 1 times
s, minus 3/8 times

00:05:13.970 --> 00:05:15.800
s is equal to 5/6.

00:05:15.800 --> 00:05:17.920
You could factor out
an s, if you like.

00:05:17.920 --> 00:05:18.750
Maybe I'll do it this way.

00:05:18.750 --> 00:05:20.680
I'll factor it onto the
left-hand side.

00:05:20.680 --> 00:05:23.440
This is the same thing
as s times 1, minus

00:05:23.440 --> 00:05:28.520
3/8 is equal to 5/6.

00:05:28.520 --> 00:05:30.730
And 1 minus 3/8, what is that?

00:05:30.730 --> 00:05:34.930
That 1, I can rewrite as 8/8.

00:05:34.930 --> 00:05:36.030
That's 1.

00:05:36.030 --> 00:05:43.980
So this is the same thing as 8/8
minus 3/8 is 5/8, times s.

00:05:43.980 --> 00:05:46.680
You could switch the order
of multiplication.

00:05:46.680 --> 00:05:51.510
5/8 times s is equal to 5/6.

00:05:51.510 --> 00:05:53.020
And you might be able to
go straight from that.

00:05:53.020 --> 00:05:56.100
If I have 1 of something minus
3/8 of that something, I have

00:05:56.100 --> 00:05:58.980
8/8 of that something minus
3/8 of that something, I'm

00:05:58.980 --> 00:06:01.340
going to have 5/8 of
that something.

00:06:01.340 --> 00:06:05.450
And now, to solve for s, I can
multiply both sides by the

00:06:05.450 --> 00:06:07.120
inverse of this coefficient.

00:06:07.120 --> 00:06:10.900
So I multiply 8 over
5 times 5/8s.

00:06:10.900 --> 00:06:12.790
If I do it to the left-hand
side, I have to do it to the

00:06:12.790 --> 00:06:13.650
right-hand side.

00:06:13.650 --> 00:06:15.370
8 over 5.

00:06:15.370 --> 00:06:18.100
I multiplied by 8 over 5 so
that those cancel out and

00:06:18.100 --> 00:06:19.380
those cancel out.

00:06:19.380 --> 00:06:23.660
And you are left with s is equal
to-- right, this is just

00:06:23.660 --> 00:06:27.750
a 1-- is equal to-- well,
the 5's we can divide.

00:06:27.750 --> 00:06:29.910
Divide the numerator and
the denominator by 5.

00:06:29.910 --> 00:06:34.140
Divide the numerator by 2 and
the denominator by 2.

00:06:34.140 --> 00:06:36.890
You're left with-- sorry, divide
the denominator by 2,

00:06:36.890 --> 00:06:38.890
you get 6 divided by 2 is 3.

00:06:38.890 --> 00:06:41.740
You're left with 4/3.

00:06:41.740 --> 00:06:45.860
s is equal to 4/3.

00:06:45.860 --> 00:06:48.010
Let's do one more of these.

00:06:48.010 --> 00:06:53.640
So here I have 5 times q minus
7 over 12 is equal to 2/3.

00:06:53.640 --> 00:06:54.570
So let me write this.

00:06:54.570 --> 00:07:00.290
And I could rewrite this as just
5 over 12 times q minus 7

00:07:00.290 --> 00:07:01.890
is equal to 2/3.

00:07:01.890 --> 00:07:03.390
And what I want to do with this
video, is to show you

00:07:03.390 --> 00:07:05.330
that I can do it two
different ways.

00:07:05.330 --> 00:07:08.210
But as long as I do legitimate
operations, I should get the

00:07:08.210 --> 00:07:09.240
same answer.

00:07:09.240 --> 00:07:11.670
So the first way I'm going
to do it, is I'm going to

00:07:11.670 --> 00:07:13.500
multiply both sides of
this equation by

00:07:13.500 --> 00:07:16.040
the inverse of 5/12.

00:07:16.040 --> 00:07:23.460
So I'm going to multiply both
sides by 12 over 5.

00:07:23.460 --> 00:07:25.200
Because I wanted to get
rid of this 5/12 on

00:07:25.200 --> 00:07:25.820
the left-hand side.

00:07:25.820 --> 00:07:28.510
It makes everything look
a little bit messy.

00:07:28.510 --> 00:07:30.470
And I multiply it by 12 over 5,
because these are going to

00:07:30.470 --> 00:07:30.960
cancel out.

00:07:30.960 --> 00:07:33.960
The 5 and the 5 cancel out, the
12 and the 12 cancel out.

00:07:33.960 --> 00:07:38.190
So the left-hand side of my
equation becomes q minus 7 is

00:07:38.190 --> 00:07:41.860
equal to the right-hand
side, 2/3 times 5/12.

00:07:41.860 --> 00:07:44.450
If you divide the 12
by 3, you get a 4.

00:07:44.450 --> 00:07:46.190
You divide the 3 by
3, you get a 1.

00:07:46.190 --> 00:07:52.080
So 2 times 4 is 8 over 5.

00:07:52.080 --> 00:07:55.960
And now we can add 7 to both
sides of this equation.

00:07:55.960 --> 00:07:59.490
So let's add-- I want to do that
in a different color--

00:07:59.490 --> 00:08:04.340
add 7 to both sides
of this equation.

00:08:04.340 --> 00:08:05.880
These two 7's cancel out.

00:08:05.880 --> 00:08:08.080
That was the whole point
of adding the 7.

00:08:08.080 --> 00:08:13.650
And you are left with q is
equal to 8/5 plus 7.

00:08:13.650 --> 00:08:22.270
Or we could write 8/5 plus
7 can be written as 35/5.

00:08:22.270 --> 00:08:26.075
And so this is going to be
equal to 8-- well, the

00:08:26.075 --> 00:08:27.720
denominator is 5.

00:08:27.720 --> 00:08:33.490
8 plus 35 is 43.

00:08:33.490 --> 00:08:37.990
So my answer, going this way,
is q is equal to 43/5.

00:08:37.990 --> 00:08:39.250
And I said I would
do it two ways.

00:08:39.250 --> 00:08:40.520
Let's do it another way.

00:08:40.520 --> 00:08:43.240
So let me write the
same problem down.

00:08:43.240 --> 00:08:48.310
So I have 5/12-- actually, let
me just do it a completely

00:08:48.310 --> 00:08:50.240
different way.

00:08:50.240 --> 00:08:51.890
Let me write it the
way they wrote it.

00:08:51.890 --> 00:08:58.590
5 times q minus 7, over
12 is equal to 2/3.

00:08:58.590 --> 00:09:01.410
Let me just get rid of the 12
first. Let me multiply both

00:09:01.410 --> 00:09:03.890
sides of this equation by 12.

00:09:03.890 --> 00:09:05.830
I just don't like that 12
sitting there, so I'm going to

00:09:05.830 --> 00:09:08.530
multiply both sides by 12.

00:09:08.530 --> 00:09:10.430
So these are going to cancel
out, and you're going to be

00:09:10.430 --> 00:09:17.440
left with 5 times q minus 7
is equal to 2/3 times 12.

00:09:17.440 --> 00:09:20.020
That's the same thing
as 24 over 3.

00:09:20.020 --> 00:09:21.620
So this is, let me write this.

00:09:21.620 --> 00:09:27.570
2 over 3 times 12 over 1 is
equal to-- if you divide that

00:09:27.570 --> 00:09:29.610
by 3, you get a 4, divide
that by 3, you get a

00:09:29.610 --> 00:09:31.590
1-- is equal to 8.

00:09:31.590 --> 00:09:34.700
So you get 5 times q minus
7 is equal to 8.

00:09:34.700 --> 00:09:37.140
And then instead of dividing
both sides by 5, which would

00:09:37.140 --> 00:09:41.090
get us pretty close to what we
were doing over here, let me

00:09:41.090 --> 00:09:43.510
distribute this 5, I just want
to show you, you can do it

00:09:43.510 --> 00:09:45.210
multiple legitimate ways.

00:09:45.210 --> 00:09:47.830
So 5 times q is 5q.

00:09:47.830 --> 00:09:53.130
5 times negative 7 is minus, or
negative 35, is equal to 8.

00:09:53.130 --> 00:09:56.030
5q minus 35 is equal to 8.

00:09:56.030 --> 00:09:58.630
Now, if I want to get rid of
that minus 35, or that

00:09:58.630 --> 00:10:01.230
negative 35, the best
way to do it is to

00:10:01.230 --> 00:10:06.750
add 35 to both sides.

00:10:06.750 --> 00:10:09.650
I did that so that these cancel
out, and I'm left with

00:10:09.650 --> 00:10:15.120
5q is equal to 8 plus
35, which is 43.

00:10:15.120 --> 00:10:19.110
Now I can multiply both sides of
this equation by 1/5, which

00:10:19.110 --> 00:10:22.720
is the same thing as dividing
both sides by 5.

00:10:22.720 --> 00:10:23.970
And these cancel out.

00:10:23.970 --> 00:10:27.340
You get q is equal
to 43 over 5.

00:10:27.340 --> 00:10:30.410
So there's a bunch of ways you
can do these problems. But as

00:10:30.410 --> 00:10:33.410
long as you do legitimate
steps, you will

00:10:33.410 --> 00:10:34.150
get the right answer.

00:10:34.150 --> 00:10:38.140
And I'll leave it to you to
verify that this truly is the

00:10:38.140 --> 00:10:39.240
right answer for q.

00:10:39.240 --> 00:10:43.310
This is the q that will
satisfy this equation.

00:10:43.310 --> 00:10:46.240
Let's do one word
problem here.

00:10:46.240 --> 00:10:51.710
Jade is stranded downtown with
only $10 to get home.

00:10:51.710 --> 00:10:57.880
Taxis cost $0.75 per mile, but
there's an additional $2.35

00:10:57.880 --> 00:10:59.380
hire charge.

00:10:59.380 --> 00:11:02.080
Write a formula and use it to
calculate how many miles she

00:11:02.080 --> 00:11:05.290
can travel with her money.

00:11:05.290 --> 00:11:05.790
All right.

00:11:05.790 --> 00:11:10.250
So the total cost of a cab ride
is going to be equal to

00:11:10.250 --> 00:11:18.960
just the initial hire charge,
which is $2.35, plus the $0.75

00:11:18.960 --> 00:11:25.010
per mile, times the
number of miles.

00:11:25.010 --> 00:11:28.780
We're letting m is equal to
the miles she travels.

00:11:28.780 --> 00:11:30.030
Miles traveled.

00:11:33.410 --> 00:11:34.710
So this is the equation.

00:11:34.710 --> 00:11:37.700
We know that she only
has $10 to get home.

00:11:37.700 --> 00:11:40.890
So her cost has to be $10.

00:11:40.890 --> 00:11:43.210
So we have to say, the
cost has to be $10.

00:11:43.210 --> 00:11:52.370
So 10 is equal to
2.35 plus 0.75m.

00:11:52.370 --> 00:11:54.320
So how do we solve for
m, or the number of

00:11:54.320 --> 00:11:56.815
miles Jade can travel?

00:11:56.815 --> 00:12:00.150
Well, we can get rid of the 2.35
on this right-hand side

00:12:00.150 --> 00:12:01.830
by subtracting that
amount from both

00:12:01.830 --> 00:12:03.860
sides of this equation.

00:12:03.860 --> 00:12:05.900
So let's do that.

00:12:05.900 --> 00:12:13.020
So let's subtract minus
2.35 from both sides.

00:12:13.020 --> 00:12:14.470
These will cancel out.

00:12:14.470 --> 00:12:15.620
That was the point.

00:12:15.620 --> 00:12:25.880
The left-hand side-- what
is 10 minus 2.35?

00:12:25.880 --> 00:12:27.390
Now, these will cancel out.

00:12:27.390 --> 00:12:29.040
Now what is 10 minus 2.35?

00:12:29.040 --> 00:12:33.070
10 minus 2 is 8.

00:12:33.070 --> 00:12:37.580
10 minus 2.3 is 7.7.

00:12:37.580 --> 00:12:41.620
So it's going to be 7.65.

00:12:41.620 --> 00:12:44.270
If you want to believe
me, let's do it.

00:12:44.270 --> 00:12:48.150
10 minus 2.35.

00:12:48.150 --> 00:12:50.220
7.65.

00:12:50.220 --> 00:12:56.220
And that is going to
be equal to 0.75m.

00:12:56.220 --> 00:12:57.480
Let me write that in
that same color.

00:12:57.480 --> 00:13:01.050
It's nice to see where different
things came from.

00:13:01.050 --> 00:13:03.360
0.75m.

00:13:03.360 --> 00:13:05.930
I have, like, five shades
of this purple here.

00:13:05.930 --> 00:13:10.770
so this is that, that is that,
and then these two guys

00:13:10.770 --> 00:13:11.970
canceled out.

00:13:11.970 --> 00:13:16.682
Now to solve for m, I can just
divide both sides by 0.75.

00:13:16.682 --> 00:13:20.340
So if I divide that side by
0.75, I have to do it to the

00:13:20.340 --> 00:13:21.350
left-hand side as well.

00:13:21.350 --> 00:13:23.430
0.75.

00:13:23.430 --> 00:13:25.990
That cancels out, so on the
right-hand side, I'm left

00:13:25.990 --> 00:13:27.380
with just an m.

00:13:27.380 --> 00:13:29.540
And on the left-hand side-- I'll
have to get my calculator

00:13:29.540 --> 00:13:37.350
out for this one-- I have 7.65
divided by 0.75, which is

00:13:37.350 --> 00:13:42.320
equal to 10.2.

00:13:42.320 --> 00:13:47.930
m is 10.2, so Jade can
travel 10.2 miles.


Simple equations: examples solving a variety of forms

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.670
Let's do some examples
of solving equations.

00:00:03.670 --> 00:00:06.110
And you're going to see the
equations in this video,

00:00:06.110 --> 00:00:08.320
they're not going to require a
lot of steps, but I really

00:00:08.320 --> 00:00:10.970
want to focus on what's actually
happening on the

00:00:10.970 --> 00:00:14.020
equation, and why it makes sense
to do the steps that

00:00:14.020 --> 00:00:14.910
we're doing.

00:00:14.910 --> 00:00:18.495
So this first equation here,
we have x-- I'll rewrite it

00:00:18.495 --> 00:00:22.890
--x plus 11 is equal to 7.

00:00:22.890 --> 00:00:25.280
So that's telling us-- And I
have this little dot here, let

00:00:25.280 --> 00:00:29.510
me eliminate, let me
clear that dot.

00:00:29.510 --> 00:00:31.950
Don't want you to think I have
some type of a decimal there.

00:00:31.950 --> 00:00:32.820
There you go.

00:00:32.820 --> 00:00:34.200
So back to the problem.

00:00:34.200 --> 00:00:35.130
So what is this saying?

00:00:35.130 --> 00:00:39.950
This is saying, some number
plus 11 is equal to 7.

00:00:39.950 --> 00:00:41.840
And you may already be able
to do this in your head.

00:00:41.840 --> 00:00:45.730
You know, what do I have to
add to 11 to get to 7?

00:00:45.730 --> 00:00:46.650
So it's going to have
to be some type

00:00:46.650 --> 00:00:47.375
of a negative number.

00:00:47.375 --> 00:00:49.780
But we're going to learn how
to do this systematically.

00:00:49.780 --> 00:00:52.860
So what we want to do is, when
we solve an equation, it

00:00:52.860 --> 00:00:54.885
should be of the form,
when we're all done,

00:00:54.885 --> 00:00:56.890
x is equal to something.

00:00:56.890 --> 00:00:57.950
And that's going to
be our answer.

00:00:57.950 --> 00:01:01.840
We'll know what number satisfies
this equation.

00:01:01.840 --> 00:01:04.470
Now, to get there, what we want
to do is to just have

00:01:04.470 --> 00:01:06.460
this x on the left-hand
side of the equation.

00:01:06.460 --> 00:01:08.000
We don't want this 11 there.

00:01:08.000 --> 00:01:11.150
We want to get rid of this 11
somehow, on the left-hand side

00:01:11.150 --> 00:01:12.220
of this equation.

00:01:12.220 --> 00:01:14.570
Now to do that, we can't just
get rid of it only on the

00:01:14.570 --> 00:01:15.310
left-hand side.

00:01:15.310 --> 00:01:18.950
You might want to just say, hey,
let me subtract 11 from

00:01:18.950 --> 00:01:20.640
the left-hand side
of the equation.

00:01:20.640 --> 00:01:22.130
But this is an equation.

00:01:22.130 --> 00:01:25.310
Anything you do to one side of
the equation, you have to do

00:01:25.310 --> 00:01:27.110
to the other side
of the equation.

00:01:27.110 --> 00:01:31.850
If x plus 11 is equal to
7, x plus 11 minus 11

00:01:31.850 --> 00:01:33.230
won't equal 7 anymore.

00:01:33.230 --> 00:01:36.820
You'll have to also subtract
11 from the 7.

00:01:36.820 --> 00:01:38.230
Now they're equal.

00:01:38.230 --> 00:01:39.950
Remember, I want to really
emphasize this.

00:01:39.950 --> 00:01:41.300
I want you to really
think about what

00:01:41.300 --> 00:01:42.840
this equality means.

00:01:42.840 --> 00:01:45.810
Equation, it comes from the
notion of equality.

00:01:45.810 --> 00:01:49.670
This is equal to this.

00:01:49.670 --> 00:01:52.640
These two things are equivalent
quantities.

00:01:52.640 --> 00:01:56.050
Now, in order for this equal to
hold true, anything I do to

00:01:56.050 --> 00:01:59.470
this, if I'm subtracting 11 from
this quantity, in order

00:01:59.470 --> 00:02:02.480
for the equality to hold true,
I also have to subtract 11

00:02:02.480 --> 00:02:06.530
from this quantity right here.
x plus 7 and 7 are equal.

00:02:06.530 --> 00:02:09.940
So they're still going to be
equal after I subtract 11.

00:02:09.940 --> 00:02:11.830
Now let's focus on the
left-hand side.

00:02:11.830 --> 00:02:17.360
Negative 11 plus x plus 11,
negative 11 plus 11, they're

00:02:17.360 --> 00:02:18.480
going to cancel out, right?

00:02:18.480 --> 00:02:20.910
11 minus 11 is 0.

00:02:20.910 --> 00:02:24.270
So you're going to be with left
with just an x is equal

00:02:24.270 --> 00:02:27.446
to 7 minus 11.

00:02:27.446 --> 00:02:29.910
Now what's 7 minus 11?

00:02:29.910 --> 00:02:32.810
You can draw a number line
out, if you like.

00:02:32.810 --> 00:02:39.030
If we draw a number line right
here-- 0, 1, 2, 3, 4, 5, 6, 7.

00:02:39.030 --> 00:02:40.880
7 is there.

00:02:40.880 --> 00:02:46.550
And if you subtract 11 from
that, you go back 1, 2, 3, 4,

00:02:46.550 --> 00:02:52.460
5, 6, 7, 8, 9, 10, 11.

00:02:52.460 --> 00:02:54.790
8, 9, 10, 11.

00:02:54.790 --> 00:02:57.320
That takes you to negative 4.

00:02:57.320 --> 00:03:01.370
7 minus 11 is negative 4.

00:03:01.370 --> 00:03:03.560
That's our first equation.

00:03:03.560 --> 00:03:06.120
Now we have 7 times some
number-- this is the new

00:03:06.120 --> 00:03:09.570
problem --7 times some number
x is equal to 21.

00:03:09.570 --> 00:03:10.860
This is going to be a different
x than this one

00:03:10.860 --> 00:03:11.750
right here.

00:03:11.750 --> 00:03:14.560
So how, once again, do we get
it in the form of x is equal

00:03:14.560 --> 00:03:15.120
to something?

00:03:15.120 --> 00:03:17.770
Remember, we've got x is
equal to something.

00:03:17.770 --> 00:03:19.460
How do we get into that form?

00:03:19.460 --> 00:03:21.000
Let me rewrite the problem.

00:03:21.000 --> 00:03:25.020
7 times something
is equal to 21.

00:03:25.020 --> 00:03:28.250
Well, I could think
of it two ways.

00:03:28.250 --> 00:03:31.360
If I divide 7 times
x by 7, what am I

00:03:31.360 --> 00:03:33.250
going to be left with?

00:03:33.250 --> 00:03:35.920
If I divide that by 7, what am
I going to be left with?

00:03:35.920 --> 00:03:38.780
7 times something divided by
7, that's just going to be

00:03:38.780 --> 00:03:39.470
that something.

00:03:39.470 --> 00:03:41.090
That's just going to
end up with an x.

00:03:41.090 --> 00:03:42.620
But if I do it on the left-hand
side of the

00:03:42.620 --> 00:03:43.500
equation, I have
to do it on the

00:03:43.500 --> 00:03:45.340
right-hand side of the equation.

00:03:45.340 --> 00:03:48.280
Remember, 7x is equal to 21.

00:03:48.280 --> 00:03:50.130
It's the same thing as 21.

00:03:50.130 --> 00:03:53.190
In order for this to be equal to
that when I divide by 7 on

00:03:53.190 --> 00:03:57.020
the left, I also have to divide
by 7 on the right.

00:03:57.020 --> 00:04:02.020
And when I do that, 7 times x
divided by 7-- that's just

00:04:02.020 --> 00:04:07.250
going to be x --and then 21
divided by 7 is equal to 3.

00:04:07.250 --> 00:04:10.710
So the solution to this equation
is x is equal to 3.

00:04:10.710 --> 00:04:11.900
And you can verify it.

00:04:11.900 --> 00:04:13.880
Take 3, plug it into
that equation.

00:04:13.880 --> 00:04:17.130
7 times 3 is indeed 21.

00:04:17.130 --> 00:04:18.329
Same thing over here.

00:04:18.329 --> 00:04:20.120
Take x is equal to negative 4.

00:04:20.120 --> 00:04:23.550
Negative 4 plus 11 is the
same thing as 11 minus

00:04:23.550 --> 00:04:26.180
4, is indeed 7.

00:04:26.180 --> 00:04:27.090
Let's do this one here.

00:04:27.090 --> 00:04:28.700
This one looks a little bit
more difficult, but you're

00:04:28.700 --> 00:04:30.700
going to see, it's
not too bad.

00:04:30.700 --> 00:04:38.020
So we have 5x over 12
is equal to 2/3.

00:04:38.020 --> 00:04:41.290
Now, how do you isolate
the x over here?

00:04:41.290 --> 00:04:44.960
What can we multiply or divide
by to isolate it?

00:04:44.960 --> 00:04:47.220
Well, what if we were to
multiply both sides of this

00:04:47.220 --> 00:04:49.850
equation by 12 over 5?

00:04:49.850 --> 00:04:51.230
What's going to happen?

00:04:51.230 --> 00:04:57.520
So if I multiply the left-hand
side by 12 over 5-- And

00:04:57.520 --> 00:04:59.630
remember, anything I do to one
side of an equation, I have to

00:04:59.630 --> 00:05:01.550
do to the other side
of an equation.

00:05:01.550 --> 00:05:04.140
So if I multiply the left side
by 12 over 5, I have to

00:05:04.140 --> 00:05:08.320
multiply the right side by 12
over 5 in order for that

00:05:08.320 --> 00:05:09.200
equality to hold true.

00:05:09.200 --> 00:05:11.470
But why would I pick
12 over 5?

00:05:11.470 --> 00:05:13.990
Because I wanted to pick a
number that cancels perfectly

00:05:13.990 --> 00:05:16.000
with the 5 over 12.

00:05:16.000 --> 00:05:16.180
Right?

00:05:16.180 --> 00:05:17.340
I have a 5 over 12 here.

00:05:17.340 --> 00:05:21.820
12 over 5 times 5 over 12, the
12's cancel out, the 5's

00:05:21.820 --> 00:05:24.390
cancel out, and you're
left with just an x.

00:05:24.390 --> 00:05:30.260
So you get x is equal to
2/3 times 12 over 5.

00:05:30.260 --> 00:05:32.910
You can divide the 12
by 3 and get a 4.

00:05:32.910 --> 00:05:35.120
You can divide the 3
by 3 and get a 1.

00:05:35.120 --> 00:05:39.160
So you get 2 times 4 is
equal to 8 over 1

00:05:39.160 --> 00:05:40.620
times 5, which is 5.

00:05:40.620 --> 00:05:42.940
So x is equal to 8/5.

00:05:42.940 --> 00:05:46.590
Now you might say, hey, Sal,
over here you divided by 7,

00:05:46.590 --> 00:05:49.130
here you're multiplying by the
inverse of the coefficient.

00:05:49.130 --> 00:05:50.390
When do I know what to do?

00:05:50.390 --> 00:05:52.080
And the answer is, you
can do either.

00:05:52.080 --> 00:05:55.790
I could have done this exact
same problem that way.

00:05:55.790 --> 00:05:59.100
I started with 7x
is equal to 21.

00:05:59.100 --> 00:06:01.630
Instead of dividing by 7, I
could have said, let me

00:06:01.630 --> 00:06:03.910
multiply by the inverse of 7.

00:06:03.910 --> 00:06:07.800
So I could have multiplied by
the inverse of 7 on both sides

00:06:07.800 --> 00:06:08.870
of the equation.

00:06:08.870 --> 00:06:12.020
This is equivalent
to dividing by 7.

00:06:12.020 --> 00:06:12.930
You can verify it.

00:06:12.930 --> 00:06:15.640
When you multiply by 1/7, that's
the exact same thing as

00:06:15.640 --> 00:06:17.130
dividing by 7.

00:06:17.130 --> 00:06:19.580
So if you work this out,
these would cancel out.

00:06:19.580 --> 00:06:21.750
This is 1/7 times 7/1.

00:06:21.750 --> 00:06:22.940
The 7's cancel out.

00:06:22.940 --> 00:06:27.890
You get x is equal to 21 over
7, which is equal to 3.

00:06:27.890 --> 00:06:33.950
Likewise, over here,
I have 5x over 12.

00:06:33.950 --> 00:06:35.150
Let me write it this way.

00:06:35.150 --> 00:06:41.360
I could write it as, 5 over
12x is equal to 2/3.

00:06:41.360 --> 00:06:42.650
This is the exact
same problem.

00:06:42.650 --> 00:06:44.990
I'm just separating the
x out a little bit.

00:06:44.990 --> 00:06:48.890
5/12 times x is 5x over 12.

00:06:48.890 --> 00:06:51.490
Now here, I could just say, let
me divide both sides by

00:06:51.490 --> 00:06:53.940
the coefficient 5 over 12.

00:06:53.940 --> 00:07:00.520
So I could have just divided
by 5 over 12.

00:07:00.520 --> 00:07:03.550
But what does it mean to
divide by 5 over 12?

00:07:03.550 --> 00:07:08.070
That's the same thing as
multiplying by 12 over 5.

00:07:08.070 --> 00:07:10.725
Dividing by a fraction is the
same thing as multiplying by

00:07:10.725 --> 00:07:11.500
its inverse.

00:07:11.500 --> 00:07:15.140
So this and this, we're doing
the exact same thing.

00:07:15.140 --> 00:07:19.750
Whether you divide by the
coefficient, like we did over

00:07:19.750 --> 00:07:22.320
here, or whether you multiply
by the inverse of the

00:07:22.320 --> 00:07:25.350
coefficient, you're doing the
exact same operation.

00:07:25.350 --> 00:07:27.760
Now let's do some of these other
problems. And this is

00:07:27.760 --> 00:07:29.990
really just to show you that the
variable isn't always x.

00:07:29.990 --> 00:07:31.180
We can solve for q.

00:07:31.180 --> 00:07:32.610
We can solve for z.

00:07:32.610 --> 00:07:33.740
We can solve for s.

00:07:33.740 --> 00:07:35.100
We can solve for anything.

00:07:35.100 --> 00:07:37.710
So let's solve these
down here.

00:07:37.710 --> 00:07:42.790
So we have-- and I'll rewrite
it --you have q minus 13 is

00:07:42.790 --> 00:07:45.540
equal to negative 13.

00:07:45.540 --> 00:07:47.380
So once again, we want
to isolate the q.

00:07:47.380 --> 00:07:49.300
We want it to be, q is
equal to something.

00:07:49.300 --> 00:07:51.880
Then we will have solved
our problem.

00:07:51.880 --> 00:07:53.220
So how can we isolate the q?

00:07:53.220 --> 00:07:55.590
How can we get rid of this 13?

00:07:55.590 --> 00:07:58.315
What if we add 13 to both
sides of this equation?

00:08:02.300 --> 00:08:06.110
And I'm adding 13, because if
I add 13 to negative 13, or

00:08:06.110 --> 00:08:08.680
minus 13 right here,
I'm going to get 0.

00:08:08.680 --> 00:08:11.910
So the left-hand side of this
equation is going to become q,

00:08:11.910 --> 00:08:13.900
and then what is the right-hand
side, right?

00:08:13.900 --> 00:08:15.720
That and that will cancel out.

00:08:15.720 --> 00:08:17.480
And the right-hand
side, negative 13

00:08:17.480 --> 00:08:19.590
plus 13, is just 0.

00:08:19.590 --> 00:08:21.775
So our solution is,
q is equal to 0.

00:08:21.775 --> 00:08:23.230
And we can check it.

00:08:23.230 --> 00:08:28.100
0 minus 13 is indeed
negative 13.

00:08:28.100 --> 00:08:29.120
So it works.

00:08:29.120 --> 00:08:30.810
Actually I forgot to check
this one over here.

00:08:30.810 --> 00:08:31.380
Let's check it.

00:08:31.380 --> 00:08:32.500
The fun thing about
algebra, you can

00:08:32.500 --> 00:08:33.919
always check your answer.

00:08:33.919 --> 00:08:36.600
We got x is equal to
8/5 over here.

00:08:36.600 --> 00:08:38.010
Let's check our answer.

00:08:38.010 --> 00:08:44.880
5 over 12 times x times 8
over 5 is equal to what?

00:08:44.880 --> 00:08:45.910
5's cancel out.

00:08:45.910 --> 00:08:47.990
You can divide the 8
by 4 and get a 2.

00:08:47.990 --> 00:08:50.130
You could divide the 12
by 4 and get a 3.

00:08:50.130 --> 00:08:51.990
It equals 2/3.

00:08:51.990 --> 00:08:53.200
So we can verify.

00:08:53.200 --> 00:08:53.950
That's the neat thing.

00:08:53.950 --> 00:08:55.600
Once you get your answer, you
can always make sure, if you

00:08:55.600 --> 00:08:58.500
have enough time, that you
got the right answer.

00:08:58.500 --> 00:09:02.180
Let's do these last two.

00:09:02.180 --> 00:09:11.330
We have z plus 1.1 is
equal to 3.0001.

00:09:11.330 --> 00:09:13.810
Once again, we want to get
rid of this 1.1 on

00:09:13.810 --> 00:09:14.840
the left-hand side.

00:09:14.840 --> 00:09:17.370
The best way to do that that I
can think of is subtracting

00:09:17.370 --> 00:09:21.970
1.1 from both sides, or adding
negative 1.1, however

00:09:21.970 --> 00:09:22.750
you want to do it.

00:09:22.750 --> 00:09:25.290
If I do it on the left, I've
got to do it on the right.

00:09:25.290 --> 00:09:27.205
Otherwise the equality won't
still hold true.

00:09:27.205 --> 00:09:30.880
If this is equal to this, in
order for the two sides to be

00:09:30.880 --> 00:09:32.530
equal, anything I do
to the left, I have

00:09:32.530 --> 00:09:34.070
to do to the right.

00:09:34.070 --> 00:09:34.630
So let's see.

00:09:34.630 --> 00:09:39.390
If I subtract 1.1 from z plus
1.1, those two cancel out, and

00:09:39.390 --> 00:09:50.960
I'm left with just a z is equal
to 3.0001 minus 1.1.

00:09:50.960 --> 00:09:51.852
So here we're going
to do a little bit

00:09:51.852 --> 00:09:53.500
of subtracting decimals.

00:09:53.500 --> 00:09:55.590
That's probably the hardest
part of this problem.

00:09:55.590 --> 00:10:00.030
So 3.0001 minus 1.1.

00:10:00.030 --> 00:10:01.320
You want to align
the decimals.

00:10:01.320 --> 00:10:02.570
Minus 1.1.

00:10:05.830 --> 00:10:06.510
Just like that.

00:10:06.510 --> 00:10:10.890
And we can even add some 0's
there, just to make it clear.

00:10:10.890 --> 00:10:13.260
And let's do our subtraction.

00:10:13.260 --> 00:10:14.730
1 minus 0 is 1.

00:10:14.730 --> 00:10:16.360
0 minus 0 is 0.

00:10:16.360 --> 00:10:18.460
0 minus 0 is 0.

00:10:18.460 --> 00:10:21.180
Uh oh, we don't have-- We
can't do 0 minus 1.

00:10:21.180 --> 00:10:23.470
We can regroup, or
we can borrow.

00:10:23.470 --> 00:10:28.840
So this will become a 10,
which is really a 1.

00:10:28.840 --> 00:10:31.090
And we're going to take
a 1 from here, so this

00:10:31.090 --> 00:10:33.120
becomes a 2, right?

00:10:33.120 --> 00:10:35.970
We borrowed from the 3, we
regrouped the numbers.

00:10:35.970 --> 00:10:39.260
We took the whole 1 and put it
in the tenths place, so it

00:10:39.260 --> 00:10:40.840
becomes 10/10.

00:10:40.840 --> 00:10:43.660
So 10 minus 1 is 9.

00:10:43.660 --> 00:10:45.100
We have our decimal space.

00:10:45.100 --> 00:10:47.460
And then 2 minus 1 is 1.

00:10:47.460 --> 00:10:53.380
So our answer is z is
equal to 1.9001.

00:10:53.380 --> 00:10:55.220
And we can verify that.

00:10:55.220 --> 00:10:56.470
What's z plus 1.1?

00:11:00.365 --> 00:11:01.980
Let's add it up.

00:11:01.980 --> 00:11:08.090
You get a 1, 00, 9.1 is 10, 1
plus 1 plus 1 is 3, we have

00:11:08.090 --> 00:11:11.870
our decimal sign, and
we got 3.0001.

00:11:11.870 --> 00:11:14.770
That's a good review of
subtracting decimals.

00:11:14.770 --> 00:11:15.290
All right.

00:11:15.290 --> 00:11:16.390
And we have this last problem.

00:11:16.390 --> 00:11:19.920
21s is equal to 3.

00:11:19.920 --> 00:11:21.890
Like we said, we could do
this two ways, which are

00:11:21.890 --> 00:11:23.530
essentially the same way.

00:11:23.530 --> 00:11:25.750
21 times some number
is equal to 3.

00:11:25.750 --> 00:11:27.670
Well, what happens if
we multiply both

00:11:27.670 --> 00:11:33.550
sides by 1 over 21?

00:11:33.550 --> 00:11:36.050
By the inverse of
the coefficient.

00:11:36.050 --> 00:11:38.050
The coefficient is just the
number that sits out in front

00:11:38.050 --> 00:11:40.450
of the variable, that's
multiplying the variable.

00:11:40.450 --> 00:11:44.990
Well, 1 over 21 times 21 is just
1, so we get s is equal

00:11:44.990 --> 00:11:48.000
to 3 over 21.

00:11:48.000 --> 00:11:51.830
And 3 over 21 is the same
thing as 1 over 7.

00:11:51.830 --> 00:11:53.180
And we can verify that.

00:11:53.180 --> 00:11:56.540
What is 21 times 1 over 7?

00:11:56.540 --> 00:11:59.360
It's equal to 21 over 7.

00:11:59.360 --> 00:12:01.370
21 divided by 7.

00:12:01.370 --> 00:12:03.000
Which is 3.

00:12:03.000 --> 00:12:04.940
And just to really hit the point
home, I could have done

00:12:04.940 --> 00:12:08.520
this problem-- Instead of
multiplying by 1 over 21--

00:12:08.520 --> 00:12:11.240
let's see, 21s is equal to 3
--I could've, instead of

00:12:11.240 --> 00:12:13.640
multiplying by 1 over 21, a
completely equivalent thing to

00:12:13.640 --> 00:12:18.000
do would have been to just
divide both sides by 21.

00:12:18.000 --> 00:12:19.290
So these would cancel out.

00:12:19.290 --> 00:12:26.590
You'd get s is equal to 3
over 21, which is 1/7.

00:12:26.590 --> 00:12:29.840
Anyway, hopefully you found
those examples useful.