Age word problems 2

WEBVTT
Kind: captions
Language: en

00:00:01.430 --> 00:00:06.220
Welcome to the second
set of presentations

00:00:06.220 --> 00:00:08.640
on age word problems.

00:00:08.640 --> 00:00:10.200
I have one typed out here.

00:00:10.200 --> 00:00:11.240
Let's see what it says.

00:00:11.240 --> 00:00:15.530
It says, Salman- that's
me- is 108 years old.

00:00:15.530 --> 00:00:19.500
And so this is clearly at some
vast time in the future.

00:00:19.500 --> 00:00:23.380
And this'll only happen if my
caloric restriction works out.

00:00:23.380 --> 00:00:26.370
But Salman is 108 years old.

00:00:26.370 --> 00:00:29.220
Jonathan is 24 years old.

00:00:29.220 --> 00:00:33.210
How many years will it take
for Salman to be exactly 4

00:00:33.210 --> 00:00:35.930
times as old as Jonathan?

00:00:35.930 --> 00:00:38.720
Well, let's figure
this one out.

00:00:38.720 --> 00:00:41.240
What we're trying to solve for
is how many years will it take.

00:00:41.240 --> 00:00:43.740
So let's use the
variable y, for years.

00:00:43.740 --> 00:00:53.880
So y equals years until, let's
just say Sal, for short.

00:00:53.880 --> 00:00:57.275
Sal is 4 times Jonathan's age.

00:01:02.910 --> 00:01:09.275
Well, if today, I, or Sal, if
Salman, it's hard to speak

00:01:09.275 --> 00:01:10.430
in the third person.

00:01:10.430 --> 00:01:16.840
If Salman is 100 years old
today, in y years, Salman

00:01:16.840 --> 00:01:25.950
will be 108 plus y is
equal to Sal in y years.

00:01:25.950 --> 00:01:28.920
We'll say y years.

00:01:28.920 --> 00:01:32.670
Sal in y years is going
to be 108 plus y.

00:01:32.670 --> 00:01:35.330
And then Jonathan in y
years, that's pretty easy.

00:01:35.330 --> 00:01:39.720
He's going to be 24 plus y.

00:01:39.720 --> 00:01:42.690
So that's, I'll say
Jon for short.

00:01:42.690 --> 00:01:45.440
In y years.

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Not years.

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y years.

00:01:49.180 --> 00:01:51.260
And what else does
the problem say?

00:01:51.260 --> 00:01:54.760
It says, in how many years
would take for Salman to be

00:01:54.760 --> 00:01:57.610
exactly-- I should put that in
another color for emphasis--

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Exactly 4 times as
old as Jonathan?

00:02:02.900 --> 00:02:04.770
Well, and the exactly
is important.

00:02:04.770 --> 00:02:08.220
Because Salman is already
more than 4 times

00:02:08.220 --> 00:02:09.340
as old as Jonathan.

00:02:09.340 --> 00:02:12.590
But they want to figure out
exactly when is Salman going to

00:02:12.590 --> 00:02:14.890
be 4 times as old as Jonathan.

00:02:14.890 --> 00:02:21.450
Well, in y years Salman is
going to be 108 plus y.

00:02:21.450 --> 00:02:22.190
So we know that.

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108 plus y.

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And after y years, he's
going to be 4 times

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as old as Jonathan.

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Exactly 4 times.

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And Jonathan's going to
be 24 plus y years old.

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And now we just
solve the equation.

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Get 108 plus y is
equal to 4 times 24.

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Well, 4 times 25 is 100.

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And we can subtract
4 from there.

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So it's 96.

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Plus 4y.

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And now we just solve
this equation.

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We get 3.

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3, I'm going to skip some
steps, because I think this

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part is easy for you.

00:03:00.390 --> 00:03:05.910
3y will equal 108 minus 96.

00:03:05.910 --> 00:03:13.950
So we get 3y is equal to,
what is that, 12, right?

00:03:13.950 --> 00:03:16.780
So y is equal to 4.

00:03:16.780 --> 00:03:21.635
So our algebra has told us that
in 4 years Salman is going

00:03:21.635 --> 00:03:24.260
to be exactly 4 times
as old as Jonathan.

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Let's see if that's true.

00:03:25.630 --> 00:03:28.480
Well, if Salman is 108
right now, in 4 years

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he's going to be 112.

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And if Jonathan is 24 right
now, in 4 years he's

00:03:33.900 --> 00:03:36.610
going to be 28.

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And let's see.

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28 times 4 is 80 plus 32.

00:03:41.780 --> 00:03:42.730
Yep, exactly.

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He'll be 112.

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Looks like that problem worked.

00:03:45.780 --> 00:03:46.920
Excellent.

00:03:46.920 --> 00:03:50.035
Let's do another one.

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Hope you didn't hear that,
it was my stomach growling.

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See how hard I work
on this site?

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Don't even eat properly.

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Let's do another problem.

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I'll type it in
green this time.

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Tarush is 5 times as old as
Arman is today, 85 years ago.

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We're dealing with
huge swathes of time.

00:04:26.620 --> 00:04:29.920
85 years ago.

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Tarush was 10 times
as old as Arman.

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How old is Arman today?

00:04:45.730 --> 00:04:47.970
Let's see if we can tackle
this quite interesting

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problem, I think.

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Settings.

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Pen tool on.

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OK.

00:04:55.320 --> 00:04:57.380
Well, I think this might be
useful to do one of those

00:04:57.380 --> 00:05:01.030
charts like we did in the
first video presentation.

00:05:01.030 --> 00:05:03.980
Well, we're trying to solve
for how old is Arman today's.

00:05:03.980 --> 00:05:05.750
So let's say Arman.

00:05:09.010 --> 00:05:09.700
And we're going to have today.

00:05:13.350 --> 00:05:14.600
And we're going to have,
what's the other time

00:05:14.600 --> 00:05:15.470
period we're dealing with.

00:05:15.470 --> 00:05:17.020
We're dealing with
85 years ago.

00:05:17.020 --> 00:05:19.096
So, let's say.

00:05:19.096 --> 00:05:23.054
85 years ago.

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In a galaxy far, far away.

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OK, Arman and Tarush.

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Well, we're trying to solve
for how old is Arman today's.

00:05:36.970 --> 00:05:39.280
So let's just make that
x, for simplicity.

00:05:39.280 --> 00:05:43.150
It says Tarush is 5 times
as old as Arman today.

00:05:43.150 --> 00:05:44.550
So let me underline that.

00:05:44.550 --> 00:05:48.310
Tarush is 5 times as
old as Arman today.

00:05:48.310 --> 00:05:53.900
So if Arman is x, that tells us
that Tarush is going to be 5x.

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85 years ago.

00:05:56.330 --> 00:06:03.210
So, 85 years ago, well, Arman
is x today, then 85 years ago,

00:06:03.210 --> 00:06:05.980
Arman would have
been x minus 85.

00:06:08.720 --> 00:06:16.230
And if Tarush is 5x today, then
Tarush would be 5x minus 85.

00:06:16.230 --> 00:06:20.150
I just subtracted 85
from the current age.

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Because we're going 85
years in the past.

00:06:22.630 --> 00:06:25.480
And now we have this extra
piece of information.

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Which tells us that 85
years ago, Tarush was 10

00:06:30.500 --> 00:06:32.420
times as old as Arman.

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So this number is going to be
10 times more than this number.

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That's what that
sentence tells us.

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85 years ago, we're
in this situation.

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Tarush, which is this, was
10 times older than Arman.

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We just write that
out algebraically.

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85 years ago, Tarush
is 5x minus 85.

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And the sentence tells us that
he was 10 times older than

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Arman, who is x minus 85.

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Now we just solve the equation.

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We get 5x minus 85 is
equal to 10x minus 850.

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And then you get-- and I'm
going to do this, skip some

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steps, just to confuse you-- 5x
is equal too-- well, I think

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I just confused myself.

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5x is equal to 850 minus 85.

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Let's see, what's 850 minus 85.

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It'll be 35 less than 800.

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So we could say that
5x-- 35 less than 800.

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30 less than 800 gets 770.

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So 765.

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And then x is equal
to, let me see.

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5 goes into 70.

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1 times 5.

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26.

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5 times 5 is 25.

00:08:08.230 --> 00:08:09.586
15.

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So x is equal to 153.

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We have some
long-lifespanned people.

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So we get the solution that
Arman is 153 years old.

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Today.

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Let's see if that makes sense.

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Well, if he's 153, then,
no, that can't be right.

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Because then he's 5.

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Huh.

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Well, actually I'm
almost out of time.

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Let me see where I might
have messed up on this.

00:08:38.190 --> 00:08:43.320
If Arman is x, Tarush is 5x.

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That makes sense.

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Oh, right, right, right.

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I think this could make sense.

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So, I thought that Tarush had
to be younger than Arman.

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But no, Tarush is 5 times 153.

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So Tarush is actually
765 years old.

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We should call him Methuselah.

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So, Tarush is 765 years old.

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And then if we go 85 years into
the past, Tarush would have

00:09:09.120 --> 00:09:11.040
been-- what did we say?

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If we go 85 years into the
past, so let's write this down.

00:09:15.760 --> 00:09:18.820
This is equal to, I'm about to
run out of time, so you might

00:09:18.820 --> 00:09:20.112
have to check this yourself.

00:09:20.112 --> 00:09:26.010
If that's equal to 153,
then 5x is equal to 765.

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These numbers are big.

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And then 153 minus 85.

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Well, 155 minus 85 would be 73.

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So this would be 72.

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And then Tarush, 85 years ago.

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No, I think I messed
up someplace.

00:09:48.670 --> 00:09:51.150
Well, actually, I only have
ten seconds before YouTube

00:09:51.150 --> 00:09:52.130
won't let me upload it.

00:09:52.130 --> 00:09:54.450
So we'll have to stop there.

00:09:54.450 --> 00:09:55.950
Thank you.


Age word problems 3

WEBVTT
Kind: captions
Language: en

00:00:02.170 --> 00:00:06.550
Welcome to the third in this
series of age word problems.

00:00:06.550 --> 00:00:09.070
And I just wanted to start
off where I left off

00:00:09.070 --> 00:00:10.210
in the second one.

00:00:10.210 --> 00:00:11.340
I got myself confused.

00:00:11.340 --> 00:00:13.370
You guys make me very nervous
while you just sit there

00:00:13.370 --> 00:00:15.180
listening and I have
to perform for you.

00:00:15.180 --> 00:00:18.180
But, it turns out we did get
the right answer, if you say

00:00:18.180 --> 00:00:24.430
Arman is 153 and Tarush is 765,
then 85 years ago Arman would

00:00:24.430 --> 00:00:28.990
have been 68 and Tarush
would have been 680.

00:00:28.990 --> 00:00:31.720
And notice that's 10 times.

00:00:31.720 --> 00:00:33.040
So it worked out.

00:00:33.040 --> 00:00:35.380
So we're ready to tackle
another problem.

00:00:35.380 --> 00:00:36.540
One more problem.

00:00:36.540 --> 00:00:37.210
Let's see.

00:00:49.140 --> 00:00:58.790
This problem says, Zack is
4 times as old as Salman.

00:00:58.790 --> 00:00:59.590
What a coincidence.

00:00:59.590 --> 00:01:02.560
Salman really shows up
in a lot of problems.

00:01:02.560 --> 00:01:05.340
So Zack is 4 times
as old as Salman.

00:01:05.340 --> 00:01:15.470
Zack is also 3 years
older then Salman.

00:01:15.470 --> 00:01:19.490
How old is Zack?

00:01:19.490 --> 00:01:21.960
Well, actually this one
actually seems fairly

00:01:21.960 --> 00:01:22.620
straightforward.

00:01:22.620 --> 00:01:24.830
We're not dealing with the
present and the future.

00:01:24.830 --> 00:01:26.170
We're just dealing
with the present.

00:01:26.170 --> 00:01:29.320
So let's see if we
can figure this out.

00:01:29.320 --> 00:01:31.530
So we're trying to figure
out how old is Zack.

00:01:31.530 --> 00:01:33.600
Let's say, z for Zack.

00:01:33.600 --> 00:01:35.125
z equals Zack.

00:01:40.460 --> 00:01:42.390
It says Zack is 4 times
as old as Salman.

00:01:45.290 --> 00:01:49.990
So, Salman, if Zack is
z, this first sentence,

00:01:49.990 --> 00:01:52.130
let me circle it.

00:01:52.130 --> 00:01:57.170
Zack is 4 times as
old as Salman.

00:01:57.170 --> 00:02:03.740
So, if Zack is 4 times as old
as Salman, that means that Sal

00:02:03.740 --> 00:02:19.020
is equal to Zack divided by 4,
But then it also says here that

00:02:19.020 --> 00:02:23.940
Zack is also 3 years
older than Salman.

00:02:23.940 --> 00:02:26.600
Well, if Zack is also 3
years older than Salman, so

00:02:26.600 --> 00:02:35.310
that means that also Sal
would be Zack minus 3.

00:02:35.310 --> 00:02:37.510
The first sentence says Zack
is 4 times as old as Salman.

00:02:37.510 --> 00:02:40.150
So that means if you take
Zack's age and divide it

00:02:40.150 --> 00:02:42.110
by 4, you get Sal's age.

00:02:42.110 --> 00:02:45.290
The second sentence says
Zack is also 3 years

00:02:45.290 --> 00:02:46.640
older than Salman.

00:02:46.640 --> 00:02:49.890
So that means if you take
Zack's age and you subtract

00:02:49.890 --> 00:02:53.360
3 from it, you get
Sal's age again.

00:02:53.360 --> 00:02:54.720
Well, we have our
equation set up.

00:02:54.720 --> 00:02:56.190
And, actually, instead of
writing Zack, I should

00:02:56.190 --> 00:02:57.440
have just written z.

00:02:57.440 --> 00:02:59.660
But the equation is
all set up for us.

00:02:59.660 --> 00:03:05.500
We get z over 4 is
equal to z minus 3.

00:03:05.500 --> 00:03:08.780
We can multiply both sides of
this equation times 4 and we'll

00:03:08.780 --> 00:03:15.690
get z is equal to 4z, remember
to distribute, minus 12.

00:03:15.690 --> 00:03:18.290
And then, we'll skip
a couple of steps.

00:03:18.290 --> 00:03:22.030
And I get 3z is equal to 12.

00:03:22.030 --> 00:03:28.820
And I get Zack is
equal to 4 years old.

00:03:28.820 --> 00:03:29.980
Let's see if that makes sense.

00:03:29.980 --> 00:03:33.460
If Zack is 4 years old, then
this first sentence says Zack

00:03:33.460 --> 00:03:35.030
is 4 times as old as Salman.

00:03:35.030 --> 00:03:39.390
So that means Sal
is 1 year old.

00:03:39.390 --> 00:03:41.700
It also says Zack is 3
years older than Salman.

00:03:41.700 --> 00:03:42.570
Well, that's consistent.

00:03:42.570 --> 00:03:44.580
4 is 3 more than 1.

00:03:44.580 --> 00:03:45.350
So we were right.

00:03:45.350 --> 00:03:48.400
Zack is exactly four years old.

00:03:48.400 --> 00:03:49.610
Hope that helps.

00:03:49.610 --> 00:03:50.090
Have fun.

00:03:50.090 --> 00:03:50.900
I think you're ready.

00:03:50.900 --> 00:03:53.820
You've now seen, at
least, every type of

00:03:53.820 --> 00:03:55.000
the age word problems.

00:03:55.000 --> 00:03:56.485
If you're still a little
confused, you might just want

00:03:56.485 --> 00:03:59.790
to re-watch the videos and
maybe pause it right after I

00:03:59.790 --> 00:04:02.060
give you the problem, and see
if you can solve it yourself.

00:04:02.060 --> 00:04:04.030
And remember, you had a lot of
practice problems that you

00:04:04.030 --> 00:04:06.330
can do on the actual
modules themselves.

00:04:06.330 --> 00:04:07.930
Have fun.


Age word problems 1

WEBVTT
Kind: captions
Language: en

00:00:01.190 --> 00:00:04.000
Welcome to the presentation
on age word problems.

00:00:04.000 --> 00:00:04.860
Let's get started.

00:00:04.860 --> 00:00:06.800
Well, I already have
one written down here.

00:00:06.800 --> 00:00:12.630
It says in 4 years, Ali will be
3 times as old as he is today.

00:00:12.630 --> 00:00:14.510
How old is Ali today?

00:00:14.510 --> 00:00:21.520
Well, let's just say that x --
that's a little too big for

00:00:21.520 --> 00:00:29.910
what I want to write -- x
equals Ali's current age.

00:00:29.910 --> 00:00:32.170
Or his age today.

00:00:32.170 --> 00:00:33.170
Please forgive my handwriting.

00:00:35.740 --> 00:00:39.530
If x is Ali's current age --
and here we say, in 4 years.

00:00:39.530 --> 00:00:41.400
So how old is Ali going
to be in 4 years?

00:00:41.400 --> 00:00:44.440
Well, if x is his current age,
in 4 years, he's going to

00:00:44.440 --> 00:00:47.020
be x plus 4 years old.

00:00:47.020 --> 00:00:56.940
So that is equal to
Ali's age in 4 years.

00:01:00.600 --> 00:01:02.155
Let's read what other
information there

00:01:02.155 --> 00:01:02.600
is in the problem.

00:01:02.600 --> 00:01:04.560
It says, in 4 years.

00:01:04.560 --> 00:01:10.040
So in this situation, that's
a 4, x plus 4, in 4 years

00:01:10.040 --> 00:01:13.150
Ali will be 3 times as
old as he is today.

00:01:13.150 --> 00:01:14.750
Well, today he's x.

00:01:14.750 --> 00:01:17.116
So we say, in 4 years.

00:01:17.116 --> 00:01:19.140
x plus 4.

00:01:19.140 --> 00:01:22.520
And that's just --
so, in 4 years.

00:01:22.520 --> 00:01:26.360
So we could translate
that into x plus 4.

00:01:26.360 --> 00:01:32.010
That's going to equal 3 times
as old as he is today.

00:01:32.010 --> 00:01:33.210
Well, we know today he's x.

00:01:33.210 --> 00:01:36.820
So that's going to
be equal to 3x.

00:01:36.820 --> 00:01:38.110
And now we just solve for x.

00:01:38.110 --> 00:01:39.350
Subtract x from both sides.

00:01:39.350 --> 00:01:41.530
You get 2x is equal to 4.

00:01:41.530 --> 00:01:43.280
x equals 2.

00:01:43.280 --> 00:01:47.100
So Ali is 2 years old today.

00:01:47.100 --> 00:01:49.990
Let's do another problem.

00:01:49.990 --> 00:01:53.340
That makes sense, because if
he's 2 today, in 4 years

00:01:53.340 --> 00:01:54.500
he's going to be 6.

00:01:54.500 --> 00:01:56.890
Which is going to be 3 times
as old as he is today.

00:01:56.890 --> 00:01:57.820
So it all works out.

00:01:57.820 --> 00:01:58.780
Let's do another problem.

00:02:01.400 --> 00:02:07.204
Let's clear this and now I'm
going to attempt to type.

00:02:07.204 --> 00:02:08.660
Hope I don't mess up.

00:02:11.610 --> 00:02:12.110
OK.

00:02:12.110 --> 00:02:24.420
The problem, is Nazrith is
4 years older then Omama.

00:02:24.420 --> 00:02:36.470
2 years ago, Nazrith was
5 times as old as Omama.

00:02:39.230 --> 00:02:44.690
How old is Nazrith today?

00:02:44.690 --> 00:02:48.106
This one seems more difficult.

00:02:48.106 --> 00:02:51.820
But I think we'll be
able to get through it.

00:02:51.820 --> 00:02:53.140
Let's figure it out.

00:02:53.140 --> 00:02:56.960
All right, so once again what
are we trying to solve for.

00:02:56.960 --> 00:02:59.270
We're trying to solve
for Nazrith today.

00:02:59.270 --> 00:03:01.450
So, let's say this is Nazrith.

00:03:01.450 --> 00:03:02.454
We'll say Naz for short.

00:03:06.100 --> 00:03:07.040
And let's say this is Omama.

00:03:11.320 --> 00:03:13.352
And this is today.

00:03:13.352 --> 00:03:16.780
I'm making a little chart.

00:03:16.780 --> 00:03:19.960
And let's say 4 years from now.

00:03:19.960 --> 00:03:21.580
4 -- oh, no, no.

00:03:21.580 --> 00:03:22.600
Not 4 years from now.

00:03:22.600 --> 00:03:24.640
I'm confusing myself.

00:03:24.640 --> 00:03:27.990
This would be 2 years ago.

00:03:27.990 --> 00:03:29.830
I'm still remembering
the last problem.

00:03:29.830 --> 00:03:31.600
2 years ago.

00:03:31.600 --> 00:03:34.980
So this is today, 2
years into the future.

00:03:34.980 --> 00:03:37.980
Let me make a little chart.

00:03:37.980 --> 00:03:42.910
It's not the cleanest chart,
but I think it'll do the job.

00:03:42.910 --> 00:03:44.500
So we're trying to solve for x.

00:03:44.500 --> 00:03:45.600
How old is Nazrith today.

00:03:45.600 --> 00:03:51.340
So let's just say Naz
today that is equal to x.

00:03:51.340 --> 00:03:55.690
It says Nazrith is 4
years older than Omama.

00:03:55.690 --> 00:03:59.420
So this piece of information up
here-- let me underline it.

00:03:59.420 --> 00:04:03.240
Nazrith is 4 years older Omama.

00:04:03.240 --> 00:04:07.530
Well, that piece of information
tells us that Omama today, if

00:04:07.530 --> 00:04:12.890
Nazrith is x, then Omama's
going to be x minus 4.

00:04:12.890 --> 00:04:17.740
Which makes sense because Naz
is 4 years older than Omama.

00:04:17.740 --> 00:04:20.090
Now we go into the past.

00:04:20.090 --> 00:04:23.430
Let's write like this.

00:04:23.430 --> 00:04:27.680
So, 2 years ago -- so if
Naz is x today, how old

00:04:27.680 --> 00:04:28.900
was she 2 years ago?

00:04:28.900 --> 00:04:32.050
Well, she was going
to be x minus 2.

00:04:32.050 --> 00:04:33.080
That makes sense.

00:04:33.080 --> 00:04:34.940
If I'm 10 years old
today, 2 years ago,

00:04:34.940 --> 00:04:35.970
I'll be 2 years less.

00:04:35.970 --> 00:04:36.960
I'd be 8 years old.

00:04:36.960 --> 00:04:40.900
So, if she's x today, 2 years
ago she was x minus 2.

00:04:40.900 --> 00:04:44.690
And if Omama is x minus 4
today, than 2 years ago

00:04:44.690 --> 00:04:46.580
she'll be 2 less than that.

00:04:46.580 --> 00:04:49.730
So she'll be x minus 4 minus 2.

00:04:49.730 --> 00:04:53.230
And that equals x minus 6.

00:04:53.230 --> 00:04:53.940
Which makes sense.

00:04:53.940 --> 00:04:57.230
And, notice that she's
still 4 years younger

00:04:57.230 --> 00:05:00.050
than Naz, or Nazrith.

00:05:00.050 --> 00:05:03.890
Now we have one final piece of
information in this problem.

00:05:03.890 --> 00:05:12.330
It says, 2 years ago Nazrith
was 5 times as old as Omama.

00:05:12.330 --> 00:05:14.980
So, we're in this situation.

00:05:14.980 --> 00:05:17.500
2 years ago -- that's what the
sentence says, 2, years ago, so

00:05:17.500 --> 00:05:21.570
it's this situation -- Nazrith
was 5 times as old as Omama.

00:05:21.570 --> 00:05:25.450
So Nazrith, which is x minus
2, was 5 times older than

00:05:25.450 --> 00:05:28.010
Omama, which is x minus 6.

00:05:28.010 --> 00:05:30.030
So let's just write that down.

00:05:30.030 --> 00:05:33.160
So, Nazrith, 2 years
ago, was x minus 2.

00:05:33.160 --> 00:05:34.160
We get that from the chart.

00:05:34.160 --> 00:05:37.050
Nazrith was x minus
2, 2 years ago.

00:05:37.050 --> 00:05:41.140
And that sentence tells us,
Nazrith was 5 times older

00:05:41.140 --> 00:05:42.670
than Omama was 2 years ago.

00:05:42.670 --> 00:05:44.860
Omama, 2 years ago,
was x minus 6.

00:05:47.780 --> 00:05:49.670
Well, now we just solve for x.

00:05:49.670 --> 00:05:56.590
x minus 2 is equal
to 5x minus 30.

00:05:56.590 --> 00:05:59.450
Now we could subtract
x from both sides.

00:05:59.450 --> 00:06:02.280
And if you get 4x, I'm going
to switch it around, to

00:06:02.280 --> 00:06:04.120
hopefully confuse you more.

00:06:04.120 --> 00:06:07.870
4x minus 30 equals minus 2.

00:06:07.870 --> 00:06:15.060
And then we get 4x is equal
to 28 and x is equal to 7.

00:06:15.060 --> 00:06:16.880
So we solve the problem.

00:06:16.880 --> 00:06:20.160
Nazrith, today is 7 years old.

00:06:20.160 --> 00:06:21.560
And does this make sense?

00:06:21.560 --> 00:06:26.620
Well, if Nazrith today is 7
years old, then Omama today

00:06:26.620 --> 00:06:28.390
is going to be 3 years old.

00:06:28.390 --> 00:06:30.530
And that makes sense, because
there's a 4 year difference.

00:06:30.530 --> 00:06:34.500
2 years ago, Nazrith
was 5 years old.

00:06:34.500 --> 00:06:38.940
And 2 years ago Omama
was only 1 year old.

00:06:38.940 --> 00:06:40.890
Because x minus 6.

00:06:40.890 --> 00:06:42.080
So it makes sense.

00:06:42.080 --> 00:06:44.570
Nazrith is 4 years older
than Omama; Nazrith is

00:06:44.570 --> 00:06:47.430
7, Omama is 3 right now.

00:06:47.430 --> 00:06:50.130
2 years ago, when Nazrith was
5 times as old as Omama.

00:06:50.130 --> 00:06:53.340
2 years ago, Nazrith
was 5 and Omama was 1.

00:06:53.340 --> 00:06:54.600
So it all works out.

00:06:54.600 --> 00:06:56.000
Pretty deep, no?

00:06:56.000 --> 00:06:57.990
Well, anyway, I think you're
ready to at least try some

00:06:57.990 --> 00:06:59.200
of these age problems.

00:06:59.200 --> 00:07:01.120
And I'm going to do some
more example problems.

00:07:01.120 --> 00:07:05.060
So that you can come back
to this video page and see

00:07:05.060 --> 00:07:07.620
more if you're still a
little bit confused.

00:07:07.620 --> 00:07:08.230
Have fun.


Algebra: Equation of a line

WEBVTT
Kind: captions
Language: en

00:00:01.110 --> 00:00:04.540
Welcome to the presentation on
finding the equation of a line.

00:00:04.540 --> 00:00:05.410
Let's get started.

00:00:05.410 --> 00:00:06.345
Say I had two points.

00:00:06.345 --> 00:00:12.360
Let's say I have the point 1
comma 2, and I have the point 3

00:00:12.360 --> 00:00:14.360
comma 4, and I want to figure
out the equation of the

00:00:14.360 --> 00:00:15.360
line through these points.

00:00:15.360 --> 00:00:19.640
So let's at least figure out
what that line looks like.

00:00:19.640 --> 00:00:32.490
So 1 comma 2 is here,
and 2, 3; 3, 4.

00:00:32.490 --> 00:00:39.130
3 comma 4 is here, and if
I want to draw a line

00:00:39.130 --> 00:00:44.050
through them, it'll look
something like that.

00:00:44.050 --> 00:00:46.375
So what we want to do is figure
out the equation of this line.

00:00:46.375 --> 00:00:53.440
Well, we know the form of an
equation of a line is y equals

00:00:53.440 --> 00:01:00.840
mx plus b, where m is the
slope, and that tells you how

00:01:00.840 --> 00:01:03.620
steep the line is, and
b is the y-intercept.

00:01:03.620 --> 00:01:05.430
And I don't know why
people chose m and b.

00:01:05.430 --> 00:01:06.980
We'll have to do some
research on that.

00:01:06.980 --> 00:01:09.270
b is the y-intercept and the
y-intercept is just where does

00:01:09.270 --> 00:01:10.250
it intersect the y-axis.

00:01:13.250 --> 00:01:15.050
And this problem, you could
actually look at it and

00:01:15.050 --> 00:01:17.940
figure it out, but let's
do it mathematically.

00:01:17.940 --> 00:01:24.390
So the equation for the slope
m: it's rise over run.

00:01:24.390 --> 00:01:27.640
Another way to view that is
for any amount that you

00:01:27.640 --> 00:01:31.680
run along the x-axis,
how much do you rise?

00:01:31.680 --> 00:01:33.350
Well, let's do
that numerically.

00:01:33.350 --> 00:01:36.430
Rise is the same thing as
change over y, and run is the

00:01:36.430 --> 00:01:37.800
same thing as change over x.

00:01:37.800 --> 00:01:41.570
Delta, this triangle, means
change, change in y.

00:01:41.570 --> 00:01:44.650
Well, change in y, let's
take the starting

00:01:44.650 --> 00:01:46.280
point to be 3 comma 4.

00:01:46.280 --> 00:01:48.980
Let's say we're going from
3 comma 4 to 2 comma 1.

00:01:48.980 --> 00:01:52.080
The change in y is 4 minus 2.

00:01:52.080 --> 00:02:01.830
We just took this 4 minus
this 2 over 3 minus 1.

00:02:01.830 --> 00:02:02.550
My phone was ringing.

00:02:07.420 --> 00:02:10.760
And that's just this
3 minus this 1.

00:02:10.760 --> 00:02:16.560
So if we just solve for it,
we get 4 minus 2 is 2, and 3

00:02:16.560 --> 00:02:21.280
minus 1 is also 2, so we get
the slope is equal to 1.

00:02:21.280 --> 00:02:24.280
And that makes sense because
when we move over 1 in x,

00:02:24.280 --> 00:02:26.270
we go up exactly 1 in y.

00:02:26.270 --> 00:02:32.790
When we move to the left 1 in
x, we move down exactly 1 in y.

00:02:32.790 --> 00:02:38.490
So now we know the equation is
y equals 1x plus b because

00:02:38.490 --> 00:02:40.880
we solved the m equals 1.

00:02:40.880 --> 00:02:42.580
And this is, of course,
the same thing as

00:02:42.580 --> 00:02:46.200
y equals x plus b.

00:02:46.200 --> 00:02:48.760
Now, all we have left
to do is solve for b.

00:02:48.760 --> 00:02:52.550
Well, how do we do that because
we have three variables here.

00:02:52.550 --> 00:02:55.460
Well, we could actually
substitute one of these pairs

00:02:55.460 --> 00:02:59.780
of points in for y and x, and
that makes sense, because

00:02:59.780 --> 00:03:02.510
these points have to
satisfy this equation.

00:03:02.510 --> 00:03:06.600
So let's take this first
pair. y is equal to 2.

00:03:06.600 --> 00:03:10.360
2 is equal to x,
which is 1 plus b.

00:03:10.360 --> 00:03:12.160
It's a pretty easy
equation to solve.

00:03:12.160 --> 00:03:16.180
We get b equals 1, so that
tells us that the equation of

00:03:16.180 --> 00:03:20.070
this line is y equals x plus 1.

00:03:20.070 --> 00:03:22.490
That's a pretty straightforward
equation, and it makes sense.

00:03:22.490 --> 00:03:25.360
The y-intercept is 1, which
is exactly here, 0 comma

00:03:25.360 --> 00:03:29.100
1, and the slope is 1, and
that's pretty obvious.

00:03:29.100 --> 00:03:31.690
For every amount that we move
to the right, we move the same

00:03:31.690 --> 00:03:34.150
amount up, so the slope is 1.

00:03:34.150 --> 00:03:35.110
Let's do another problem.

00:03:38.090 --> 00:03:40.200
Let's say I wanted to find the
equation of the line between

00:03:40.200 --> 00:03:49.350
the points negative 3 comma
5 and 2 comma negative 6.

00:03:49.350 --> 00:03:51.260
Well, we do the same thing.

00:03:51.260 --> 00:03:55.690
m is equal to change in
y over change in x.

00:03:55.690 --> 00:03:58.630
So let's take this as
the starting point.

00:03:58.630 --> 00:04:01.955
So say negative 6 minus 5.

00:04:01.955 --> 00:04:08.300
So we just took negative
6 minus 5 over 2

00:04:08.300 --> 00:04:11.840
minus negative 3.

00:04:11.840 --> 00:04:13.650
You've got to be real careful
to get the signs right.

00:04:13.650 --> 00:04:16.840
So it's 2 minus negative 3.

00:04:16.840 --> 00:04:22.485
Negative 6 minus 5 is minus 11,
and 2 minus negative 3, well,

00:04:22.485 --> 00:04:26.350
that's the same thing as two
plus plus 3, so that's 5.

00:04:26.350 --> 00:04:30.370
So we have the slope is
equal to negative 11/5.

00:04:30.370 --> 00:04:35.450
And notice that if on the
numerator we use negative 6 as

00:04:35.450 --> 00:04:37.630
the starting point, that in the
denominator, we have to use

00:04:37.630 --> 00:04:39.940
2 as the starting point.

00:04:39.940 --> 00:04:41.090
We could have done it
the other way around.

00:04:41.090 --> 00:04:48.240
We could have said 5 minus
negative 6 over negative 3

00:04:48.240 --> 00:04:51.070
minus 2, in which case we would
have gotten-- this would have

00:04:51.070 --> 00:04:54.570
been 11 over negative 5.

00:04:54.570 --> 00:04:58.305
So as long as you-- if you use
the negative 6 first, then you

00:04:58.305 --> 00:05:00.840
have to use the 2 first, or if
you use the 5 first, then

00:05:00.840 --> 00:05:01.885
you have to use the
negative 3 first.

00:05:01.885 --> 00:05:04.080
I hope I'm not completely
confusing you guys.

00:05:04.080 --> 00:05:07.500
Well, anyway, we know the slope
is negative 11/5, so the

00:05:07.500 --> 00:05:15.680
equation of this line so far is
y equals minus 11/5x plus b.

00:05:15.680 --> 00:05:18.410
Now we can take one of these
pairs on the top and substitute

00:05:18.410 --> 00:05:20.090
back and solve for b.

00:05:20.090 --> 00:05:21.760
Let's take the first pair.

00:05:21.760 --> 00:05:23.390
So 5 is y.

00:05:23.390 --> 00:05:33.320
So we say 5 equals negative
3, so it's negative 11/5

00:05:33.320 --> 00:05:35.750
times negative 3, right?

00:05:35.750 --> 00:05:40.220
I just put the x
in for x plus b.

00:05:40.220 --> 00:05:48.450
So just simplifying that, I get
5 is equal to 33/5 plus b, or b

00:05:48.450 --> 00:06:01.840
is equal to 5 minus 33/5, and
this equals 25 minus 33/5.

00:06:01.840 --> 00:06:05.460
25 minus 33 is minus 8/5.

00:06:07.960 --> 00:06:10.580
So the equation of this line,
and this one wasn't as clean as

00:06:10.580 --> 00:06:13.340
the other one, obviously, is--
let me do it in another color

00:06:13.340 --> 00:06:23.800
for emphasis-- y equals
minus 11/5x minus 8/5.

00:06:26.400 --> 00:06:29.530
Hopefully, those two examples
will give you enough of an idea

00:06:29.530 --> 00:06:34.570
to do the figuring out the
equation of a line problems.

00:06:34.570 --> 00:06:36.820
And if you have problems with
this, you might just want to

00:06:36.820 --> 00:06:39.750
try just the slope of the line
problems or the y-intercept

00:06:39.750 --> 00:06:41.170
problems separately.

00:06:41.170 --> 00:06:43.160
I hope you have fun.

00:06:43.160 --> 00:06:44.460
Bye.


Level 3 exponents

WEBVTT
Kind: captions
Language: en

00:00:01.120 --> 00:00:03.160
Welcome to level
three exponents.

00:00:03.160 --> 00:00:05.200
Let's get started.

00:00:05.200 --> 00:00:12.590
So if I asked you what 4 to the
1/2 power is, your immediate

00:00:12.590 --> 00:00:14.660
inclination is to view this
probably as like a

00:00:14.660 --> 00:00:18.800
multiplication problem and try
to multiply it somehow or add

00:00:18.800 --> 00:00:19.750
them or something.

00:00:19.750 --> 00:00:21.430
And you always have to
remind yourself this

00:00:21.430 --> 00:00:22.340
is not multiplication.

00:00:22.340 --> 00:00:24.730
I know when I first learned it,
I always was tempted to do

00:00:24.730 --> 00:00:26.620
something with multiplication.

00:00:26.620 --> 00:00:29.820
Well, something to the 1/2
power might not be intuitive to

00:00:29.820 --> 00:00:32.870
you, but it actually turns out
that this means the same thing

00:00:32.870 --> 00:00:36.260
as the square root of 4.

00:00:36.260 --> 00:00:40.390
Or another way, what times
itself is equal to 4,

00:00:40.390 --> 00:00:43.600
and we know that the
square root of 4 is 2.

00:00:43.600 --> 00:00:47.090
It could actually be a positive
2 or a negative 2 because we

00:00:47.090 --> 00:00:49.830
know that either of those
numbers when they're

00:00:49.830 --> 00:00:51.530
squared could equal 4.

00:00:51.530 --> 00:00:54.090
But for the sake of this one
we'll assume it's always

00:00:54.090 --> 00:00:55.110
the positive square root.

00:00:55.110 --> 00:00:58.610
So 4 to the 1/2 is equal to 2.

00:00:58.610 --> 00:01:05.510
Similarly, 9 to the 1/2,
well that would be 3.

00:01:05.510 --> 00:01:09.710
16 to the 1/2 -- oops.

00:01:09.710 --> 00:01:13.270
16 to the 1/2, my subconscious
gave away the answer.

00:01:13.270 --> 00:01:15.920
16 to the 1/2 power is 4.

00:01:15.920 --> 00:01:22.140
25 to the 1/2 power is 5.

00:01:22.140 --> 00:01:24.170
I think that might make
sense to you now.

00:01:24.170 --> 00:01:27.740
So what does it mean when
something is to the 1/3 power?

00:01:27.740 --> 00:01:32.950
Well, if I say 8 to the
1/3 power, you might

00:01:32.950 --> 00:01:34.260
already catch on.

00:01:34.260 --> 00:01:36.522
In the 1/2 power we said
something times something

00:01:36.522 --> 00:01:38.190
is equal to 4.

00:01:38.190 --> 00:01:41.600
Well in the 1/3 power we have
to say that something to the

00:01:41.600 --> 00:01:43.530
third power is equal to 8.

00:01:43.530 --> 00:01:46.730
And if you've been practicing
your exponents you know that 2

00:01:46.730 --> 00:01:50.060
to the third power
is equal to 8.

00:01:50.060 --> 00:01:55.060
So we know that 8 to
the 1/3 is equal to 2.

00:01:55.060 --> 00:02:02.906
Similarly, 27 to the
1/3 is equal to 3.

00:02:02.906 --> 00:02:08.750
And 64 to the 1/3
is equal to 4.

00:02:08.750 --> 00:02:10.920
You might notice that I'm
picking particular numbers,

00:02:10.920 --> 00:02:13.050
like 8 and 27, 64.

00:02:13.050 --> 00:02:15.060
That's because they have
clean, cube roots.

00:02:15.060 --> 00:02:17.050
And then by the way, when
something is to the 1/3 power

00:02:17.050 --> 00:02:20.480
that's the same thing as
saying the cube root.

00:02:20.480 --> 00:02:22.190
I just used terminology
without explaining it,

00:02:22.190 --> 00:02:25.250
which is very bad.

00:02:25.250 --> 00:02:28.900
So I just used 8 and 27 and 64
because when I raised them to

00:02:28.900 --> 00:02:31.010
the fractional exponents, they
actually come out to

00:02:31.010 --> 00:02:31.740
be clean numbers.

00:02:31.740 --> 00:02:34.290
You could use a calculator and
do something like 5 to the 1/3

00:02:34.290 --> 00:02:36.710
power and you'll get
some weird decimal.

00:02:36.710 --> 00:02:37.695
Let's do some more problems.

00:02:41.980 --> 00:02:49.530
So we know that 9 to
the 1/2 is equal to 3.

00:02:49.530 --> 00:02:54.930
Well what do you think 9
to the 2/3 is equal to?

00:02:54.930 --> 00:03:00.970
Well, it turns out that this is
equivalent to 9 to the -- oops,

00:03:00.970 --> 00:03:05.580
I actually didn't want to do
those -- what do you think 9

00:03:05.580 --> 00:03:08.860
to the 3/2 is equivalent to?

00:03:08.860 --> 00:03:14.650
Well this is the same thing
as 9 to the 1/2 power

00:03:14.650 --> 00:03:15.580
to the third power.

00:03:15.580 --> 00:03:20.400
And I'll do a whole
presentation on the actual

00:03:20.400 --> 00:03:24.020
principles of exponents, but
it actually turns out you

00:03:24.020 --> 00:03:24.800
could just multiply.

00:03:24.800 --> 00:03:26.920
When you have one exponent
to another is when you can

00:03:26.920 --> 00:03:28.920
multiply the two and that's
where you get 3/2 .

00:03:28.920 --> 00:03:32.950
But 9 to the 1/2 we know is 3.

00:03:32.950 --> 00:03:36.690
And you're raising it to the
third power, so that equals 27.

00:03:36.690 --> 00:03:38.740
I'm sure at this point
I have confused you.

00:03:38.740 --> 00:03:42.050
Let's do more of these.

00:03:42.050 --> 00:03:47.370
So you know at this point
that 16 to the 1/4 power --

00:03:47.370 --> 00:03:48.270
think about what that is.

00:03:48.270 --> 00:03:51.500
That means that some number
to the fourth power is 16.

00:03:51.500 --> 00:03:54.030
If you've been practicing your
level one exponents, you'll

00:03:54.030 --> 00:03:58.820
probably know that well, that
equals 2, because 2 times 2

00:03:58.820 --> 00:04:01.920
times 2 times 2, well
that equals 16.

00:04:01.920 --> 00:04:06.320
So we know that 16 to
the 1/4 is equal to 2.

00:04:06.320 --> 00:04:11.740
So what do you think 16
to the 2/4 is equal to?

00:04:11.740 --> 00:04:14.270
Well, we already know from that
last problem that that's the

00:04:14.270 --> 00:04:21.710
same thing as 16 to the 1/4
squared -- that's the 2 on both

00:04:21.710 --> 00:04:26.340
sides -- and we know 16 to the
1/4 is 2, so that equals 2

00:04:26.340 --> 00:04:29.600
squared and that equals 4.

00:04:29.600 --> 00:04:32.790
And it all works out because we
know from fractions another way

00:04:32.790 --> 00:04:35.200
to write the fraction
2/4 is to write 1/2.

00:04:35.200 --> 00:04:40.270
So this is the same thing
as 16 to the 1/2 power.

00:04:40.270 --> 00:04:45.410
16 to the 1/2 power, well
that's just equal to 4.

00:04:45.410 --> 00:04:48.790
Now I'm going to mix it up real
good and do some negative

00:04:48.790 --> 00:04:50.790
fractional exponents.

00:04:50.790 --> 00:04:56.680
So what if I were to tell you
16 to the negative 1/2 power?

00:04:56.680 --> 00:04:59.400
Well this might seem very
daunting at first, but as we

00:04:59.400 --> 00:05:01.460
know with the negative
exponents level three,

00:05:01.460 --> 00:05:04.170
immediately we just say well
this is the same thing as 1

00:05:04.170 --> 00:05:11.170
over 16 to the positive 1/2.

00:05:11.170 --> 00:05:17.910
And that's the same thing as 1
to the 1/2 over 16 to the 1/2.

00:05:17.910 --> 00:05:20.640
Well the square root
of 1 is easy, it's 1.

00:05:20.640 --> 00:05:23.130
And 16 to the 1/2 is 4.

00:05:23.130 --> 00:05:24.300
So that wasn't too bad.

00:05:24.300 --> 00:05:26.500
It's a little daunting when you
see a negative exponent, but

00:05:26.500 --> 00:05:29.640
immediately when you see that
negative, just flip the 16 and

00:05:29.640 --> 00:05:33.320
then work it out like a regular
fractional exponent problem.

00:05:33.320 --> 00:05:34.070
Let's do another one.

00:05:38.480 --> 00:05:51.980
8 over 27 to the negative 1/3.

00:05:51.980 --> 00:05:53.660
Immediately when we
see that negative, we

00:05:53.660 --> 00:05:54.560
want to just flip it.

00:05:54.560 --> 00:06:03.440
So we'll say that equals 27
over 8 to the 1/3, and that

00:06:03.440 --> 00:06:11.060
equals 27 to the 1/3
over 8 to the 1/3.

00:06:11.060 --> 00:06:15.860
And we know that 27 to the
1/3, well that equals 3.

00:06:15.860 --> 00:06:20.720
And 8 to the 1/3,
well that equals 2.

00:06:20.720 --> 00:06:26.275
So we've got 8 over 27 to
the negative 1/3 is 3/2.

00:06:26.275 --> 00:06:28.170
The first problem probably
looked very intimidating to

00:06:28.170 --> 00:06:30.280
you, but it only took us two
steps to get there and as you

00:06:30.280 --> 00:06:32.330
do more practice, hopefully
it'll seem more and

00:06:32.330 --> 00:06:34.070
more intuitive to you.

00:06:34.070 --> 00:06:36.760
Let me give you
another problem.

00:06:36.760 --> 00:06:40.890
What's negative 8 to the
negative third power?

00:06:43.990 --> 00:06:44.710
Let me change that.

00:06:44.710 --> 00:06:51.250
What's negative 8 to the
negative 1/3 power?

00:06:51.250 --> 00:06:53.430
Once again, at first this might
confuse you, but when you see

00:06:53.430 --> 00:06:55.590
that negative in the exponent,
we just take the reciprocal of

00:06:55.590 --> 00:06:59.690
the base, so that we say that
that is equal to negative 1

00:06:59.690 --> 00:07:06.680
over 8 to the 1/3 power.

00:07:06.680 --> 00:07:13.650
And we say well that is equal
to, we could write it as 1

00:07:13.650 --> 00:07:16.770
over negative 8 to the 1/3.

00:07:16.770 --> 00:07:21.280
We say what number times itself
3 times is equal to negative 8?

00:07:21.280 --> 00:07:24.540
Well, we know from intuition
there's no real mechanical way

00:07:24.540 --> 00:07:28.440
to do this, but we know that
negative 2 times negative 2 is

00:07:28.440 --> 00:07:31.060
4 times negative
2 is negative 8.

00:07:31.060 --> 00:07:34.340
So we know that this is
equal to 1 over negative

00:07:34.340 --> 00:07:37.956
2 or negative 1/2.

00:07:37.956 --> 00:07:41.175
So negative eight to
the negative 1/3 is

00:07:41.175 --> 00:07:45.210
equal to minus 1/2.

00:07:45.210 --> 00:07:48.760
Let's do another one, one more
problem just to thoroughly

00:07:48.760 --> 00:07:51.070
melt your brain.

00:07:51.070 --> 00:08:03.260
Let's say 9 over 4 to
the negative 3 over 2.

00:08:03.260 --> 00:08:06.170
Well, immediately we see
that negative exponent,

00:08:06.170 --> 00:08:07.670
let's flip the base.

00:08:07.670 --> 00:08:13.250
We get 4 over 9 to the 3/2 .

00:08:13.250 --> 00:08:22.510
Well we know that that equals
4 over 9 to the 1/2, and all

00:08:22.510 --> 00:08:25.700
of that to the third power.

00:08:25.700 --> 00:08:28.100
4 over 9 to the 1/2, I think at
this point you know that's the

00:08:28.100 --> 00:08:31.470
same thing as 4 to the 1/2
which is 2 over 9 to

00:08:31.470 --> 00:08:33.400
the 1/2 which is 3.

00:08:33.400 --> 00:08:37.520
Now we have to just raise
everything to the third power.

00:08:37.520 --> 00:08:39.950
And that's the same thing as
2 the third power which is

00:08:39.950 --> 00:08:45.400
8 divided by 3 to
the third power.

00:08:45.400 --> 00:08:48.030
Well that's 27.

00:08:48.030 --> 00:08:52.470
There we have 9/4 to the
negative 3/2 power is

00:08:52.470 --> 00:08:55.090
equal to 8 over 27.

00:08:55.090 --> 00:08:57.410
Now hopefully you can at
least do these problems.

00:08:57.410 --> 00:09:00.920
You probably don't have a good
intuitive sense for exactly

00:09:00.920 --> 00:09:04.650
what a negative 3/2 power is,
and hopefully I can cover that

00:09:04.650 --> 00:09:06.580
for you in future modules.

00:09:06.580 --> 00:09:08.710
But I think you're ready to
try some of the level three

00:09:08.710 --> 00:09:10.310
exponent practice problems.

00:09:10.310 --> 00:09:11.830
Have fun.


Level 1 multiplying expressions

WEBVTT
Kind: captions
Language: en

00:00:01.100 --> 00:00:03.570
Welcome to the presentation
on multiplying expressions.

00:00:03.570 --> 00:00:05.260
Let's get started.

00:00:05.260 --> 00:00:11.540
So if I were to ask you what x
plus 2 times, let me use a

00:00:11.540 --> 00:00:20.610
different color, times x plus 3
equals, at first you'd be like,

00:00:20.610 --> 00:00:22.450
huh, that's a little strange.

00:00:22.450 --> 00:00:24.180
But it turns out that you
actually already know

00:00:24.180 --> 00:00:24.940
how to do this problem.

00:00:24.940 --> 00:00:27.060
And it's just the
distributive property.

00:00:27.060 --> 00:00:30.040
Because, if I just had -- let
me write a different problem

00:00:30.040 --> 00:00:33.603
here -- if I just wrote a let
me just make it clear, this is

00:00:33.603 --> 00:00:41.400
a separate problem -- if I just
wrote a times x plus 3, you

00:00:41.400 --> 00:00:58.060
know that that's just a x plus,
you could say a 3, or

00:00:58.060 --> 00:01:00.510
another way to say that
would be 3a, right?

00:01:00.510 --> 00:01:04.020
But as you see, all we did is
distribute it, this a times 3.

00:01:04.020 --> 00:01:07.290
All we did is distribute the
a times the x and the 3.

00:01:07.290 --> 00:01:08.763
Well, we're going to do the
same thing here, but instead

00:01:08.763 --> 00:01:10.630
of a we have x plus 2.

00:01:10.630 --> 00:01:12.050
So let's do that.

00:01:12.050 --> 00:01:18.830
So this would be -- let me
switch to green -- we get x

00:01:18.830 --> 00:01:22.995
plus 2, and since I'm using
green, I'm just going

00:01:22.995 --> 00:01:24.482
to stay with green.

00:01:24.482 --> 00:01:28.400
x plus 2, let me
switch to orange.

00:01:28.400 --> 00:01:33.980
x plus 2 times x plus
x plus 2 times 3.

00:01:33.980 --> 00:01:35.310
Does that make sense?

00:01:35.310 --> 00:01:36.720
If that looks a little
confusing, just pretend

00:01:36.720 --> 00:01:39.550
like that x plus 2 is
the a in this example.

00:01:39.550 --> 00:01:42.080
And all we did is, we
distributed it across

00:01:42.080 --> 00:01:44.110
that x plus three.

00:01:44.110 --> 00:01:46.980
And now it becomes a pretty
straightforward problem.

00:01:46.980 --> 00:01:50.140
What is -- we do distribution
again with each part

00:01:50.140 --> 00:01:51.420
of this problem.

00:01:51.420 --> 00:01:54.740
So, let me stay in orange.

00:01:54.740 --> 00:01:56.950
What is x times x?

00:01:56.950 --> 00:02:00.820
Well that's x squared.

00:02:00.820 --> 00:02:04.850
And then x times 2
-- well, that's 2x.

00:02:04.850 --> 00:02:06.860
So we did this left-hand
side already.

00:02:06.860 --> 00:02:08.680
Now we do this right-hand side.

00:02:08.680 --> 00:02:10.730
What's 3 times x?

00:02:10.730 --> 00:02:13.650
Well that's 3x.

00:02:13.650 --> 00:02:14.940
What's 3 times 2?

00:02:14.940 --> 00:02:17.680
Well, that's 6.

00:02:17.680 --> 00:02:19.680
And now we're almost done.

00:02:19.680 --> 00:02:21.670
We could say, oh, we have a 2x
here and a 3x here, we can

00:02:21.670 --> 00:02:23.700
simplify and add
those together.

00:02:23.700 --> 00:02:26.070
And we know that that
equals x squared.

00:02:26.070 --> 00:02:28.000
2x plus 3x is equal to 5x.

00:02:28.000 --> 00:02:32.930
x squared plus 5x plus 6.

00:02:32.930 --> 00:02:34.890
So all we did, there's really
nothing new to learn here.

00:02:34.890 --> 00:02:39.460
We just distributed the x plus
2 times each term of x plus 3.

00:02:39.460 --> 00:02:40.630
And we got the second step.

00:02:40.630 --> 00:02:43.480
And then we distributed this x
times x plus 2, and this 3

00:02:43.480 --> 00:02:45.660
times x plus 2, and simplify.

00:02:45.660 --> 00:02:46.810
Let's do a couple of problems.

00:02:46.810 --> 00:02:51.970
And hopefully it'll
hit the point home.

00:02:51.970 --> 00:03:07.180
Let's say I had 5x plus
9 times 4x minus 2.

00:03:07.180 --> 00:03:16.590
Once again, that
equals 5x plus 9.

00:03:16.590 --> 00:03:18.120
I'm just distributing it.

00:03:18.120 --> 00:03:20.640
I want to stay in my color that
I'm using, so that's why I'm

00:03:20.640 --> 00:03:22.670
just writing it out like this.

00:03:22.670 --> 00:03:27.660
That equals 5x plus 9 times 4x.

00:03:27.660 --> 00:03:32.350
And then plus, 5x plus
9 times minus 2.

00:03:32.350 --> 00:03:36.570
That minus 2 is here
and this 4x is here.

00:03:36.570 --> 00:03:37.830
And now we just
multiply it out.

00:03:37.830 --> 00:03:45.170
5x times 4x is 20x squared.

00:03:45.170 --> 00:03:46.720
Hope that makes sense to
you; let me write it

00:03:46.720 --> 00:03:47.756
down in this corner.

00:03:47.756 --> 00:03:55.050
5x times 4x is the same thing
as 5 times 4 times x times x.

00:03:55.050 --> 00:03:57.040
That equals 20x squared.

00:03:57.040 --> 00:03:58.840
Hopefully that makes
sense to you.

00:03:58.840 --> 00:04:00.290
So, going back to the problem.

00:04:00.290 --> 00:04:05.830
5x times 4x is 20x squared
plus 9 times 4x is 36x.

00:04:08.760 --> 00:04:13.040
Plus minus 2 times 5x.

00:04:13.040 --> 00:04:17.000
So that's minus 10x.

00:04:17.000 --> 00:04:20.860
And then 9 times negative
2, well, that's minus 18.

00:04:20.860 --> 00:04:22.470
And we're almost done.

00:04:22.470 --> 00:04:24.696
So we get 20x squared.

00:04:24.696 --> 00:04:27.310
And we have 36x.

00:04:27.310 --> 00:04:28.990
Minus 10x.

00:04:28.990 --> 00:04:34.040
So that's plus 26x minus 18.

00:04:34.040 --> 00:04:34.520
There.

00:04:34.520 --> 00:04:36.430
We're done.

00:04:36.430 --> 00:04:37.325
Let's do another problem.

00:04:42.110 --> 00:04:49.710
Let's do 2x plus y.

00:04:49.710 --> 00:04:52.640
Whoops, some parentheses.

00:04:52.640 --> 00:04:55.040
I'll stay in one color, since
I think you understand

00:04:55.040 --> 00:04:56.170
what we're doing.

00:04:56.170 --> 00:05:03.210
Times 3x plus 2y.

00:05:03.210 --> 00:05:06.790
Well, once again, this is the
same thing as -- and we could

00:05:06.790 --> 00:05:08.100
just do it in a little
different way, a little

00:05:08.100 --> 00:05:08.840
different order.

00:05:08.840 --> 00:05:10.350
But we could distribute
it like this.

00:05:10.350 --> 00:05:15.960
We could say that this
is 3x times 2x plus y.

00:05:15.960 --> 00:05:21.570
Plus 2y times 2x plus y.

00:05:21.570 --> 00:05:22.050
See what I did?

00:05:22.050 --> 00:05:23.280
I just switched the
order this time.

00:05:23.280 --> 00:05:24.410
Just to mix things up.

00:05:24.410 --> 00:05:26.060
This 3x is there.

00:05:26.060 --> 00:05:27.140
This 2y is here.

00:05:27.140 --> 00:05:31.320
And we just distributed it
along 2 -- and we just, we

00:05:31.320 --> 00:05:35.260
distributed it 2x plus y
along each of these numbers.

00:05:35.260 --> 00:05:36.540
And now we just
multiply it out.

00:05:36.540 --> 00:05:40.630
3x times 2x is 6x squared.

00:05:40.630 --> 00:05:46.970
3x times y -- that's 3xy.

00:05:46.970 --> 00:05:50.930
3xy is just another way of
saying 3 times x times y.

00:05:50.930 --> 00:05:56.850
Plus, now, 2 times y times 2y
times 2x, well, that's 4 y x.

00:05:59.800 --> 00:06:01.320
4 y x.

00:06:01.320 --> 00:06:04.450
2 times 2 times y times x.

00:06:04.450 --> 00:06:08.020
Plus 2y times y.

00:06:08.020 --> 00:06:11.370
Well, that's just 2y squared.

00:06:11.370 --> 00:06:13.590
Now, can we simplify
anything here?

00:06:13.590 --> 00:06:14.980
Think about it a little bit.

00:06:14.980 --> 00:06:17.960
Well, it turns out that x y and
y are actually the same thing.

00:06:17.960 --> 00:06:19.570
They're just
switching the order.

00:06:19.570 --> 00:06:24.300
So you could just rewrite
this as 6x squared plus

00:06:24.300 --> 00:06:29.110
7 x y plus 2y squared.

00:06:29.110 --> 00:06:31.560
And we could have just as
easily have written it as

00:06:31.560 --> 00:06:39.471
6x squared plus 7y
x plus 2y squared.

00:06:39.471 --> 00:06:41.420
I hope I'm not confusing you.

00:06:41.420 --> 00:06:42.910
Let's do one more, real fast.

00:06:45.800 --> 00:06:53.100
Let's do 2x plus 2 squared.

00:06:53.100 --> 00:06:55.870
Well, you might be temptd to
just square each of these

00:06:55.870 --> 00:06:57.180
terms, but you've got to be
careful because you

00:06:57.180 --> 00:06:58.340
have this x here.

00:06:58.340 --> 00:07:01.840
So it actually turns out
that this is equal to 2x

00:07:01.840 --> 00:07:06.230
plus 2 times 2x plus 2.

00:07:06.230 --> 00:07:08.740
Any number squared is just
that number times itself.

00:07:08.740 --> 00:07:11.950
So any expression like 2x plus
2 squared, is just that

00:07:11.950 --> 00:07:14.010
expression times itself.

00:07:14.010 --> 00:07:16.980
And now we can just do it as
-- this is the same thing

00:07:16.980 --> 00:07:22.310
as 2x times 2x plus 2.

00:07:22.310 --> 00:07:26.070
Plus 2 times 2x plus 2.

00:07:26.070 --> 00:07:27.330
We just multiply
everything out.

00:07:27.330 --> 00:07:31.240
2x times 2x, that's 4x squared.

00:07:31.240 --> 00:07:35.530
2x times 2, that's 4x.

00:07:35.530 --> 00:07:39.690
Plus 2 times 2x is 4x again.

00:07:39.690 --> 00:07:41.802
And then 2 times 2,
well, that's 4.

00:07:41.802 --> 00:07:43.770
And we're almost there.

00:07:43.770 --> 00:07:45.750
We just can add up
these two terms.

00:07:45.750 --> 00:07:51.140
4x squared plus 8x plus 4.

00:07:51.140 --> 00:07:52.520
And we're done.

00:07:52.520 --> 00:07:55.900
I think you're ready now to try
some level one multiplying

00:07:55.900 --> 00:07:57.220
expression problems.

00:07:57.220 --> 00:07:58.930
Hope you have fun.


Level 2 Exponents

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.582
Welcome to the presentation
on the Level 2 exponents.

00:00:03.582 --> 00:00:06.480
In Level 2 exponents, the only
thing we're going to add to

00:00:06.480 --> 00:00:09.740
the mix now is the concept
of a negative exponent.

00:00:09.740 --> 00:00:14.180
So we learned already that 2
the third power, well, that

00:00:14.180 --> 00:00:17.140
just equals 2 times 2 times 2.

00:00:17.140 --> 00:00:19.110
Hopefully, by now, that's
second nature to you,

00:00:19.110 --> 00:00:21.320
and that equals 8.

00:00:21.320 --> 00:00:24.960
Now I'm going to teach you
what 2 to the negative

00:00:24.960 --> 00:00:26.630
third power is.

00:00:26.630 --> 00:00:28.490
I know a lot of you are going
to say, oh, no, that's negative

00:00:28.490 --> 00:00:31.100
8, but whenever you see
exponents, you always have to

00:00:31.100 --> 00:00:34.260
remind yourself, exponents
are not multiplication.

00:00:34.260 --> 00:00:36.650
I know there's that temptation
to say, well, 2 times negative

00:00:36.650 --> 00:00:39.720
3 is negative 6, so maybe 2 to
the negative third power

00:00:39.720 --> 00:00:42.300
is negative 8, but
that's not the case.

00:00:42.300 --> 00:00:45.920
And I'll explain in a future
module why we use this

00:00:45.920 --> 00:00:49.220
convention, but 2 to the
negative third power, it

00:00:49.220 --> 00:00:57.580
turns out, is equivalent to
1/2 to the third power.

00:00:57.580 --> 00:01:00.280
So it turns out that the
negative exponent, what it does

00:01:00.280 --> 00:01:03.440
is it means to take the inverse
of the base-- we'll call the

00:01:03.440 --> 00:01:06.760
number 2 as the base-- and
take that to the positive

00:01:06.760 --> 00:01:08.070
version of the exponent.

00:01:08.070 --> 00:01:10.020
And 1/2 to the third power?

00:01:10.020 --> 00:01:16.336
Well, we already learned that's
1/2 times 1/2 times 1/2,

00:01:16.336 --> 00:01:19.170
and that equals 1/8.

00:01:19.170 --> 00:01:22.600
So we say 2 to the
negative third power.

00:01:22.600 --> 00:01:24.040
That didn't come out right.

00:01:24.040 --> 00:01:29.300
2 to the negative third
power is equal to 1/8.

00:01:29.300 --> 00:01:31.810
Let's do another one.

00:01:31.810 --> 00:01:38.530
Let's say 3 to the
negative 2 power.

00:01:38.530 --> 00:01:41.300
Once again, immediately when
we see that negative in the

00:01:41.300 --> 00:01:43.420
exponent, the easiest thing to
do is just immediately take

00:01:43.420 --> 00:01:44.882
the reciprocal of the base.

00:01:44.882 --> 00:01:46.780
So we take 1/3.

00:01:46.780 --> 00:01:49.910
And we raise that to
the positive 2 power.

00:01:49.910 --> 00:01:51.040
And that's easy enough.

00:01:51.040 --> 00:01:55.261
1/3 squared, well,
that's equal to 1/9.

00:01:55.261 --> 00:01:56.700
Let's do some more problems.

00:02:00.170 --> 00:02:12.270
What if I had 2/3 to the
negative third power?

00:02:12.270 --> 00:02:14.960
Once again, just to make it
simple, whenever I see that

00:02:14.960 --> 00:02:16.900
negative in the exponent,
I want to get rid of it.

00:02:16.900 --> 00:02:20.100
So I immediately take the
reciprocal of the base.

00:02:20.100 --> 00:02:25.430
The reciprocal of 2/3 is 3/2,
and I raise that to the

00:02:25.430 --> 00:02:27.180
positive third power.

00:02:27.180 --> 00:02:29.495
So what changed between the
left and the right side?

00:02:29.495 --> 00:02:33.300
The 2/3 I flipped, and
I turned the negative

00:02:33.300 --> 00:02:34.530
3 into a positive 3.

00:02:34.530 --> 00:02:37.510
And now this just becomes
a Level 1 exponent.

00:02:37.510 --> 00:02:52.710
This equals 3/2 times 3/2 times
3/2, and that equals 27/8.

00:02:52.710 --> 00:02:53.970
So that's interesting.

00:02:53.970 --> 00:02:58.150
2/3 to the negative
3 is equal to 27/8.

00:02:58.150 --> 00:02:59.090
Let's do some more.

00:03:02.740 --> 00:03:12.190
Let's do 4/7 to the negative 1.

00:03:12.190 --> 00:03:16.870
Once again, we have a negative
number in the exponent.

00:03:16.870 --> 00:03:21.700
That's the same thing as taking
the reciprocal of the base

00:03:21.700 --> 00:03:23.720
and raising it to the
positive exponent.

00:03:23.720 --> 00:03:26.425
Well, 7/4 to the 1, any
number to the first power

00:03:26.425 --> 00:03:27.570
is just the same number.

00:03:27.570 --> 00:03:29.380
It's equal to 7/4.

00:03:29.380 --> 00:03:31.770
So when take it to the negative
1 power, all you're essentially

00:03:31.770 --> 00:03:34.490
doing is getting the
reciprocal of the number.

00:03:34.490 --> 00:03:37.440
Let's do some more problems.

00:03:37.440 --> 00:03:41.400
2 to the negative 5.

00:03:41.400 --> 00:03:46.120
Once again, we take the
reciprocal of 2, and we say

00:03:46.120 --> 00:03:51.680
1/2, and now that can be raised
to the fifth power, and that

00:03:51.680 --> 00:04:00.520
equals 1/2 times 1/2 times
1/2 times 1/2 times 1/2,

00:04:00.520 --> 00:04:04.750
and that equals 1/32.

00:04:04.750 --> 00:04:07.030
Another way we could have
viewed 2 to the negative fifth

00:04:07.030 --> 00:04:12.040
is that, 2 to the negative
fifth, we could have said that

00:04:12.040 --> 00:04:16.210
equals 1 over 2 to the fifth.

00:04:16.210 --> 00:04:19.000
And we know that 2 to the
fifth is 32, so that would've

00:04:19.000 --> 00:04:20.380
been the same thing.

00:04:20.380 --> 00:04:23.380
Two ways to do it, pretty much
just changing the order of when

00:04:23.380 --> 00:04:25.180
you flip versus when you
actually calculate

00:04:25.180 --> 00:04:27.210
the exponent.

00:04:27.210 --> 00:04:31.110
Let me do two or
three more problems.

00:04:31.110 --> 00:04:33.360
And after I write down each of
these problems, you might just

00:04:33.360 --> 00:04:35.780
want to pause it and see if you
can do the problem yourself,

00:04:35.780 --> 00:04:38.100
and then compare your
answer to mine.

00:04:38.100 --> 00:04:46.760
So let's say I had negative 4
to the negative third power.

00:04:46.760 --> 00:04:49.090
Immediately, I like to get rid
of the negative in the

00:04:49.090 --> 00:04:56.120
exponent, and I know that that
equals minus 1/4 to the third

00:04:56.120 --> 00:05:03.500
power, and that equals minus
1/4 times minus 1/4

00:05:03.500 --> 00:05:06.110
times minus 1/4.

00:05:06.110 --> 00:05:08.630
The negative times a negative
is a positive, but then we're

00:05:08.630 --> 00:05:11.810
multiplying that times another
negative, so we get a negative.

00:05:11.810 --> 00:05:14.680
1 times 1 times 1 is 1.

00:05:14.680 --> 00:05:18.360
4 times 4 is 16 times 4 is 64.

00:05:18.360 --> 00:05:23.340
So it equals negative 1/64.

00:05:23.340 --> 00:05:24.340
Let's do another problem.

00:05:27.210 --> 00:05:29.100
Let me think of a good number.

00:05:29.100 --> 00:05:31.150
8/9.

00:05:31.150 --> 00:05:33.270
Let's make it negative.

00:05:33.270 --> 00:05:40.230
Negative 8/9 to the
negative second power.

00:05:40.230 --> 00:05:44.750
Well, once again, that
equals negative 9/8.

00:05:44.750 --> 00:05:48.560
Notice, I immediately just took
the reciprocal of the base to

00:05:48.560 --> 00:05:54.630
the positive 2 power, and now
that equals negative 9/8

00:05:54.630 --> 00:05:57.480
times negative 9/8.

00:05:57.480 --> 00:06:00.180
A negative times a negative
is a positive, so we

00:06:00.180 --> 00:06:05.230
get 9 times 9 is 81/64.

00:06:05.230 --> 00:06:06.260
I think you get the point now.

00:06:06.260 --> 00:06:08.730
The only new thing we've
learned, really, is that when

00:06:08.730 --> 00:06:12.790
you have a negative exponent,
it's the same thing as taking

00:06:12.790 --> 00:06:14.710
the reciprocal of the base
and raising it to the

00:06:14.710 --> 00:06:16.250
positive exponent.

00:06:16.250 --> 00:06:19.150
Hopefully, that last statement
didn't confuse you more, did

00:06:19.150 --> 00:06:21.630
more good than damage, but
I think you're ready to

00:06:21.630 --> 00:06:22.620
try some problems now.

00:06:22.620 --> 00:06:24.150
Have fun!


Solving a quadratic by factoring

WEBVTT
Kind: captions
Language: en

00:00:01.090 --> 00:00:03.050
Welcome to solving a
quadratic by factoring.

00:00:03.050 --> 00:00:04.970
Let's start doing
some problems.

00:00:04.970 --> 00:00:13.480
So, let's say I had a function
f of x is equal to x

00:00:13.480 --> 00:00:18.410
squared plus 6x plus 8.

00:00:18.410 --> 00:00:23.810
Now if I were to graph f of
x, the graph is going to

00:00:23.810 --> 00:00:24.790
look something like this.

00:00:24.790 --> 00:00:27.810
I don't know exactly what it's
going to look like, but it's

00:00:27.810 --> 00:00:31.420
going to be a parabola and it's
going to intersect the x-axis

00:00:31.420 --> 00:00:34.440
at a couple of points,
here and here.

00:00:34.440 --> 00:00:36.460
And what we're going to try
to do is determine what

00:00:36.460 --> 00:00:38.480
those two points are.

00:00:38.480 --> 00:00:40.800
So first of all, when a
function intersects the

00:00:40.800 --> 00:00:44.220
x-axis, that means f
of x is equal to zero.

00:00:44.220 --> 00:00:47.820
Because this is f of x-axis,
similar to the y-axis.

00:00:47.820 --> 00:00:49.760
So here f of x is 0.

00:00:49.760 --> 00:00:52.420
So in order to solve this
equation we set f of x to 0,

00:00:52.420 --> 00:01:01.480
and we get x squared plus
6x plus 8 is equal to 0.

00:01:01.480 --> 00:01:04.120
Now this might look like you
could solve it pretty easily,

00:01:04.120 --> 00:01:07.060
but that x squared term messes
things up and you could

00:01:07.060 --> 00:01:09.130
try it out for yourself.

00:01:09.130 --> 00:01:10.680
So we're going to
do is factor this.

00:01:10.680 --> 00:01:13.660
And we're going to say that x
squared plus 6x plus 8, but

00:01:13.660 --> 00:01:19.240
this can be written as x plus
something times x

00:01:19.240 --> 00:01:20.890
plus something.

00:01:20.890 --> 00:01:23.200
It will still equal that,
except that's equal to 0.

00:01:23.200 --> 00:01:25.650
Now in this presentation, I'm
going to just show you the

00:01:25.650 --> 00:01:27.980
systematic or you could say the
mechanical way of doing this.

00:01:27.980 --> 00:01:30.720
I'm going to give you another
presentation on why this works.

00:01:30.720 --> 00:01:33.420
And you might want to just
multiply out the answers we

00:01:33.420 --> 00:01:36.140
get in and multiply out
the expressions and

00:01:36.140 --> 00:01:38.400
see why it works.

00:01:38.400 --> 00:01:41.160
And the message we're going
to use is, we look at the

00:01:41.160 --> 00:01:43.405
coefficient on this x term, 6.

00:01:43.405 --> 00:01:46.450
And we say what two
numbers will add up to 6.

00:01:46.450 --> 00:01:49.840
And when those same two
numbers are multiplied

00:01:49.840 --> 00:01:51.650
you're going to get 8.

00:01:51.650 --> 00:01:53.290
Well let's just think
about the factors of 8.

00:01:53.290 --> 00:02:00.160
The factors of 8 are
one to 4 and 8.

00:02:00.160 --> 00:02:05.670
well 1 times 8 is 8, but 1 plus
8 is 9, so that doesn't work.

00:02:05.670 --> 00:02:09.610
2 times 4 is 8, and
2 plus 4 is 6.

00:02:09.610 --> 00:02:11.040
So that works.

00:02:11.040 --> 00:02:17.560
So we could just say x plus 2
and x plus 4 is equal to 0.

00:02:17.560 --> 00:02:21.150
Now if two expressions or two
numbers times each other equals

00:02:21.150 --> 00:02:24.470
0, that means that one of those
two numbers or both of

00:02:24.470 --> 00:02:25.990
them must equal 0.

00:02:25.990 --> 00:02:37.785
So now we can say that x
plus 2 equals 0, and x

00:02:37.785 --> 00:02:41.880
plus 4 is equal to zero.

00:02:41.880 --> 00:02:43.750
Well, this is a very
simple equation.

00:02:43.750 --> 00:02:47.680
We subtract 2 from both sides
and we get x equals negative 2.

00:02:47.680 --> 00:02:53.030
And here we get x
equals minus 4.

00:02:53.030 --> 00:02:55.020
And if we substitute
either of these into the

00:02:55.020 --> 00:02:57.470
original equation, we'll
see that it works.

00:02:57.470 --> 00:03:00.650
Minus 2-- so let's just try it
with minus 2 and I'll leave

00:03:00.650 --> 00:03:07.060
minus 4 up to you --so minus 2
squared plus 6 times

00:03:07.060 --> 00:03:09.940
minus 2 plus 8.

00:03:09.940 --> 00:03:18.070
Minus 2 squared is 4, minus
12-- 6 times minus 2 --plus 8.

00:03:18.070 --> 00:03:21.170
And sure enough that equals 0.

00:03:21.170 --> 00:03:23.030
And if you did the same thing
with negative 4, you'd

00:03:23.030 --> 00:03:24.810
also see that works.

00:03:24.810 --> 00:03:26.600
And you might be saying,
wow, this is interesting.

00:03:26.600 --> 00:03:30.180
This is an equation that
has two solutions.

00:03:30.180 --> 00:03:33.040
Well, if you think about it, it
makes sense because the graph

00:03:33.040 --> 00:03:39.130
of f of x is intersecting the
x-axis in two different places.

00:03:39.130 --> 00:03:40.150
Let's do another problem.

00:03:44.470 --> 00:03:56.580
Let's say I had f of x
is equal to 2 x squared

00:03:56.580 --> 00:04:01.830
plus 20x plus 50.

00:04:01.830 --> 00:04:04.170
So if we want to figure out
where it intersects the x-axis,

00:04:04.170 --> 00:04:07.470
we just set f of x equal to 0,
and I'll just swap the left and

00:04:07.470 --> 00:04:09.665
right sides of the equation.

00:04:09.665 --> 00:04:20.520
And I get 2x squared plus
20x plus 50 equals 0.

00:04:20.520 --> 00:04:22.890
Now, what's a little different
this time from last time, is

00:04:22.890 --> 00:04:25.910
here the coefficient on that x
squared is actually a 2 instead

00:04:25.910 --> 00:04:28.140
of a 1, and I like
it to be a 1.

00:04:28.140 --> 00:04:30.680
So let's divide the whole
equation, both the left

00:04:30.680 --> 00:04:33.150
and right sides, by 2.

00:04:33.150 --> 00:04:42.460
I get x squared plus
10x plus 25 equals 0.

00:04:42.460 --> 00:04:45.786
So all I did is I multiplied
1/2 times-- this is the same

00:04:45.786 --> 00:04:48.220
thing as dividing
by 2 --times 1/2.

00:04:48.220 --> 00:04:50.940
And of course 0 times 1/2 is 0.

00:04:50.940 --> 00:04:52.580
Now we are ready to do what
we did before, and you

00:04:52.580 --> 00:04:54.660
might want to pause it
and try it yourself.

00:04:54.660 --> 00:05:00.080
We're going to say x plus
something times x plus

00:05:00.080 --> 00:05:05.450
something is equal to 0 and
those two somethings, they

00:05:05.450 --> 00:05:08.300
should add up to 10, and
when you multiply them,

00:05:08.300 --> 00:05:09.940
they should be 25.

00:05:09.940 --> 00:05:11.690
Let's think about
the factors of 25.

00:05:11.690 --> 00:05:16.120
You have 1, 5, and 25.

00:05:16.120 --> 00:05:18.100
Well 1 times 25 is 25.

00:05:18.100 --> 00:05:22.260
1 plus 25 is 26, not 10.

00:05:22.260 --> 00:05:30.170
5 times 5 is 25, and 5 plus 5
is 10, so 5 actually works.

00:05:30.170 --> 00:05:34.640
It actually turns out that
both of these numbers are 5.

00:05:34.640 --> 00:05:39.690
So you get x plus 5 equals
0 or x plus 5 equals 0.

00:05:39.690 --> 00:05:42.882
So you just have to
really write it once.

00:05:42.882 --> 00:05:45.590
So you get x equals negative 5.

00:05:45.590 --> 00:05:47.030
So how do you think
about this graphically?

00:05:47.030 --> 00:05:50.290
I just told you that these
equations can intersect the

00:05:50.290 --> 00:05:53.470
x-axis in two places, but
this only has one solution.

00:05:53.470 --> 00:05:54.920
Well, this solution
would look like.

00:05:58.960 --> 00:06:05.080
If this is x equals negative 5,
we'd have a parabola that just

00:06:05.080 --> 00:06:08.020
touches right there, and
then comes back up.

00:06:08.020 --> 00:06:09.880
And instead of intersecting
in two places it only

00:06:09.880 --> 00:06:13.580
intersects right there
at x equals negative 5.

00:06:13.580 --> 00:06:17.110
And now as an exercise just to
prove to you that I'm not

00:06:17.110 --> 00:06:24.280
teaching you incorrectly, let's
multiply x plus 5 times x plus

00:06:24.280 --> 00:06:29.370
5 just to show you that it
equals what it should equal.

00:06:29.370 --> 00:06:32.730
So we just say that this is the
same thing is x times x plus

00:06:32.730 --> 00:06:39.950
5 plus 5 times x plus 5.

00:06:39.950 --> 00:06:47.200
x squared plus 5x
plus 5x plus 25.

00:06:47.200 --> 00:06:52.410
And that's x squared
plus 10x plus 25.

00:06:52.410 --> 00:06:54.780
So, it equals what we
said it should equal.

00:06:54.780 --> 00:06:57.320
And I'm going to once again do
another module where I explain

00:06:57.320 --> 00:06:59.870
this a little bit more.

00:06:59.870 --> 00:07:03.220
Let's do one more problem.

00:07:03.220 --> 00:07:04.930
And this one I am just
going to cut to the chase.

00:07:04.930 --> 00:07:15.250
Let's just solve x squared
minus x minus 30 is equal to 0.

00:07:15.250 --> 00:07:19.550
Once again, two numbers when we
add them they equal-- whats the

00:07:19.550 --> 00:07:21.850
coefficient here,
it's negative 1.

00:07:21.850 --> 00:07:26.090
So we could say those two
numbers are a plus b equals

00:07:26.090 --> 00:07:33.760
minus 1 and a times b
will equal minus 30.

00:07:33.760 --> 00:07:37.050
Well let's just think about
what all the factors are of 30.

00:07:37.050 --> 00:07:46.590
1, 2, 3, 5, 6, 10, 15, 30.

00:07:46.590 --> 00:07:50.930
Well, something interesting is
happening this time though.

00:07:50.930 --> 00:07:55.320
Since a times b is negative
30, one of these numbers

00:07:55.320 --> 00:07:56.000
have to be negative.

00:07:56.000 --> 00:07:59.370
They both can't be negative,
because if they're both

00:07:59.370 --> 00:08:01.800
negative then this would
be a positive 30.

00:08:01.800 --> 00:08:04.950
a times b is negative 30.

00:08:04.950 --> 00:08:07.620
So actually we're going to have
to say, two of these factors,

00:08:07.620 --> 00:08:12.030
the difference between them
should be negative 1.

00:08:12.030 --> 00:08:13.930
Well, if we look at all of
these, all these numbers

00:08:13.930 --> 00:08:17.850
obviously when you pair them
up, they multiply out to 30.

00:08:17.850 --> 00:08:22.200
But the only ones that have a
difference of 1 is 5 and 6.

00:08:22.200 --> 00:08:25.700
And since it's a negative 1,
it's going to be-- and I know

00:08:25.700 --> 00:08:27.770
I'm going very fast with this
and I'll do more example

00:08:27.770 --> 00:08:33.280
problems --this would be x
minus 6 times x plus

00:08:33.280 --> 00:08:36.010
5 is equal to 0.

00:08:36.010 --> 00:08:37.850
So how did I think about that?

00:08:37.850 --> 00:08:42.160
Negative 6 times 5
is negative 30.

00:08:42.160 --> 00:08:46.180
Negative 6 plus 5
is negative 1.

00:08:46.180 --> 00:08:47.550
So it works out.

00:08:47.550 --> 00:08:50.000
And the more and more you do
these practices-- I know it

00:08:50.000 --> 00:08:51.570
seems a little confusing
right now --it'll make

00:08:51.570 --> 00:08:52.900
a lot more sense.

00:08:52.900 --> 00:09:00.010
So you get x equals 6 or
x equals negative 5.

00:09:00.010 --> 00:09:04.150
I think at this point you're
ready to try some solving

00:09:04.150 --> 00:09:06.610
quadratics by factoring and
I'll do a couple more modules

00:09:06.610 --> 00:09:08.980
as soon as you get some
more practice problems.

00:09:08.980 --> 00:09:10.520
Have fun.


Algebra: Linear equations 1

WEBVTT
Kind: captions
Language: en

00:00:01.230 --> 00:00:03.860
Welcome to level one
linear equations.

00:00:03.860 --> 00:00:05.990
So let's start
doing some problems.

00:00:05.990 --> 00:00:16.190
So let's say I had the equation
5-- a big fat 5, 5x equals 20.

00:00:16.190 --> 00:00:18.540
So at first this might look
a little unfamiliar for you,

00:00:18.540 --> 00:00:19.914
but if I were to
rephrase this, I

00:00:19.914 --> 00:00:22.920
think you'll realize this
is a pretty easy problem.

00:00:22.920 --> 00:00:31.170
This is the same thing as
saying 5 times question mark

00:00:31.170 --> 00:00:32.920
equals 20.

00:00:32.920 --> 00:00:34.920
And the reason we do the
notation a little bit--

00:00:34.920 --> 00:00:37.400
we write the 5 next
to the x, because when

00:00:37.400 --> 00:00:39.300
you write a number right
next to a variable,

00:00:39.300 --> 00:00:41.030
you assume that you're
multiplying them.

00:00:41.030 --> 00:00:44.060
So this is just
saying 5 times x,

00:00:44.060 --> 00:00:46.420
so instead of a question
mark, we're writing an x.

00:00:46.420 --> 00:00:49.520
So 5 times x is equal to 20.

00:00:49.520 --> 00:00:51.520
Now, most of you all could
do that in your head.

00:00:51.520 --> 00:00:53.990
You could say, well, what
number times 5 is equal to 20?

00:00:53.990 --> 00:00:56.440
Well, it equals 4.

00:00:56.440 --> 00:00:58.960
But I'll show you a way to do
it systematically just in case

00:00:58.960 --> 00:01:01.570
that 5 was a more
complicated number.

00:01:01.570 --> 00:01:05.800
So let me make my pen
a little thinner, OK.

00:01:05.800 --> 00:01:12.434
So rewriting it, if
I had 5x equals 20,

00:01:12.434 --> 00:01:14.350
we could do two things
and they're essentially

00:01:14.350 --> 00:01:15.100
the same thing.

00:01:15.100 --> 00:01:18.000
We could say we just divide
both sides of this equation

00:01:18.000 --> 00:01:23.620
by 5, in which case, the left
hand side, those two 5's will

00:01:23.620 --> 00:01:25.490
cancel out, we'll get x.

00:01:25.490 --> 00:01:28.590
And the right hand side,
20 divided by 5 is 4,

00:01:28.590 --> 00:01:29.870
and we would have solved it.

00:01:29.870 --> 00:01:32.260
Another way to do it, and this
is actually the exact same way,

00:01:32.260 --> 00:01:34.010
we're just phrasing
it a little different.

00:01:34.010 --> 00:01:40.950
If you said 5x equals 20,
instead of dividing by 5,

00:01:40.950 --> 00:01:42.500
we could multiply by 1/5.

00:01:45.450 --> 00:01:48.620
And if you look at that, you can
realize that multiplying by 1/5

00:01:48.620 --> 00:01:51.159
is the same thing
as dividing by 5,

00:01:51.159 --> 00:01:52.950
if you know the difference
between dividing

00:01:52.950 --> 00:01:54.560
and multiplying fractions.

00:01:54.560 --> 00:01:56.710
And then that gets the
same thing, 1/5 times 5

00:01:56.710 --> 00:02:01.830
is 1, so you're just
left with an x equals 4.

00:02:01.830 --> 00:02:03.880
I tend to focus a
little bit more on this

00:02:03.880 --> 00:02:07.120
because when we start having
fractions instead of a 5,

00:02:07.120 --> 00:02:09.980
it's easier just to think about
multiplying by the reciprocal.

00:02:09.980 --> 00:02:11.730
Actually, let's do one
of those right now.

00:02:14.260 --> 00:02:26.925
So let's say I had negative
3/4 times x equals 10/13.

00:02:30.060 --> 00:02:31.330
Now, this is a harder problem.

00:02:31.330 --> 00:02:32.820
I can't do this one in my head.

00:02:32.820 --> 00:02:36.890
We're saying negative
3/4 times some number

00:02:36.890 --> 00:02:39.412
x is equal to 10/13.

00:02:39.412 --> 00:02:41.870
If someone came up to you on
the street and asked you that,

00:02:41.870 --> 00:02:46.040
I think you'd be like me,
and you'd be pretty stumped.

00:02:46.040 --> 00:02:47.980
But let's work it
out algebraically.

00:02:47.980 --> 00:02:49.710
Well, we do the same thing.

00:02:49.710 --> 00:02:53.930
We multiply both sides
by the coefficient on x.

00:02:53.930 --> 00:02:57.390
So the coefficient, all that
is, all that fancy word means,

00:02:57.390 --> 00:03:00.820
is the number that's
being multiplied by x.

00:03:00.820 --> 00:03:04.390
So what's the
reciprocal of minus 3/4.

00:03:04.390 --> 00:03:12.470
Well, it's minus 4/3 times,
and dot is another way

00:03:12.470 --> 00:03:14.070
to use times, and
you're probably

00:03:14.070 --> 00:03:17.624
wondering why in algebra, there
are all these other conventions

00:03:17.624 --> 00:03:19.040
for doing times
as opposed to just

00:03:19.040 --> 00:03:20.840
the traditional
multiplication sign.

00:03:20.840 --> 00:03:23.800
And the main reason is, I think,
just a regular multiplication

00:03:23.800 --> 00:03:25.860
sign gets confused
with the variable x,

00:03:25.860 --> 00:03:28.770
so they thought of either using
a dot if you're multiplying two

00:03:28.770 --> 00:03:31.170
constants, or just writing
it next to a variable

00:03:31.170 --> 00:03:33.790
to imply you're
multiplying a variable.

00:03:33.790 --> 00:03:36.460
So if we multiply the left
hand side by negative 4/3,

00:03:36.460 --> 00:03:38.090
we also have to
do the same thing

00:03:38.090 --> 00:03:42.280
to the right hand
side, minus 4/3.

00:03:42.280 --> 00:03:44.850
The left hand side, the
minus 4/3 and the 3/4,

00:03:44.850 --> 00:03:46.280
they cancel out.

00:03:46.280 --> 00:03:48.920
You could work it out on
your own to see that they do.

00:03:48.920 --> 00:03:53.610
They equal 1, so we're just
left with x is equal to 10 times

00:03:53.610 --> 00:04:01.740
minus 4 is minus 40, 13 times
3, well, that's equal to 39.

00:04:01.740 --> 00:04:04.180
So we get x is equal
to minus 40/39.

00:04:07.100 --> 00:04:08.930
And I like to leave
my fractions improper

00:04:08.930 --> 00:04:11.170
because it's easier
to deal with them.

00:04:11.170 --> 00:04:14.185
But you could also view
that-- that's minus--

00:04:14.185 --> 00:04:16.310
if you wanted to write it
as a mixed number, that's

00:04:16.310 --> 00:04:17.130
minus 1 and 1/39.

00:04:19.839 --> 00:04:22.220
I tend to keep it like this.

00:04:22.220 --> 00:04:24.160
Let's check to make
sure that's right.

00:04:24.160 --> 00:04:26.770
The cool thing about algebra is
you can always get your answer

00:04:26.770 --> 00:04:28.230
and put it back into
the original equation

00:04:28.230 --> 00:04:29.355
to make sure you are right.

00:04:29.355 --> 00:04:34.860
So the original equation
was minus 3/4 times x,

00:04:34.860 --> 00:04:38.940
and here we'll substitute
the x back into the equation.

00:04:38.940 --> 00:04:43.150
Wherever we saw x, we'll
now put our answer.

00:04:43.150 --> 00:04:49.550
So it's minus 40/39, and
our original equation

00:04:49.550 --> 00:04:52.786
said that equals 10/13.

00:04:52.786 --> 00:04:54.170
Well, and once
again, when I just

00:04:54.170 --> 00:04:57.260
write the 3/4 right next to
the parentheses like that,

00:04:57.260 --> 00:05:00.060
that's just another
way of writing times.

00:05:00.060 --> 00:05:08.132
So minus 3 times minus
40, it is minus 100--

00:05:08.132 --> 00:05:10.340
Actually, we could do
something a little bit simpler.

00:05:10.340 --> 00:05:16.370
This 4 becomes a 1
and this becomes a 10.

00:05:16.370 --> 00:05:18.740
If you remember when you're
multiplying fractions,

00:05:18.740 --> 00:05:21.480
you can simplify it like that.

00:05:21.480 --> 00:05:26.390
So it actually becomes
minus-- actually, plus 30,

00:05:26.390 --> 00:05:30.970
because we have a minus times
a minus and 3 times 10, over,

00:05:30.970 --> 00:05:34.550
the 4 is now 1, so all
we have left is 39.

00:05:34.550 --> 00:05:39.250
And 30/39, if we divide the
top and the bottom by 3,

00:05:39.250 --> 00:05:45.130
we get 10 over 13, which
is the same thing as what

00:05:45.130 --> 00:05:47.050
the equation said
we would get, so we

00:05:47.050 --> 00:05:49.597
know that we've got
the right answer.

00:05:49.597 --> 00:05:50.680
Let's do one more problem.

00:05:54.750 --> 00:06:01.532
Minus 5/6x is equal to 7/8.

00:06:01.532 --> 00:06:02.990
And if you want to
try this problem

00:06:02.990 --> 00:06:05.570
yourself, now's a
good time to pause,

00:06:05.570 --> 00:06:08.400
and I'm going to start
doing the problem right now.

00:06:08.400 --> 00:06:10.010
So same thing.

00:06:10.010 --> 00:06:12.090
What's the reciprocal
of minus 5/6?

00:06:12.090 --> 00:06:15.700
Well, it's minus 6/5.

00:06:15.700 --> 00:06:16.710
We multiply that.

00:06:16.710 --> 00:06:18.620
If you do that on
the left hand side,

00:06:18.620 --> 00:06:21.750
we have to do it on the
right hand side as well.

00:06:21.750 --> 00:06:23.430
Minus 6/5.

00:06:23.430 --> 00:06:26.920
The left hand side, the
minus 6/5 and the minus 5/6

00:06:26.920 --> 00:06:27.660
cancel out.

00:06:27.660 --> 00:06:30.030
We're just left with x.

00:06:30.030 --> 00:06:33.150
And the right hand
side, we have,

00:06:33.150 --> 00:06:37.740
well, we can divide both
the 6 and the 8 by 2,

00:06:37.740 --> 00:06:40.330
so this 6 becomes negative 3.

00:06:40.330 --> 00:06:42.882
This becomes 4.

00:06:42.882 --> 00:06:45.575
7 times negative
3 is minus 21/20.

00:06:50.087 --> 00:06:52.170
And assuming I haven't
made any careless mistakes,

00:06:52.170 --> 00:06:53.045
that should be right.

00:06:53.045 --> 00:06:55.340
Actually, let's just
check that real quick.

00:06:55.340 --> 00:06:59.580
Minus 5/6 times minus 21/20.

00:07:02.350 --> 00:07:06.672
Well, that equals
5, make that into 1.

00:07:06.672 --> 00:07:08.940
Turn this into a 4.

00:07:08.940 --> 00:07:10.205
Make this into a 2.

00:07:10.205 --> 00:07:12.680
Make this into a 7.

00:07:12.680 --> 00:07:14.280
Negative times
negative is positive.

00:07:14.280 --> 00:07:15.710
So you have 7.

00:07:15.710 --> 00:07:17.640
2 times 4 is 8.

00:07:17.640 --> 00:07:19.615
And that's what we
said we would get.

00:07:19.615 --> 00:07:21.290
So we got it right.

00:07:21.290 --> 00:07:22.710
I think you're
ready at this point

00:07:22.710 --> 00:07:25.220
to try some level one equations.

00:07:25.220 --> 00:07:27.040
Have fun.


Algebra: Linear equations 2

WEBVTT
Kind: captions
Language: en

00:00:01.340 --> 00:00:03.440
Welcome to level two
linear equations.

00:00:03.440 --> 00:00:05.330
Let's do a problem.

00:00:05.330 --> 00:00:14.130
2x plus 3 is equal to minus 15.

00:00:14.130 --> 00:00:17.260
Throw the minus in there to
make it a little bit tougher.

00:00:17.260 --> 00:00:18.970
So the first thing we
want to do whenever

00:00:18.970 --> 00:00:20.850
we do any linear
equation, is we want

00:00:20.850 --> 00:00:23.190
to get all of the
variable terms on one hand

00:00:23.190 --> 00:00:25.180
side of the equation and
all the constant terms

00:00:25.180 --> 00:00:25.930
on the other side.

00:00:25.930 --> 00:00:27.860
And it doesn't really
matter, although I

00:00:27.860 --> 00:00:30.970
tend to get my variables on the
left hand side of the equation.

00:00:30.970 --> 00:00:34.530
Well, my variables are
already on the left hand

00:00:34.530 --> 00:00:36.570
side of the equation
but I have this plus 3

00:00:36.570 --> 00:00:39.020
that I somehow want to
move to the right hand

00:00:39.020 --> 00:00:40.470
side of the equation.

00:00:40.470 --> 00:00:43.900
And the way I can-- you can put
it in quotes, move the 3 is I

00:00:43.900 --> 00:00:48.080
can subtract 3 from both
sides of this equation.

00:00:48.080 --> 00:00:51.690
And look at that carefully as
to why you think that works.

00:00:51.690 --> 00:00:53.690
Because if I subtract 3
from the left hand side,

00:00:53.690 --> 00:00:57.350
clearly this negative 3 that I'm
subtracting and the original 3

00:00:57.350 --> 00:00:59.690
will cancel out and become 0.

00:00:59.690 --> 00:01:02.250
and as long as I do whatever
I do on the left hand side,

00:01:02.250 --> 00:01:03.958
as long as I do it on
the right hand side

00:01:03.958 --> 00:01:07.526
as well, because whatever you do
on one side of the equal side,

00:01:07.526 --> 00:01:08.900
you have to do to
the other side,

00:01:08.900 --> 00:01:11.130
then I'm making a
valid operation.

00:01:11.130 --> 00:01:15.600
So this will simplify to 2x,
because the 3's cancel out.

00:01:15.600 --> 00:01:17.200
They become just 0.

00:01:17.200 --> 00:01:20.320
Equals minus 15 minus 3.

00:01:20.320 --> 00:01:22.595
Well, that's minus 18.

00:01:25.200 --> 00:01:28.200
And now, we're just at
a level one problem,

00:01:28.200 --> 00:01:29.840
and you can just
multiply both sides

00:01:29.840 --> 00:01:31.630
of this equation
times the reciprocal

00:01:31.630 --> 00:01:32.890
on the coefficient of 2x.

00:01:35.850 --> 00:01:37.274
I mean, some people
would just say

00:01:37.274 --> 00:01:39.190
that we're dividing by
2, which is essentially

00:01:39.190 --> 00:01:40.140
what we're doing.

00:01:40.140 --> 00:01:42.020
I like to always go
with the reciprocal,

00:01:42.020 --> 00:01:44.410
because if this
2 was a fraction,

00:01:44.410 --> 00:01:46.270
it's easier to think
about it that way.

00:01:46.270 --> 00:01:49.250
But either way, you either
multiply by the reciprocal,

00:01:49.250 --> 00:01:50.250
or divide by the number.

00:01:50.250 --> 00:01:51.580
It's the same thing.

00:01:51.580 --> 00:01:52.590
So 1/2 times 2x.

00:01:52.590 --> 00:01:54.930
Well, that's just 1x.

00:01:54.930 --> 00:02:00.780
So you get x equals,
and then minus 18/2.

00:02:00.780 --> 00:02:05.660
And minus 18/2, well,
that just equals minus 9.

00:02:05.660 --> 00:02:07.060
Let's do another problem.

00:02:07.060 --> 00:02:10.530
And actually, well, if we wanted
to check it, we could say,

00:02:10.530 --> 00:02:17.070
well, the original problem
was 2x plus 3 equals minus 15.

00:02:17.070 --> 00:02:22.160
So we could say 2
times minus 9 plus 3.

00:02:22.160 --> 00:02:25.530
2 times minus 9 is
minus 18 plus 3.

00:02:25.530 --> 00:02:29.090
Well, that's equal
to minus 15, which

00:02:29.090 --> 00:02:31.120
is equal to what the
original equation said,

00:02:31.120 --> 00:02:32.120
so we know that's right.

00:02:32.120 --> 00:02:33.620
That's the neat
thing about algebra.

00:02:33.620 --> 00:02:35.279
You can always check your work.

00:02:35.279 --> 00:02:36.320
Let's do another problem.

00:02:36.320 --> 00:02:38.560
I'm going to put some
fractions in this time,

00:02:38.560 --> 00:02:43.360
just to show you that it
can get a little bit hairy.

00:02:43.360 --> 00:02:57.135
So let's say I had minus 1/2x
plus 3/4 is equal to 5/6.

00:02:57.135 --> 00:02:58.260
So we'll do the same thing.

00:02:58.260 --> 00:03:01.460
First, we just want to get
this 3/4 out of the left hand

00:03:01.460 --> 00:03:03.846
side of the equation,
and actually,

00:03:03.846 --> 00:03:05.720
if you want to try
working this out yourself,

00:03:05.720 --> 00:03:06.650
you might want to
pause the video

00:03:06.650 --> 00:03:09.240
and then play it once you're
ready to see how I do it.

00:03:09.240 --> 00:03:12.360
Anyway, let me move forward
assuming you haven't paused it.

00:03:12.360 --> 00:03:14.730
If we want to get rid
of this 3/4, all we do

00:03:14.730 --> 00:03:18.175
is we subtract 3/4 from
both sides of this equation.

00:03:20.740 --> 00:03:23.240
Minus 3/4.

00:03:23.240 --> 00:03:26.990
Well, the left hand side,
the two 3/4 will just cancel.

00:03:26.990 --> 00:03:33.640
We get minus 1/2x equals, and
then on the right hand side,

00:03:33.640 --> 00:03:38.200
we just have to do this
fraction addition or fraction

00:03:38.200 --> 00:03:38.880
subtraction.

00:03:38.880 --> 00:03:44.120
So the least common
multiple of 6 and 4 is 12.

00:03:44.120 --> 00:03:53.470
So this becomes 5/6 6 is
10/12 minus 3/4 is 9/12,

00:03:53.470 --> 00:04:01.900
so we get minus 1/2x
is equal to 1/12.

00:04:01.900 --> 00:04:04.190
Hopefully, I didn't make
a mistake over here.

00:04:04.190 --> 00:04:06.490
And if that step confused
you, I went a little fast,

00:04:06.490 --> 00:04:08.156
you might just want
to review the adding

00:04:08.156 --> 00:04:10.009
and subtraction of fractions.

00:04:10.009 --> 00:04:11.300
So going back to where we were.

00:04:11.300 --> 00:04:13.400
So now all we have to do
is, well, the coefficient

00:04:13.400 --> 00:04:16.850
on the x term is minus 1/2,
and this is now a level one

00:04:16.850 --> 00:04:17.490
problem.

00:04:17.490 --> 00:04:19.490
So to solve for x,
we just multiply

00:04:19.490 --> 00:04:23.160
both sides by the reciprocal
of this minus 1/2x,

00:04:23.160 --> 00:04:29.570
and that's minus 2/1 times
minus 1/2x on that side,

00:04:29.570 --> 00:04:34.880
and then that's times minus 2/1.

00:04:34.880 --> 00:04:37.430
The left hand side, and
you're used to this by now,

00:04:37.430 --> 00:04:40.180
simplifies to x.

00:04:40.180 --> 00:04:45.510
The right hand side
becomes minus 2/12,

00:04:45.510 --> 00:04:49.160
and we could simplify
that further to minus 1/6.

00:04:49.160 --> 00:04:55.750
Well, let's check that just
to make sure we got it right.

00:04:55.750 --> 00:04:58.840
So let's try to
remember that minus 1/6.

00:04:58.840 --> 00:05:03.640
So the original
problem was minus 1/2x,

00:05:03.640 --> 00:05:08.270
so here we can substitute
the minus 1/6, plus 3/4.

00:05:08.270 --> 00:05:12.150
I just wrote only the left hand
side of the original problem.

00:05:12.150 --> 00:05:15.640
So minus 1/2 times
minus 1/6, well, that's

00:05:15.640 --> 00:05:23.150
positive 1/12 plus 3/4.

00:05:23.150 --> 00:05:25.730
Well, that's the
same thing as 12,

00:05:25.730 --> 00:05:29.270
the 1 stays the same, plus 9.

00:05:29.270 --> 00:05:33.530
1 plus 9 is 10 over 12.

00:05:33.530 --> 00:05:36.650
And that is equal
to 5/6, which is

00:05:36.650 --> 00:05:38.020
what our original problem was.

00:05:38.020 --> 00:05:38.730
Our original problem was this.

00:05:38.730 --> 00:05:39.850
This stuff I wrote later.

00:05:39.850 --> 00:05:43.150
So it's 5/6, so the
problem checks out.

00:05:43.150 --> 00:05:46.750
So hopefully, you're now
ready to try some level two

00:05:46.750 --> 00:05:48.664
problems on your own.

00:05:48.664 --> 00:05:50.330
I might add some other
example problems.

00:05:50.330 --> 00:05:55.000
But the only extra step here
relative to level one problems

00:05:55.000 --> 00:05:58.062
is you'll have this
constant term that you

00:05:58.062 --> 00:06:00.930
need to add or subtract from
both sides of this equation,

00:06:00.930 --> 00:06:03.720
and you'll essentially turn
it into a level one problem.

00:06:03.720 --> 00:06:05.329
Have fun.