Super Yoga plans: Solving systems by elimination

WEBVTT
Kind: captions
Language: en

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While I was doing my tiny
little bit of analysis

00:00:03.450 --> 00:00:07.220
to figure out whether the basic
or the trial plan made sense,

00:00:07.220 --> 00:00:10.890
and we did this whole
analysis of saying well,

00:00:10.890 --> 00:00:13.400
at what number of sessions
does the trial plan

00:00:13.400 --> 00:00:15.292
give me the same cost
as the basic plan?

00:00:15.292 --> 00:00:17.000
And I did this whole
calculation and said

00:00:17.000 --> 00:00:18.190
it was at five sessions.

00:00:18.190 --> 00:00:22.150
I didn't realize it, but the guy
at the front desk of the yoga

00:00:22.150 --> 00:00:23.884
studio was watching
me the whole time.

00:00:23.884 --> 00:00:25.550
And then all of a
sudden he said, "Well,

00:00:25.550 --> 00:00:26.990
that's all cute
what you did Sal.

00:00:26.990 --> 00:00:28.156
I understand what you said."

00:00:28.156 --> 00:00:30.650
There's some number
of sessions where

00:00:30.650 --> 00:00:34.120
12s is equal to
20 plus 8s, where

00:00:34.120 --> 00:00:36.870
we get the same exact cost.

00:00:36.870 --> 00:00:40.990
And so you said 20 plus
8x is equal to 12 plus s.

00:00:40.990 --> 00:00:45.490
Or another way you could say it
is the C that 20 plus 8s gets

00:00:45.490 --> 00:00:48.610
is going to be the same
as the C that 12s gets.

00:00:48.610 --> 00:00:51.690
So we just equate them
and then you solved for s.

00:00:51.690 --> 00:00:53.664
And then you solved
for the cost.

00:00:53.664 --> 00:00:55.830
But the guy says, "I have
different way of doing it,

00:00:55.830 --> 00:00:58.461
and I'm curious to see if I'm
going to get the same answer."

00:00:58.461 --> 00:01:00.002
So here's a way of
thinking about it.

00:01:00.002 --> 00:01:00.810
He says, well look.

00:01:00.810 --> 00:01:01.809
Look at this right here.

00:01:05.660 --> 00:01:13.100
We know from our basic plan
that the total monthly cost

00:01:13.100 --> 00:01:20.330
is actually going to be $20
upfront fee plus $8 per session

00:01:20.330 --> 00:01:22.450
times the number of sessions.

00:01:22.450 --> 00:01:23.790
So we know that.

00:01:23.790 --> 00:01:27.080
And the whole thing we
wanted to think about

00:01:27.080 --> 00:01:32.020
is at what number of sessions
do we get the same cost for both

00:01:32.020 --> 00:01:34.580
the basic and the trial plan?

00:01:34.580 --> 00:01:35.920
So he says, I have an idea.

00:01:35.920 --> 00:01:39.080
I have an idea of turning this
into an equation of only one

00:01:39.080 --> 00:01:39.580
unknown.

00:01:39.580 --> 00:01:41.460
Right now it's an
equation of two unknowns.

00:01:41.460 --> 00:01:44.940
I have a certain cost that's
dependent on a certain number

00:01:44.940 --> 00:01:47.330
of sessions.

00:01:47.330 --> 00:01:50.607
He thinks he's got a way of
getting rid of the C. He says,

00:01:50.607 --> 00:01:52.690
what I'm going to do is
I'm going to just subtract

00:01:52.690 --> 00:01:54.230
a C from the left-hand side.

00:02:01.109 --> 00:02:03.400
And then I quickly point to
him, I was like, look look.

00:02:06.509 --> 00:02:08.800
You can't just do one thing
to one side of an equation.

00:02:08.800 --> 00:02:10.967
Then the equality
won't be true anymore.

00:02:10.967 --> 00:02:12.550
And he's like, oh
no, no, don't worry.

00:02:12.550 --> 00:02:14.920
I'm going to do the same
thing on the other side

00:02:14.920 --> 00:02:15.844
of the equation.

00:02:15.844 --> 00:02:17.260
And so I tell him,
well, what good

00:02:17.260 --> 00:02:19.136
would that be, subtracting
C from both sides?

00:02:19.136 --> 00:02:20.593
Because then you're
going to end up

00:02:20.593 --> 00:02:22.260
with a C on the right-hand side.

00:02:22.260 --> 00:02:25.869
You're going to end up
with 20 plus 8s minus C. So

00:02:25.869 --> 00:02:27.410
you're not going to
get rid of the C,

00:02:27.410 --> 00:02:29.076
you're still going
to have two unknowns.

00:02:29.076 --> 00:02:30.300
He's like, no, no, no Sal.

00:02:30.300 --> 00:02:31.490
You don't get it.

00:02:31.490 --> 00:02:34.620
I'm going to subtract C
from the left-hand side,

00:02:34.620 --> 00:02:37.660
but I know that we're
looking for an s where

00:02:37.660 --> 00:02:44.290
12s is going to give us
the same as whatever C is.

00:02:44.290 --> 00:02:47.930
Whether I use the basic plan or
whether I use the trial plan,

00:02:47.930 --> 00:02:50.180
I'm going to get the same cost.

00:02:50.180 --> 00:02:52.960
So I'll subtract
the cost, whatever

00:02:52.960 --> 00:02:55.880
that cost is, that same
cost that either the basic

00:02:55.880 --> 00:02:58.490
or the trial plan will
give me, I'll subtract that

00:02:58.490 --> 00:03:02.380
from the left-hand side, because
that'll cancel out with that.

00:03:02.380 --> 00:03:05.660
But from the right-hand
side, I'll just subtract 12s.

00:03:09.250 --> 00:03:11.260
So he says, I'm just
going to subtract C

00:03:11.260 --> 00:03:13.200
from the left-hand
side and I'm going

00:03:13.200 --> 00:03:19.060
to subtract 12s from
the right-hand side.

00:03:19.060 --> 00:03:23.980
So I'm wondering, is
this a valid thing to do?

00:03:23.980 --> 00:03:27.270
He's essentially
subtracting this equation

00:03:27.270 --> 00:03:28.860
from that equation.

00:03:28.860 --> 00:03:31.840
He's subtracting the C
from the left-hand side

00:03:31.840 --> 00:03:35.970
and he's subtracting this
12s from the right-hand side.

00:03:35.970 --> 00:03:39.317
Does that make any sense?

00:03:39.317 --> 00:03:40.650
Well, let's just think about it.

00:03:40.650 --> 00:03:44.530
We know that if we have an
equation, if we add or subtract

00:03:44.530 --> 00:03:47.200
one thing to one side, we
have to do the same thing

00:03:47.200 --> 00:03:48.670
to the right-hand side.

00:03:48.670 --> 00:03:52.460
What he's doing,
he's saying, look.

00:03:52.460 --> 00:03:55.000
For either plan we use,
we know that there's

00:03:55.000 --> 00:03:57.010
some number of
sessions where we're

00:03:57.010 --> 00:03:59.000
going to get the same cost.

00:03:59.000 --> 00:04:02.530
So let's subtract that same
cost from the left-hand side.

00:04:02.530 --> 00:04:05.170
And that's going to be
the same thing as 12s.

00:04:05.170 --> 00:04:06.919
We know that this
has to be true.

00:04:06.919 --> 00:04:08.460
We know that both
of these statements

00:04:08.460 --> 00:04:10.912
have to be true at
this point of equality.

00:04:10.912 --> 00:04:11.620
So he says, look.

00:04:11.620 --> 00:04:15.640
If these two things are equal
to each other, subtracting a 12s

00:04:15.640 --> 00:04:18.769
is completely equivalent
to subtracting a C.

00:04:18.769 --> 00:04:21.779
So this actually is
a valid operation.

00:04:21.779 --> 00:04:24.120
He is subtracting
the same thing.

00:04:24.120 --> 00:04:25.970
He's subtracting a
C over here and he's

00:04:25.970 --> 00:04:28.045
subtracting a 12s over here.

00:04:28.045 --> 00:04:29.920
We know they're the same
thing, because we're

00:04:29.920 --> 00:04:31.730
putting that constraint on us.

00:04:31.730 --> 00:04:36.110
We know that both C has
to be equal to 20 plus 8s

00:04:36.110 --> 00:04:39.180
and C has to be equal to 12s.

00:04:39.180 --> 00:04:41.660
Because C has to
be equal to 12s,

00:04:41.660 --> 00:04:46.390
we can subtract C from one
side and 12s from the other.

00:04:46.390 --> 00:04:48.740
But let's see if this
actually gets us anywhere.

00:04:48.740 --> 00:04:52.090
So if we subtract C from one
side and 12s from the other,

00:04:52.090 --> 00:04:55.150
what do we actually get?

00:04:55.150 --> 00:04:58.340
Well, let's think it through.

00:04:58.340 --> 00:05:01.710
On the left-hand side,
if I have C minus C, well

00:05:01.710 --> 00:05:02.946
I just end up with 0.

00:05:02.946 --> 00:05:04.570
They just completely
negate each other,

00:05:04.570 --> 00:05:06.940
and there's nothing left
here on the left-hand side.

00:05:06.940 --> 00:05:09.450
So I just have 0 on
the left-hand side.

00:05:09.450 --> 00:05:14.597
On the right-hand side,
I have the 20 still.

00:05:14.597 --> 00:05:16.430
And then I have 8 of
something, and then I'm

00:05:16.430 --> 00:05:18.750
subtracting 12 of something.

00:05:18.750 --> 00:05:24.890
So 8s's minus 12s's is going
to give me negative 4s's.

00:05:24.890 --> 00:05:29.050
So that's going to
give me negative 4s's.

00:05:29.050 --> 00:05:30.670
So this is interesting now.

00:05:30.670 --> 00:05:33.310
I now have one equation
and one unknown.

00:05:33.310 --> 00:05:34.930
That one unknown is s.

00:05:34.930 --> 00:05:38.050
And so I should be able
to solve for that s.

00:05:38.050 --> 00:05:41.050
But how can I do it?

00:05:41.050 --> 00:05:43.420
Well, I always like
to isolate my variable

00:05:43.420 --> 00:05:45.140
on one side of the equation.

00:05:45.140 --> 00:05:47.850
And so now, I'll isolate
it on the left-hand side.

00:05:47.850 --> 00:05:50.430
And as you know, there's
many, many ways of doing this.

00:05:50.430 --> 00:05:52.990
And so, what can I do
if I want to get the s's

00:05:52.990 --> 00:05:55.160
on the left-hand side
and get rid of them

00:05:55.160 --> 00:05:58.010
on the right-hand side?

00:05:58.010 --> 00:06:00.660
Well, if I want to get rid of
them on the right-hand side,

00:06:00.660 --> 00:06:02.620
I'm subtracting 4s's here.

00:06:02.620 --> 00:06:04.760
I just have to add 4s's.

00:06:04.760 --> 00:06:07.800
But once again, I can't just
do that to the right-hand side.

00:06:07.800 --> 00:06:09.830
Then the equality
won't be true anymore.

00:06:09.830 --> 00:06:13.720
I have to also do that
to the left-hand side.

00:06:13.720 --> 00:06:15.350
And so what will that give me?

00:06:17.980 --> 00:06:21.070
Well, we get 0 plus 4s
on the left-hand side,

00:06:21.070 --> 00:06:22.910
which gives us just 4s.

00:06:22.910 --> 00:06:26.300
And that's going to be equal
to-- On the right-hand side,

00:06:26.300 --> 00:06:29.500
negative 4s and 4s,
they negate each other.

00:06:29.500 --> 00:06:32.000
We're left with just 20.

00:06:32.000 --> 00:06:36.020
So we're left with
4s is equal to 20.

00:06:36.020 --> 00:06:38.870
To solve for s, we
just divide by 4.

00:06:38.870 --> 00:06:40.510
We have 4 times something.

00:06:40.510 --> 00:06:43.560
The special number of
sessions is equal to 20.

00:06:43.560 --> 00:06:45.492
Well, 4 times
something divided by 4

00:06:45.492 --> 00:06:46.950
is just going to
be that something.

00:06:46.950 --> 00:06:49.120
But we can't just divide
the left side by 4.

00:06:49.120 --> 00:06:52.360
We also have to divide
the right-hand side by 4.

00:06:52.360 --> 00:06:56.880
So we're left with s is
equal to 20 divided by 4,

00:06:56.880 --> 00:07:02.530
or s is equal to 5
sessions, which is exactly

00:07:02.530 --> 00:07:04.370
the answer we got before.

00:07:04.370 --> 00:07:06.610
And then we can go back
into either of the plans

00:07:06.610 --> 00:07:10.180
and see that the cost
will still be $60.

00:07:10.180 --> 00:07:13.130
And so what's neat here
is that both myself

00:07:13.130 --> 00:07:15.630
and the guy at the front
desk of the yoga studio,

00:07:15.630 --> 00:07:17.790
we had very, very
different approaches

00:07:17.790 --> 00:07:19.170
to solving this problem.

00:07:19.170 --> 00:07:23.380
But because what each of us
did was logically correct--

00:07:23.380 --> 00:07:25.450
and this is what's really
neat about algebra--

00:07:25.450 --> 00:07:27.930
we got to the same conclusion.

00:07:27.930 --> 00:07:30.660
As long as we do
logically correct things,

00:07:30.660 --> 00:07:34.640
we will always get
to the same answer.


Super Yoga plans: Solving systems by substitution

WEBVTT
Kind: captions
Language: en

00:00:00.236 --> 00:00:11.370
In the last video we began to visually explore the relationship between the number of sessions I attend and my monthly cost given the different plans.

00:00:11.370 --> 00:00:24.571
So right here, in this blue line we visualized it for the trial plan, and then this orange dotted line we visualized it for the basic plan.

00:00:24.571 --> 00:00:30.861
And it meets where it intuitively, hopefully represents what we already understood about the different plans.

00:00:30.861 --> 00:00:39.858
The trial plan, we don't pay anything upfront and every time we add a session, so every time we move to the right here we move to the right one session.

00:00:39.858 --> 00:00:44.375
We add twelve dollars to out cost, we move up by twelve.

00:00:44.375 --> 00:00:49.635
Add another session, move up by 12, add another session move up by 12.

00:00:49.635 --> 00:00:53.302
On the basic plan we did have an upfront cost on 20 dollars

00:00:53.302 --> 00:00:57.574
but the line was less steep, we had to pay less for each extra session.

00:00:57.574 --> 00:01:02.175
Here when we added an extra session we only had to add 8 dollars to our total fees.

00:01:02.175 --> 00:01:05.174
Add another session, you only have to add 8 dollars.

00:01:05.174 --> 00:01:12.711
So the orange line starts at a higher point but it is less steep, and we see because of that they clearly intersect.

00:01:12.711 --> 00:01:17.303
And they intersect right over here.

00:01:17.303 --> 00:01:28.595
So my question to you is, "What is the significance of that point of intersection?"

00:01:28.595 --> 00:01:30.436
Well lets think about it a little bit.

00:01:30.436 --> 00:01:34.509
How much do each of the plans cost if I only attend one session?

00:01:34.509 --> 00:01:43.434
So we see here from the trial plan it will cost us 12 dollars while on the basic plan one session will cost us 28 dollars.

00:01:43.434 --> 00:01:48.342
And you don't even have to look at these tables we set up. You can see this visually.

00:01:48.342 --> 00:01:54.102
So if we go up one session the trial plan is below the basic plan.

00:01:54.102 --> 00:01:57.710
If you go up 2 sessions the trial plan is still below the basic plan.

00:01:57.710 --> 00:02:01.008
Although they are not as far apart.

00:02:01.008 --> 00:02:07.170
Three sessions, still the basic plan is above the trial plan but they are getting closer and closer together.

00:02:07.170 --> 00:02:10.345
All the way until you get to this point over here.

00:02:10.345 --> 00:02:19.770
This point is essentially the number of sessions, it looks like five sessions,

00:02:19.770 --> 00:02:26.930
it looks like the number of sessions regardless of the plan we choose the cost is the same.

00:02:26.930 --> 00:02:29.842
It looks like that cost is right around 60 dollars.

00:02:29.842 --> 00:02:38.513
Once we go beyond that point, all of a sudden the basic plan becomes cheaper than the trial plan.

00:02:38.513 --> 00:02:46.308
It looks like if we were to take 6 sessions the basic plan would give a lower price than the trial plan.

00:02:46.308 --> 00:02:51.932
But how do we actually figure out what this point is, and what number of sessions, and what dollar value?

00:02:51.932 --> 00:02:56.641
I just eyeballed it right now which is useful with these visual graphs.

00:02:56.641 --> 00:03:02.839
But what if we wanted to get the exact value and the value is 5.1 or 60.25

00:03:02.839 --> 00:03:06.342
How do we get the exact same value?

00:03:06.342 --> 00:03:16.703
Well, one way to think about it is, were trying to find out s, or the number of sessions, so that regardless of which plan we choose we have the exact same cost.

00:03:16.703 --> 00:03:26.507
So if we pick the basic plan our cost is going to be 20 dollars plus 8 times s.

00:03:26.507 --> 00:03:37.768
And if we pick the trial plan the cost is going to be 12 times s

00:03:37.768 --> 00:03:43.508
And were trying to find that unknown s where this value is going to be the same as this value.

00:03:43.508 --> 00:03:51.769
where that unknown value of sessions, where regardless of which plan i choose i will pay the same price

00:03:51.769 --> 00:03:58.513
so we're curious about the s where 12s is equal to 20 plus 8s.

00:03:58.513 --> 00:04:00.429
So we set up this equation.

00:04:00.429 --> 00:04:03.976
the equation has one unknown and we should be able to solve this.

00:04:03.976 --> 00:04:06.369
So lets think about how we can do it.

00:04:06.369 --> 00:04:12.906
What is the first general idea if we want to figure out the unknown s.

00:04:12.906 --> 00:04:19.437
There are many ways to approach theses Algebra problems which is what makes them fun

00:04:19.437 --> 00:04:24.507
I try to isolate the s on one side of the equation.

00:04:24.507 --> 00:04:27.847
For me, i want to isolate it on the right side of the equation.

00:04:27.847 --> 00:04:28.900
Since i already has this 12s there and this 20 on the left side.

00:04:28.900 --> 00:04:32.100
I could have done other things.

00:04:32.100 --> 00:04:37.704
So what is the best way to get rid of this 8s from the left side.

00:04:37.704 --> 00:04:42.302
Well the easiest thing i can think of is subtract 8s from lefthand side

00:04:42.302 --> 00:04:46.437
But i can't just do that, if I do then this equality won't hold.

00:04:46.437 --> 00:04:52.369
These things were equal to each other and if i just subtract from one side then the left side wont be equal to the right side

00:04:52.369 --> 00:04:59.755
In order for them to be equal i have to do the same thing on the right side

00:04:59.755 --> 00:05:11.678
Once I do that on the left side these two characters negate each other and i am left with 20 on the left side.

00:05:11.678 --> 00:05:24.181
And on the right i am left with 12s and take away 8s and i am left with 4s.

00:05:24.181 --> 00:05:29.514
So it is equal to 4 times s.

00:05:29.514 --> 00:05:34.173
And so now we are pretty close. I just want an s on the right side.

00:05:34.173 --> 00:05:40.176
What can I do to this equation, so i can have just an s on the righthand side.

00:05:40.176 --> 00:05:46.702
Well the easiest thing to do is divide both sides by 4

00:05:46.702 --> 00:05:49.436
I can't do it just to one side.

00:05:49.436 --> 00:05:52.834
When i divide both sides by 4, what do i get for s?

00:05:52.834 --> 00:05:56.109
Well on the right hand side 4s divided by 4 is just going to be s.

00:05:56.109 --> 00:06:08.700
And thats going to be equal to 20 divided by 4 which is 5

00:06:08.700 --> 00:06:11.302
So we eyeballed it and it looked like 5 sessions.

00:06:11.302 --> 00:06:17.592
Now we know for sure that at 5 sessions the cost of either plan is going to be the same.

00:06:17.592 --> 00:06:21.173
But what is the cost of either plan?

00:06:21.173 --> 00:06:25.103
Well we should be able to go to either plan because the cost will be the same.

00:06:25.103 --> 00:06:31.636
So is we look at out trial plan and we say 5 sessions, how much will that cost?

00:06:31.636 --> 00:06:35.263
well the cost is going to be 12 times 5

00:06:35.263 --> 00:06:44.640
so the cost is going to be equal to 12,12 times 5 which is going to be 60 dollars

00:06:44.640 --> 00:06:47.102
the cost is going to be 60 dollars.

00:06:47.102 --> 00:06:53.309
So my question to you is do we even have to try out the 5 in this equation right over here?

00:06:53.309 --> 00:06:56.968
See what it would cost under the basic plan.

00:06:56.968 --> 00:07:04.929
Well we wouldn't because the whole reason we got 5 sessions is because we said this is the number f sessions where we get the same cost

00:07:04.929 --> 00:07:08.342
For the basic plan or the trial plan

00:07:08.342 --> 00:07:11.091
but if you're curious i do encourage you to find out.

00:07:11.091 --> 00:07:15.375
Substitute a 5 in for s and see that you get

00:07:15.375 --> 00:07:21.708
you'll get 20 plus 8 time 5 which is 20 plus 40 which is 60 dollars

00:07:21.708 --> 00:07:27.510
So with either plan at 5 sessions the total cost is going to be 60 dollars.

00:07:27.510 --> 00:07:29.701
Then is you add sessions after that,

00:07:29.701 --> 00:07:36.845
Because the trial plan is more steep each incremental session after that is going to cost you more.

00:07:36.845 --> 00:07:40.838
You start to see that the trial plan begins to get more and more expensive.

00:07:40.838 --> 00:07:44.503
So going back to my question of which gym membership i'm going to use.

00:07:44.503 --> 00:07:46.339
All of a sudden i have an interesting answer

00:07:46.339 --> 00:07:51.344
If i plan on attending on average less that 5 sessions

00:07:51.344 --> 00:07:59.091
it probably make sense, less that 5 sessions a month, it makes sense for me to use the trial plan.

00:07:59.091 --> 00:08:05.044
If i plan to attend more that 5 sessions a month the basic plan is going to be cheaper.

00:08:05.044 --> 00:08:09.044
If i plan to attend exactly 5 sessions a month it doesn't matter which plan i acutally use


Super Yoga plans: Plotting points

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Kind: captions
Language: en

00:00:00.650 --> 00:00:03.390
We were able to already answer
some interesting questions

00:00:03.390 --> 00:00:06.580
about the different plans
at the Super Yoga Studio,

00:00:06.580 --> 00:00:08.490
using the equations we set up.

00:00:08.490 --> 00:00:10.180
But I'm a very visual
guy, and I want

00:00:10.180 --> 00:00:12.820
to make sure that I
analyze the problem fully

00:00:12.820 --> 00:00:15.790
before I spend my
hard earned dollars.

00:00:15.790 --> 00:00:17.870
And what I want to
do now is really

00:00:17.870 --> 00:00:22.220
see a visual representation for
the relationship between how

00:00:22.220 --> 00:00:25.400
many sessions I attend, and
how much it's going to cost me.

00:00:25.400 --> 00:00:29.360
And I'm going to do it for the
trial plan and the basic plan.

00:00:29.360 --> 00:00:33.310
And to do that, down here
on the horizontal axis,

00:00:33.310 --> 00:00:36.120
I'm going to plot the number
of sessions that I attend.

00:00:36.120 --> 00:00:38.850
So s, that is equal to
the number of sessions

00:00:38.850 --> 00:00:40.320
I attend per month.

00:00:40.320 --> 00:00:41.830
And then here on
the vertical axis

00:00:41.830 --> 00:00:44.840
I'm going to plot how much
it's going to cost me,

00:00:44.840 --> 00:00:47.070
depending on how many
sessions I attend.

00:00:47.070 --> 00:00:50.197
So c is equal to
my monthly cost.

00:00:50.197 --> 00:00:52.280
So let's think about these
points right over here.

00:00:52.280 --> 00:00:54.613
And then even also think about
how this equation relates

00:00:54.613 --> 00:00:56.890
to what we might end up drawing.

00:00:56.890 --> 00:01:03.910
So the first point here is
that if I go to no sessions,

00:01:03.910 --> 00:01:06.350
it's going to cost me $0.00.

00:01:06.350 --> 00:01:10.060
So where would I plot
that right over here?

00:01:10.060 --> 00:01:12.630
Well, if I attend no sessions,
and I should have marked off

00:01:12.630 --> 00:01:15.790
a 0 right over here, it's
going to cost me $0.00.

00:01:15.790 --> 00:01:19.430
So it's this point,
just like that.

00:01:19.430 --> 00:01:21.760
Now let's think about
if I attend one session.

00:01:21.760 --> 00:01:24.200
If I attend one session,
it'll be $12.00.

00:01:24.200 --> 00:01:27.580
Where would I plot that?

00:01:27.580 --> 00:01:30.740
Well, one session is
going to cost me $12.00,

00:01:30.740 --> 00:01:34.580
see this is $15.00, so $12.00 is
going to be right around there.

00:01:34.580 --> 00:01:36.660
So I'll put it right over there.

00:01:36.660 --> 00:01:41.710
And if I attend two sessions,
where would I plot that?

00:01:41.710 --> 00:01:45.550
Well, two sessions are
going to cost me $24.00,

00:01:45.550 --> 00:01:49.630
so we get the right about there.

00:01:49.630 --> 00:01:51.242
And then what about
three sessions?

00:01:51.242 --> 00:01:53.700
And I could keep going four,
five, six, on, and on, and on.

00:01:53.700 --> 00:01:55.074
But let's just
try three, I think

00:01:55.074 --> 00:01:57.960
you'll get the general idea.

00:01:57.960 --> 00:02:03.617
Three sessions cost $36.00.

00:02:03.617 --> 00:02:06.075
And you see that all of these
points seem to sit on a line.

00:02:06.075 --> 00:02:10.300
And I can attempt, I'll attempt
my best, to draw that line.

00:02:10.300 --> 00:02:11.950
And this line
essentially describes

00:02:11.950 --> 00:02:15.100
all of the possible
combinations of cost

00:02:15.100 --> 00:02:16.417
for a given number of sessions.

00:02:16.417 --> 00:02:18.000
So any point here,
especially if we're

00:02:18.000 --> 00:02:20.600
assuming that we allow
fractional sessions.

00:02:20.600 --> 00:02:24.370
That's not conventional
practice for most yoga studios,

00:02:24.370 --> 00:02:25.870
but this one is a
super yoga studio,

00:02:25.870 --> 00:02:29.030
so it allows you to attend four
and one half sessions, or one

00:02:29.030 --> 00:02:31.750
and one half sessions, or
exactly half a session.

00:02:31.750 --> 00:02:35.720
So if you attend half a session
it's going to cost you $6.00

00:02:35.720 --> 00:02:38.870
Another way to think about it
is if you attend zero sessions

00:02:38.870 --> 00:02:43.180
it'll be $0.00, but then
as you add a session,

00:02:43.180 --> 00:02:46.790
so you go to the right one step,
it's going to cost you $12.00

00:02:46.790 --> 00:02:47.440
more.

00:02:47.440 --> 00:02:50.810
So then you're going
to go up $12.00.

00:02:50.810 --> 00:02:53.320
So we went to the right
one, we went up 12.

00:02:53.320 --> 00:02:56.569
If we go to the right one again,
we're going to go up 12 again.

00:02:56.569 --> 00:02:58.110
Because every session
we add is going

00:02:58.110 --> 00:02:59.680
to cost us 12 extra dollars.

00:02:59.680 --> 00:03:03.270
We go from 0 to 12 to 24.

00:03:03.270 --> 00:03:07.170
If you add another session
you go up 12 again.

00:03:07.170 --> 00:03:09.486
If you take away a
session, you'll go down 12.

00:03:09.486 --> 00:03:11.110
And so it's a kind
of a way of thinking

00:03:11.110 --> 00:03:12.720
about how steep this line is.

00:03:12.720 --> 00:03:16.790
Every time you go one to the
right, you're going 12 up.

00:03:16.790 --> 00:03:19.530
Now let's think about the
same thing for the basic plan,

00:03:19.530 --> 00:03:22.080
how we might actually plot that.

00:03:22.080 --> 00:03:26.460
So I'll start with this
point right over here.

00:03:26.460 --> 00:03:30.570
And the basic plan, when I go
to zero sessions it costs me

00:03:30.570 --> 00:03:31.410
$20.00.

00:03:31.410 --> 00:03:35.560
Where would I plot
that on my graph?

00:03:35.560 --> 00:03:38.750
So zero sessions are
going to cost me $20.00.

00:03:38.750 --> 00:03:40.870
That's that point
right over there.

00:03:40.870 --> 00:03:43.700
What about one session.

00:03:43.700 --> 00:03:49.850
Well, one session is
going to cost me $28.00.

00:03:49.850 --> 00:03:53.210
And another way to think about
it, you pay 20 no matter what,

00:03:53.210 --> 00:03:56.730
and as you add one session,
your total cost went up by 8.

00:03:59.390 --> 00:04:01.360
This scale and this
scale is not the same.

00:04:01.360 --> 00:04:03.010
If I did it to
scale, then you would

00:04:03.010 --> 00:04:04.870
see these lines would
look a lot steeper,

00:04:04.870 --> 00:04:07.370
and I would have to make this
axis go a lot, lot taller.

00:04:07.370 --> 00:04:08.700
But hopefully you get the idea.

00:04:08.700 --> 00:04:11.966
You add one session,
we went up by 8.

00:04:11.966 --> 00:04:14.140
You add another
session, so that was

00:04:14.140 --> 00:04:16.670
128, that's that point
right over there.

00:04:16.670 --> 00:04:21.560
So if you have two sessions,
it's going to cost us $36.00.

00:04:21.560 --> 00:04:24.160
Where would we plot that?

00:04:24.160 --> 00:04:26.970
Two sessions are
going to cost $36.00.

00:04:26.970 --> 00:04:29.510
So that is right over there.

00:04:29.510 --> 00:04:31.280
Or another way to think of it.

00:04:31.280 --> 00:04:34.390
We were at 28, we
added another session,

00:04:34.390 --> 00:04:36.940
so we're going to
add $8 to our cost.

00:04:36.940 --> 00:04:41.040
So 28 plus 8 gets us to $36.00

00:04:41.040 --> 00:04:43.851
Now let's do this other point
once again, we can keep going.

00:04:43.851 --> 00:04:45.475
Where would we plot
the three sessions.

00:04:47.980 --> 00:04:51.710
So if we attend three sessions,
it's going to cost us $44.00.

00:04:51.710 --> 00:04:56.800
So that's right about there.

00:04:56.800 --> 00:05:00.220
Once again, we added the session
and we added $8.00 to our cost.

00:05:00.220 --> 00:05:07.850
And you see these two
seem to sit on a line.

00:05:07.850 --> 00:05:10.680
Although I didn't draw
it as well as I can,

00:05:10.680 --> 00:05:16.950
so these two seem
to sit on a line.

00:05:16.950 --> 00:05:19.110
Now this by itself
is interesting.

00:05:19.110 --> 00:05:21.190
But what might be
even more interesting

00:05:21.190 --> 00:05:23.410
is where these two
lines intersect.

00:05:23.410 --> 00:05:26.570
We see that they have different
steepnesses, so at some point

00:05:26.570 --> 00:05:28.149
they should intersect.

00:05:28.149 --> 00:05:30.065
And what's exciting
about, well, actually I'll

00:05:30.065 --> 00:05:31.023
let you think about it.

00:05:31.023 --> 00:05:34.750
What is exciting about where
these two lines intersect?

00:05:34.750 --> 00:05:36.520
What does it tell us
about the sessions,

00:05:36.520 --> 00:05:40.572
and the cost at those points?

00:05:40.572 --> 00:05:42.030
And I'll leave you
there, and we'll

00:05:42.030 --> 00:05:45.090
explore this more
in the next video.


Variables and equations word problem: Which Super Yoga plan is best?

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:02.950
Now that we've set
up our equations,

00:00:02.950 --> 00:00:05.660
let's see if we can answer some
more interesting questions.

00:00:05.660 --> 00:00:06.820
Just as a warm-up,
we're going to do

00:00:06.820 --> 00:00:08.200
a little bit of what we
did in the last video.

00:00:08.200 --> 00:00:09.783
But I'll do it with
the equations now.

00:00:09.783 --> 00:00:13.070
The equations really are just
a mathematical representation

00:00:13.070 --> 00:00:16.422
of the same information
that we had up here.

00:00:16.422 --> 00:00:17.880
So as a starting
point, let's think

00:00:17.880 --> 00:00:20.940
about how much,
under the basic plan,

00:00:20.940 --> 00:00:26.720
it'll cost for us to attend
3 sessions in a given month.

00:00:26.720 --> 00:00:29.020
Well, we've got our
equation right over here.

00:00:29.020 --> 00:00:30.970
Our total cost in
that month is going

00:00:30.970 --> 00:00:40.140
to be equal to 20 plus 8
times the number of sessions.

00:00:40.140 --> 00:00:43.540
The number of sessions is S.
S is 3 in this circumstances,

00:00:43.540 --> 00:00:46.600
so we'll replace
the S with the 3.

00:00:46.600 --> 00:00:48.700
And so you get our cost
in this circumstance

00:00:48.700 --> 00:00:56.150
is going to be 20 plus 8 times
3, or 20 plus 24, or $44.

00:00:56.150 --> 00:00:58.150
Let me erase this
right over here.

00:00:58.150 --> 00:01:00.260
So this is going to cost us $44.

00:01:03.900 --> 00:01:06.230
Now let's do the same
thing under the trial plan.

00:01:06.230 --> 00:01:11.620
How much is it going to cost
us to attend 3 sessions?

00:01:11.620 --> 00:01:13.320
Well, this is more
straightforward,

00:01:13.320 --> 00:01:15.110
or slightly more
straightforward.

00:01:15.110 --> 00:01:20.370
Our total cost is going to be
equal to 12 times the number

00:01:20.370 --> 00:01:21.000
of sessions.

00:01:21.000 --> 00:01:24.130
The number of sessions in this
circumstance are 3 sessions,

00:01:24.130 --> 00:01:27.160
so it's going to be 12 times 3.

00:01:27.160 --> 00:01:31.670
Our total cost is going
to be equal to $36.

00:01:31.670 --> 00:01:33.560
Let me write that down.

00:01:33.560 --> 00:01:36.700
It's going to be equal to $36.

00:01:36.700 --> 00:01:39.550
But we could also answer
more interesting questions

00:01:39.550 --> 00:01:41.110
with these equations.

00:01:41.110 --> 00:01:54.650
For example, how many sessions
can I attend in a month for,

00:01:54.650 --> 00:01:57.550
oh, I don't know, let's
throw out something, $54.

00:02:00.590 --> 00:02:01.960
Let's say that's just my budget.

00:02:01.960 --> 00:02:04.410
When I look at my salary
and how much money

00:02:04.410 --> 00:02:06.380
I have budgeted to
health and fitness,

00:02:06.380 --> 00:02:07.930
this is what I can afford.

00:02:07.930 --> 00:02:10.690
And so based on that, I want
to ask, how many sessions can I

00:02:10.690 --> 00:02:13.690
attend in each of
these situations?

00:02:13.690 --> 00:02:15.750
First, let's think about
it for the trial plan.

00:02:15.750 --> 00:02:19.920
How many sessions
can I attend for $54?

00:02:19.920 --> 00:02:24.650
Now let's just assume that this
is some type of a special yoga

00:02:24.650 --> 00:02:27.030
studio where you can even
attend half a session.

00:02:27.030 --> 00:02:30.020
So if you attend half a
session, they'll charge you $6.

00:02:30.020 --> 00:02:32.640
If you attend a fourth of a
session, they'll charge you $3.

00:02:32.640 --> 00:02:35.490
So they'll actually keep
track of exactly what fraction

00:02:35.490 --> 00:02:36.740
of a session you're attending.

00:02:36.740 --> 00:02:40.430
You don't have to just attend
whole number of sessions.

00:02:40.430 --> 00:02:44.030
You could attend 2 and 1/2, and
they'll bill you accordingly.

00:02:44.030 --> 00:02:48.500
With that said, how many
sessions under the trial plan

00:02:48.500 --> 00:02:49.870
can I attend for $54?

00:02:52.600 --> 00:02:54.780
Well, let's think about
this a little bit.

00:02:54.780 --> 00:02:57.080
My cost now is going to be $54.

00:02:57.080 --> 00:03:00.470
$54 is C. It's my monthly cost.

00:03:00.470 --> 00:03:06.472
So 54 is going to be equal
to $12 per session times

00:03:06.472 --> 00:03:07.430
the number of sessions.

00:03:10.800 --> 00:03:12.180
Now I set up this equation.

00:03:12.180 --> 00:03:13.890
In order to answer
this question,

00:03:13.890 --> 00:03:16.750
I just need to figure out
what's the number of sessions

00:03:16.750 --> 00:03:18.170
that satisfy it.

00:03:18.170 --> 00:03:21.200
Now this S is an
unknown variable.

00:03:21.200 --> 00:03:25.320
What number of sessions
times 12 is equal to 54?

00:03:25.320 --> 00:03:27.160
Well, you might say,
well, if I could

00:03:27.160 --> 00:03:29.519
get just a S on the
right-hand side here,

00:03:29.519 --> 00:03:30.560
then I'll have an answer.

00:03:30.560 --> 00:03:33.200
S will be equal to some value.

00:03:33.200 --> 00:03:35.730
And if I want just a S
on the right-hand side,

00:03:35.730 --> 00:03:39.550
I'd ideally just like
to divide this by 12.

00:03:39.550 --> 00:03:41.910
12S divided by 12 is
just going to give me

00:03:41.910 --> 00:03:44.490
a S. It's just going to give
me the number of sessions.

00:03:44.490 --> 00:03:46.370
But if I have an equal
sign, I can't just

00:03:46.370 --> 00:03:48.730
willy-nilly divide
one side by 12

00:03:48.730 --> 00:03:52.630
without doing the exact same
thing to the other side.

00:03:52.630 --> 00:03:54.780
If that is equal to
that, in order for them

00:03:54.780 --> 00:03:57.620
to both still be equal, I
have to do the same thing

00:03:57.620 --> 00:03:59.020
to both sides.

00:03:59.020 --> 00:04:02.786
So on the right-hand side,
I'm left with just a S.

00:04:02.786 --> 00:04:08.160
And on the left-hand side,
I have 54 divided by 12.

00:04:08.160 --> 00:04:10.800
Let me think about that.

00:04:10.800 --> 00:04:12.150
I'll just work it out.

00:04:12.150 --> 00:04:17.730
12 goes into 54 4 times.

00:04:17.730 --> 00:04:20.350
4 times 12 is 48.

00:04:20.350 --> 00:04:22.510
You're left with
a remainder of 6.

00:04:22.510 --> 00:04:26.690
So you could say
12 goes into 54,

00:04:26.690 --> 00:04:35.010
or you could say 54/12 is equal
to 4 and 6/12 or 4 and 1/2.

00:04:35.010 --> 00:04:39.220
So the number of sessions that
I could attend are 4 and 1/2.

00:04:39.220 --> 00:04:43.170
If I have $54 to spend, I
could attend 4 and 1/2 sessions

00:04:43.170 --> 00:04:45.070
under the trial plan.

00:04:45.070 --> 00:04:46.740
If this was a yoga
studio that said,

00:04:46.740 --> 00:04:49.085
oh, you can't attend
half sessions,

00:04:49.085 --> 00:04:50.460
then you could
say, well, at most

00:04:50.460 --> 00:04:52.835
I could attend four sessions,
because five sessions would

00:04:52.835 --> 00:04:54.500
get me above my budget.

00:04:54.500 --> 00:04:56.820
But I said that you can
attend fractional sessions,

00:04:56.820 --> 00:04:58.870
so we're going to say 4 and 1/2.

00:04:58.870 --> 00:05:05.190
Now let's answer the same
question for the basic plan.

00:05:05.190 --> 00:05:09.255
Under the basic plan, how many
sessions can I attend for $54?

00:05:12.920 --> 00:05:15.540
Well, once again,
this is going to be

00:05:15.540 --> 00:05:17.420
our total cost for that month.

00:05:17.420 --> 00:05:25.390
So we say $54 is equal to
20 plus 8 times the number

00:05:25.390 --> 00:05:25.910
of sessions.

00:05:31.360 --> 00:05:34.450
And once again, I'd like
to just have sessions

00:05:34.450 --> 00:05:36.680
maybe on the right-hand
side of this equation.

00:05:36.680 --> 00:05:39.346
So I could say, oh, sessions are
going to be equal to something.

00:05:39.346 --> 00:05:41.500
This is the number of
sessions that I could attend.

00:05:41.500 --> 00:05:43.750
But I have this 20
here and this 8 here,

00:05:43.750 --> 00:05:46.656
and so I have to think about
how I can get rid of them.

00:05:46.656 --> 00:05:48.280
The first thing I've
got to think about

00:05:48.280 --> 00:05:51.550
is, how can I get rid of this
20 from the right-hand side

00:05:51.550 --> 00:05:52.540
of this equation?

00:05:52.540 --> 00:05:56.570
What am I going to have to
do to the entire equation?

00:05:56.570 --> 00:05:58.850
Well, the simplest thing
is I could subtract 20

00:05:58.850 --> 00:06:00.800
from the right-hand side.

00:06:00.800 --> 00:06:02.870
But if I subtract 20
from the right-hand side,

00:06:02.870 --> 00:06:05.310
I also have to subtract
20 from the left-hand side

00:06:05.310 --> 00:06:08.060
in order to maintain
the equality.

00:06:08.060 --> 00:06:12.620
So I'm left with, on the
left-hand side, 54 minus 20

00:06:12.620 --> 00:06:14.376
gets me 34.

00:06:14.376 --> 00:06:16.000
And that's going to
be equal to-- well,

00:06:16.000 --> 00:06:17.450
the whole point
of subtracting 20

00:06:17.450 --> 00:06:19.630
is so that these
negate each other.

00:06:19.630 --> 00:06:21.090
And then I have just an 8S.

00:06:23.660 --> 00:06:27.970
So 8 times the number of
sessions has to be equal to 34

00:06:27.970 --> 00:06:30.420
in order for 8 times the
number of sessions plus 20

00:06:30.420 --> 00:06:32.200
to be equal to 54.

00:06:32.200 --> 00:06:34.660
And then you could add 20 to
both sides of this equation

00:06:34.660 --> 00:06:36.490
to get our original.

00:06:36.490 --> 00:06:37.320
We're almost done.

00:06:37.320 --> 00:06:39.670
We just want to isolate
the S right here.

00:06:39.670 --> 00:06:41.630
And you can imagine
the easiest way would

00:06:41.630 --> 00:06:44.592
be to divide the
right-hand side by 8.

00:06:44.592 --> 00:06:46.550
But if we're dividing
the right-hand side by 8,

00:06:46.550 --> 00:06:50.120
we have to do the
same to the left.

00:06:50.120 --> 00:06:52.670
And so what do we get for
the number of sessions

00:06:52.670 --> 00:06:55.150
that we would have to attend?

00:06:55.150 --> 00:06:57.370
Well, these cancel out.

00:06:57.370 --> 00:07:00.330
And so you get the
number of sessions

00:07:00.330 --> 00:07:04.330
is equal to 34 divied by 8.

00:07:04.330 --> 00:07:06.640
34 divided by 8 is
equal to, let's see,

00:07:06.640 --> 00:07:11.460
8 goes into 34 four times with
a remainder of 2, so 4 and 2/8,

00:07:11.460 --> 00:07:14.000
or it's 4 and 1/4.

00:07:14.000 --> 00:07:21.370
So I could attend 4 and 1/4
sessions under the basic plan.

00:07:21.370 --> 00:07:25.540
Assuming I can do fractional
sessions, for $54,

00:07:25.540 --> 00:07:28.410
which one can I attend
more sessions in,

00:07:28.410 --> 00:07:32.280
the basic plan or
the trial plan?

00:07:32.280 --> 00:07:35.740
Well, for $54 I could attend
a slightly longer amount

00:07:35.740 --> 00:07:39.069
of sessions under
the trial plan.


Variables and equations word problem: Putting them to work for Super Yoga

WEBVTT
Kind: captions
Language: en

00:00:00.754 --> 00:00:02.903
I'm in the mood to improve my flexibility

00:00:02.903 --> 00:00:06.433
a little bit so I decide to take some Yoga classes

00:00:06.433 --> 00:00:10.172
so I show up at the local yoga place called Super Yoga

00:00:10.172 --> 00:00:12.701
and ask them, "how much does it cost?"

00:00:12.701 --> 00:00:15.737
And they say, "we've got a basic plan"

00:00:15.737 --> 00:00:17.675
"and we've got a trial plan."

00:00:17.675 --> 00:00:19.368
The trial plan, if you just want to try things out,

00:00:19.368 --> 00:00:21.231
you can come to any of the sessions

00:00:21.231 --> 00:00:24.822
and it's going to cost you $12 per 1hr session.

00:00:24.822 --> 00:00:27.035
but if you like what you're doing here

00:00:27.035 --> 00:00:29.075
you might want to get a monthly membership

00:00:29.075 --> 00:00:30.362
That'll be $20/month,

00:00:30.362 --> 00:00:32.452
that you can view that as the basic plan

00:00:32.452 --> 00:00:35.420
It's $20/month and then you get a discount per session

00:00:35.420 --> 00:00:37.569
It'll only be $8 per session

00:00:37.569 --> 00:00:41.590
So this seems interesting, but I'm a little bit confused.

00:00:41.590 --> 00:00:44.084
Which plan should I take?

00:00:44.084 --> 00:00:47.570
So the first place, I might start is to think about

00:00:47.570 --> 00:00:48.956
how much I would pay

00:00:48.956 --> 00:00:52.390
depending on how many sessions I actually take

00:00:52.420 --> 00:00:54.101
And to do a little short hand here

00:00:54.101 --> 00:00:56.282
let's just define some variables

00:00:56.282 --> 00:01:01.763
Let's say that S = "Number of Sessions" I attend per month

00:01:01.763 --> 00:01:07.741
Number of sessions per month that I attend

00:01:07.741 --> 00:01:09.941
I decided to attend at Super Yoga

00:01:09.941 --> 00:01:15.155
And let's say that C = "Total Monthly Cost"

00:01:15.616 --> 00:01:18.549
My Total Monthly Cost

00:01:18.933 --> 00:01:20.941
So with these variables defined this way,

00:01:20.941 --> 00:01:22.849
let's think about how much I would pay

00:01:22.849 --> 00:01:24.141
under each of these plans

00:01:24.141 --> 00:01:26.527
depending on how many sessions I would attend

00:01:26.527 --> 00:01:29.605
So first, let's think about the, we'll start with the Trial Plan

00:01:29.605 --> 00:01:31.147
cause that seems a little simplier

00:01:31.147 --> 00:01:32.755
and I'll draw a little column here

00:01:32.755 --> 00:01:38.768
I have a cost, actually, let me see, I'll draw my number of sessions

00:01:39.807 --> 00:01:44.243
Number of sessions and then I have my cost

00:01:44.243 --> 00:01:47.135
and then I'm going to draw a little table here

00:01:47.135 --> 00:01:50.048
A little table, that'll be for my trial plan

00:01:50.048 --> 00:01:51.570
and then let's also do the same thing,

00:01:51.570 --> 00:01:53.093
since we're doing it for the trial plan

00:01:53.093 --> 00:01:55.507
let's do it for the basic plan so that we can compare

00:01:55.845 --> 00:01:57.884
Let's do it for the basic plan

00:01:57.884 --> 00:01:59.923
so I have the Number of sessions I attend

00:01:59.923 --> 00:02:03.768
and my Total Cost

00:02:04.521 --> 00:02:08.887
So let's first think about, if I decide to attend no sessions

00:02:08.887 --> 00:02:12.744
So if I decided to attend no sessions

00:02:12.744 --> 00:02:15.567
under my trial plan, what will be my cost?

00:02:16.905 --> 00:02:20.833
Well, $12 per session, 12 x no sessions

00:02:20.833 --> 00:02:22.369
Well, I'm not going to have to pay anything

00:02:22.369 --> 00:02:25.898
My cost is going to be zero

00:02:25.898 --> 00:02:28.036
Now what about that same question

00:02:28.036 --> 00:02:29.091
under the basic plan

00:02:29.091 --> 00:02:33.102
If I have the basic plan, but in a given month

00:02:33.102 --> 00:02:35.485
I attend no sessions,

00:02:35.485 --> 00:02:38.036
I don't go to the gym, I don't go to this yoga gym at all

00:02:38.036 --> 00:02:40.100
How much am I going to have to pay?

00:02:40.946 --> 00:02:42.615
Well, it's $8 per sessions

00:02:42.615 --> 00:02:44.230
I didn't have to go to any sessions

00:02:44.230 --> 00:02:45.562
So I'm not going to have to pay anything

00:02:45.562 --> 00:02:46.894
on a per session basis

00:02:46.894 --> 00:02:49.418
but I will have to pay that $20/month

00:02:49.418 --> 00:02:53.239
So I will have to pay $20

00:02:53.239 --> 00:02:55.076
even though I didn't even attend

00:02:55.691 --> 00:02:57.923
That doesn't seem so good in that scenario

00:02:57.923 --> 00:03:00.365
But let's keep working through other scenarios

00:03:00.365 --> 00:03:04.286
Let's think about the scenario where I attend 1 session

00:03:05.024 --> 00:03:07.249
Where I attend 1 session

00:03:07.249 --> 00:03:11.011
Under the trial plan, how much will I have to pay?

00:03:11.380 --> 00:03:15.741
Well, it's $12/session x 1 session

00:03:15.741 --> 00:03:21.561
I'm going to pay $12

00:03:21.637 --> 00:03:24.476
Let's think about that same scenario under the basic plan

00:03:24.476 --> 00:03:27.011
Under the basic plan, if I attend 1 session

00:03:27.011 --> 00:03:32.070
Well, it's $8/session x 1 session

00:03:32.070 --> 00:03:33.173
$8 for that

00:03:33.173 --> 00:03:35.257
Plus just the basic monthly

00:03:35.257 --> 00:03:37.967
So I'm just going to have to pay $20 + ($8 x 1)

00:03:37.967 --> 00:03:39.209
So $28.

00:03:39.209 --> 00:03:43.981
I'm going to have to pay $28

00:03:43.981 --> 00:03:46.703
So still that trial plan still looks pretty good

00:03:46.703 --> 00:03:49.392
even if I attend 1 session

00:03:49.392 --> 00:03:50.303
Let me make it clear

00:03:50.303 --> 00:03:51.303
That's in dollars

00:03:51.303 --> 00:03:53.506
and that over here is $28

00:03:53.506 --> 00:03:54.560
and I could keep going

00:03:54.560 --> 00:03:55.615
and I encourage you to keep going

00:03:55.615 --> 00:03:56.902
but let's try 1 more

00:03:56.902 --> 00:03:59.433
just to see how, just to get a feel for the numbers here

00:03:59.433 --> 00:04:02.989
If I attend 2 sessions under the trial plan

00:04:02.989 --> 00:04:05.589
How much am I going to pay?

00:04:05.589 --> 00:04:08.368
Well, it's $12/session x 2 sessions

00:04:08.368 --> 00:04:13.595
i'm going to pay $24

00:04:14.588 --> 00:04:16.289
Let's think about the basic plan

00:04:16.289 --> 00:04:17.969
if I attend 2 sessions

00:04:17.969 --> 00:04:20.800
-- let me do that in a yellow colour --

00:04:20.800 --> 00:04:22.911
if I attend 2 sessions

00:04:23.096 --> 00:04:27.205
2 x $8/session, that's going to $16

00:04:27.205 --> 00:04:30.152
plus the $20 I'm going to have to spend every month

00:04:30.152 --> 00:04:35.415
So it's going to 2 x $8

00:04:35.815 --> 00:04:38.679
+ $20

00:04:38.679 --> 00:04:41.389
= $16 + $20

00:04:41.389 --> 00:04:44.100
= $36

00:04:44.100 --> 00:04:46.507
So at least for the scenarios that we set up here

00:04:46.507 --> 00:04:49.151
if I attend 0 or 1 or 2 sessions

00:04:49.151 --> 00:04:51.552
the trial plan seems to be winning out

00:04:51.552 --> 00:04:52.777
but I want to explore at what point

00:04:52.777 --> 00:04:56.161
does the trial plan actually become a little bit worse

00:04:56.161 --> 00:04:57.278
But before we do that

00:04:57.278 --> 00:04:59.893
let's think about if we can represent this

00:04:59.893 --> 00:05:00.970
this a little algebraically

00:05:00.970 --> 00:05:03.173
because it's going to allow us to be a little bit more precise

00:05:03.173 --> 00:05:06.036
with coming up with our answers

00:05:06.036 --> 00:05:09.168
So if we say that, S is the "Number of Sessions per Month"

00:05:09.168 --> 00:05:11.257
and C is the "Monthly Cost"

00:05:11.257 --> 00:05:15.303
How can we express the trial plan as an equation?

00:05:16.810 --> 00:05:19.092
Well, we could say our Total Cost

00:05:19.092 --> 00:05:20.911
our Total Monthly Cost

00:05:20.911 --> 00:05:22.694
so this is for our trial plan right over here

00:05:22.694 --> 00:05:24.278
let me draw a dotted line over here to show

00:05:24.278 --> 00:05:25.968
well, the dotted line goes around there

00:05:25.968 --> 00:05:29.814
so under the trial plan our total cost is going to be equal to

00:05:29.814 --> 00:05:37.559
well, it's $12/session x # of sessions

00:05:37.559 --> 00:05:41.864
times S

00:05:41.864 --> 00:05:46.159
So under the trial plan I could say my total cost is equal to

00:05:46.159 --> 00:05:48.902
$12 x # of Sessions

00:05:48.902 --> 00:05:52.101
$12/session x the Number of Sessions

00:05:52.101 --> 00:05:53.948
Let's so the same thing with the basic plan

00:05:53.948 --> 00:05:56.748
How can we express that as an equation?

00:05:58.701 --> 00:06:00.752
Well, we have our total cost

00:06:00.752 --> 00:06:03.166
our total cost is going to be equal to

00:06:03.166 --> 00:06:05.507
well, regardless of what we do

00:06:05.507 --> 00:06:07.676
any given month, we're going to have to pay

00:06:07.676 --> 00:06:09.633
the $20/month

00:06:09.633 --> 00:06:11.368
we're going to have to pay that 20

00:06:11.368 --> 00:06:12.270
So no matter what we do,

00:06:12.270 --> 00:06:13.172
we're going to have to pay that 20

00:06:13.172 --> 00:06:14.488
just from the get-go

00:06:14.488 --> 00:06:17.923
and then we're going to pay $8/session

00:06:17.923 --> 00:06:34.005
so it's going to be $20 + ($8 x # of Sessions)

00:06:34.005 --> 00:06:35.793
so that's interesting and you can see

00:06:35.793 --> 00:06:38.430
if you put S is 0 here,

00:06:38.430 --> 00:06:39.580
if you make S = 0

00:06:39.626 --> 00:06:42.300
you get 20 + (8 x 0) which is 20

00:06:42.300 --> 00:06:46.547
if you say S is 1, you get 20 + (8 x 1)

00:06:46.547 --> 00:06:47.492
which is 28

00:06:47.492 --> 00:06:49.967
so you see that each of these S's and C's

00:06:49.967 --> 00:06:52.369
they satisfy this equation

00:06:52.369 --> 00:06:53.786
Same thing over here

00:06:53.786 --> 00:06:55.204
and we can keep trying more and more

00:06:55.204 --> 00:06:56.832
What's neat about these equations is

00:06:56.832 --> 00:06:59.100
just this equation encapsulates

00:06:59.100 --> 00:07:01.369
all of the possible combinations here

00:07:01.369 --> 00:07:03.594
and just this equation encapsulates

00:07:03.594 --> 00:07:08.623
all the possible combinations there

00:07:08.623 --> 00:07:10.189
and so for the next few videos,

00:07:10.189 --> 00:07:11.483
what I want to do is explore

00:07:11.483 --> 00:07:13.444
How can we use these equations

00:07:13.444 --> 00:07:15.405
to come up with more insights as to

00:07:15.405 --> 00:07:18.533
which plan is better for me


Synthetic division example 2

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.975
Let's do another synthetic
division example.

00:00:02.975 --> 00:00:04.670
And in another
video, we actually

00:00:04.670 --> 00:00:08.310
have the why this works relative
to algebraic long division.

00:00:08.310 --> 00:00:10.040
But here it's going
to be another just,

00:00:10.040 --> 00:00:11.460
let's go through
the process of it

00:00:11.460 --> 00:00:13.450
just so that you get
comfortable with it.

00:00:13.450 --> 00:00:16.770
And now is a good chance to
give it a shot, to actually try

00:00:16.770 --> 00:00:21.200
to simplify this
rational expression.

00:00:21.200 --> 00:00:23.370
So let's think about
this step by step.

00:00:23.370 --> 00:00:24.910
So the first thing
I want to do is

00:00:24.910 --> 00:00:28.190
write all of the coefficients
of the numerator.

00:00:28.190 --> 00:00:31.910
So I have a 2.

00:00:31.910 --> 00:00:33.480
Oh, I have to be careful here.

00:00:33.480 --> 00:00:35.870
Because the 2 is the
coefficient for x to the fifth,

00:00:35.870 --> 00:00:37.860
I have no x to the fourth term.

00:00:41.350 --> 00:00:42.120
Let me start over.

00:00:42.120 --> 00:00:45.500
So I have the 2 from
2x to the fifth.

00:00:45.500 --> 00:00:47.010
And then I have no
x to the fourth.

00:00:47.010 --> 00:00:48.910
So it's really 0x to the fourth.

00:00:48.910 --> 00:00:51.090
So I'll put a 0
as the coefficient

00:00:51.090 --> 00:00:52.910
for the x to the fourth term.

00:00:52.910 --> 00:00:56.540
And then I have a negative
1 times x to the third.

00:00:56.540 --> 00:00:58.965
And then I have a positive
3 times x squared.

00:01:01.690 --> 00:01:03.320
Negative 2 times x.

00:01:06.800 --> 00:01:10.410
And then I have a constant
term, or zero degree term, of 7.

00:01:10.410 --> 00:01:12.920
I just have a positive 7.

00:01:12.920 --> 00:01:17.360
And now let me just draw my
little funky synthetic division

00:01:17.360 --> 00:01:19.870
operator-looking symbol.

00:01:19.870 --> 00:01:23.040
And remember, the type of
synthetic division we're doing,

00:01:23.040 --> 00:01:27.030
it only applies when we are
dividing by an x plus or minus

00:01:27.030 --> 00:01:28.190
something.

00:01:28.190 --> 00:01:29.720
There's a slightly
different process

00:01:29.720 --> 00:01:33.840
you would have to do if it
was 3x or if was negative 1x

00:01:33.840 --> 00:01:35.740
or if it was 5x squared.

00:01:35.740 --> 00:01:39.250
This only works when we have
x plus or minus something.

00:01:39.250 --> 00:01:42.200
In this case we have x minus 3.

00:01:42.200 --> 00:01:44.150
So we have the negative 3 here.

00:01:47.496 --> 00:01:48.870
And the process
we show-- there's

00:01:48.870 --> 00:01:51.950
other ways of doing it-- is
you take the negative of this.

00:01:51.950 --> 00:01:56.640
So the negative of
negative 3 is positive 3.

00:01:56.640 --> 00:02:00.200
And now we're ready to perform
our synthetic division.

00:02:00.200 --> 00:02:05.950
So we'll bring down
this 2 and then multiply

00:02:05.950 --> 00:02:07.780
the 2 times the 3.

00:02:07.780 --> 00:02:11.900
2 time 3 gives us 6.

00:02:11.900 --> 00:02:15.680
0 plus 6 is 6.

00:02:15.680 --> 00:02:21.520
And then we multiply that times
the 3, and we get positive 18.

00:02:21.520 --> 00:02:28.300
Negative 1 plus 18 is 17.

00:02:28.300 --> 00:02:32.226
Multiply that times the 3.

00:02:32.226 --> 00:02:37.690
17 times 3 is 51.

00:02:37.690 --> 00:02:40.350
3 plus 51 is 54.

00:02:43.210 --> 00:02:44.700
Multiply that times 3.

00:02:44.700 --> 00:02:46.556
The numbers are getting
kind of large now.

00:02:46.556 --> 00:02:47.680
So that's going to be what?

00:02:47.680 --> 00:02:49.562
50 times 3 is 150.

00:02:49.562 --> 00:02:51.440
4 times 3 is 12.

00:02:51.440 --> 00:02:55.020
So this is going to be 162.

00:02:55.020 --> 00:03:02.200
Negative 2 plus 162 is 160.

00:03:02.200 --> 00:03:08.890
And then finally, 160
times 3 is going to be 480.

00:03:08.890 --> 00:03:15.330
And you add 480 to
7, and you get 487.

00:03:15.330 --> 00:03:19.650
And you can think of it, I only
have one term or one number

00:03:19.650 --> 00:03:22.060
to the left-hand side
of this bar here.

00:03:22.060 --> 00:03:25.050
Or I'm just doing the standard,
traditional x plus or minus

00:03:25.050 --> 00:03:29.660
something version of synthetic
division, I should say.

00:03:29.660 --> 00:03:31.810
So I can separate
this out, and now I've

00:03:31.810 --> 00:03:33.380
essentially gotten my answer.

00:03:33.380 --> 00:03:36.089
And it looks like voodoo,
and it kind of is voodoo.

00:03:36.089 --> 00:03:37.630
And that's why I
don't like to do it,

00:03:37.630 --> 00:03:39.764
because you're just
memorizing an algorithm.

00:03:39.764 --> 00:03:41.680
But there are other
videos why we explain why.

00:03:41.680 --> 00:03:43.660
And it can be fast
and convenient

00:03:43.660 --> 00:03:46.820
and paper saving very often,
like you see right here.

00:03:46.820 --> 00:03:48.520
But then we have
our final answer.

00:03:48.520 --> 00:03:51.560
It's going to be-- and
let me work backwards.

00:03:51.560 --> 00:03:53.270
So I'll start with
our remainder.

00:03:53.270 --> 00:03:57.170
So our remainder is 487.

00:03:57.170 --> 00:04:00.300
And it's going to be
487 over x minus 3.

00:04:04.950 --> 00:04:06.970
And so this is
our constant term.

00:04:06.970 --> 00:04:13.540
And so you're going to have plus
160 plus 487 over x minus 3.

00:04:13.540 --> 00:04:15.250
Now this is our x term.

00:04:15.250 --> 00:04:19.730
So it's going to be
54x plus all of this.

00:04:19.730 --> 00:04:21.930
This is going to be
our x squared term.

00:04:21.930 --> 00:04:27.720
So this is going to be 17x
squared plus 54x plus 160

00:04:27.720 --> 00:04:28.790
and all of that.

00:04:28.790 --> 00:04:31.170
Then this is going to
be x to the third term.

00:04:31.170 --> 00:04:35.320
So this is going to be 6x to
the third plus all of that.

00:04:35.320 --> 00:04:38.000
And then finally, this is
our x to the fourth term--

00:04:38.000 --> 00:04:39.240
2x to the fourth.

00:04:39.240 --> 00:04:42.700
And let me erase this.

00:04:42.700 --> 00:04:46.620
So then I have my x
to the fourth term.

00:04:46.620 --> 00:04:50.180
So it is 2x to the fourth.

00:04:50.180 --> 00:04:51.520
And we are done.

00:04:51.520 --> 00:04:54.540
This thing simplifies
to this right over here.

00:04:54.540 --> 00:04:58.000
And I encourage you to verify it
with traditional algebraic long

00:04:58.000 --> 00:04:59.199
division.


Why synthetic division works

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.320
What I want to do
now is simplify

00:00:02.320 --> 00:00:04.560
the exact same
expression but do it

00:00:04.560 --> 00:00:07.360
with traditional
algebraic long division.

00:00:07.360 --> 00:00:10.820
And hopefully we'll see why
synthetic division actually

00:00:10.820 --> 00:00:12.260
gives us the exact same result.

00:00:12.260 --> 00:00:13.760
We'll be able to
see the connections

00:00:13.760 --> 00:00:17.370
between synthetic division
and algebraic long division.

00:00:17.370 --> 00:00:18.770
So let's get started.

00:00:18.770 --> 00:00:20.550
So if we're doing
algebraic long division

00:00:20.550 --> 00:00:22.466
and set it up right over
here, the first thing

00:00:22.466 --> 00:00:24.300
we want to think about
is how many times

00:00:24.300 --> 00:00:27.310
does the highest degree
term here, which is our x,

00:00:27.310 --> 00:00:29.220
go into the highest
degree term here,

00:00:29.220 --> 00:00:31.800
which is our 3x to
the third power.

00:00:31.800 --> 00:00:37.640
Well x goes into 3x to the
third power 3x squared times.

00:00:37.640 --> 00:00:39.710
So we'll write it in
the x squared place,

00:00:39.710 --> 00:00:43.230
3x squared times.

00:00:43.230 --> 00:00:45.450
Now you might already
see your parallel.

00:00:45.450 --> 00:00:47.230
When we did the
synthetic division,

00:00:47.230 --> 00:00:50.510
we dropped this 3
straight down, and this 3

00:00:50.510 --> 00:00:52.780
represented 3x squared.

00:00:52.780 --> 00:00:55.730
So this 3 and this
3x squared are really

00:00:55.730 --> 00:00:57.630
representing the same thing.

00:00:57.630 --> 00:00:59.284
But you might be
saying, well here

00:00:59.284 --> 00:01:00.450
we have to do some thinking.

00:01:00.450 --> 00:01:05.360
We had to say x goes into 3x
to the third 3x squared times.

00:01:05.360 --> 00:01:08.630
Here, we just mindlessly
dropped this 3 straight down.

00:01:08.630 --> 00:01:10.450
How did that work?

00:01:10.450 --> 00:01:14.410
The reason why we were able to
mindlessly drop this 3 straight

00:01:14.410 --> 00:01:16.790
down is because
we assumed, to do

00:01:16.790 --> 00:01:18.930
this basic type of
synthetic division,

00:01:18.930 --> 00:01:21.380
that we had just an
x right over here.

00:01:21.380 --> 00:01:22.550
We didn't have a 3x.

00:01:22.550 --> 00:01:23.540
We didn't have a 4x.

00:01:23.540 --> 00:01:24.840
We didn't have an x squared.

00:01:24.840 --> 00:01:26.680
We just had an x.

00:01:26.680 --> 00:01:31.170
And if you divide an x into
whatever this highest degree

00:01:31.170 --> 00:01:33.280
term is, your
coefficient is going

00:01:33.280 --> 00:01:35.130
to be the exact same thing.

00:01:35.130 --> 00:01:37.370
It's just going to
be one degree lower.

00:01:37.370 --> 00:01:39.900
So you go from a 3x
to the third to 3x

00:01:39.900 --> 00:01:42.360
squared-- so the exact
same coefficient.

00:01:42.360 --> 00:01:46.000
And now, in our little synthetic
division right over here,

00:01:46.000 --> 00:01:48.650
that right over there
is the x squared term.

00:01:48.650 --> 00:01:51.370
So you went from 3x to
the third to 3x squared.

00:01:51.370 --> 00:01:53.750
We essentially divided it by x.

00:01:53.750 --> 00:01:56.550
And we could blindly do it
because we knew, we assumed,

00:01:56.550 --> 00:01:59.590
that we were dealing
with just a 1x

00:01:59.590 --> 00:02:01.520
to the first right over here.

00:02:01.520 --> 00:02:03.950
But let's keep on going
and see the parallels

00:02:03.950 --> 00:02:07.260
and see why we're essentially
doing the exact same thing.

00:02:07.260 --> 00:02:09.620
Now let's take this 3x
squared and multiply

00:02:09.620 --> 00:02:11.380
that times x plus 4.

00:02:11.380 --> 00:02:15.105
So 3x squared times
x-- I'll do it

00:02:15.105 --> 00:02:19.060
in that white color-- it's
going to be 3x to the third.

00:02:19.060 --> 00:02:26.690
And 3x squared times 4 is
going to be 12x squared.

00:02:26.690 --> 00:02:29.200
And now we'll want
to subtract this.

00:02:29.200 --> 00:02:31.540
So now we subtract.

00:02:31.540 --> 00:02:34.850
We subtract this out.

00:02:34.850 --> 00:02:37.200
These guys cancel
out, and you have 4x

00:02:37.200 --> 00:02:39.920
squared minus 12x squared.

00:02:39.920 --> 00:02:45.070
And so that will give
you negative 8x squared.

00:02:45.070 --> 00:02:47.640
So once again, you're probably
seeing some parallels.

00:02:47.640 --> 00:02:49.760
You had the 4x
squared over here.

00:02:49.760 --> 00:02:51.770
You have the 4x
squared over there.

00:02:51.770 --> 00:02:54.530
We just wrote the coefficient,
but that's what it represented.

00:02:54.530 --> 00:02:56.870
4x squared, we
wrote the 4 there.

00:02:56.870 --> 00:03:00.240
Then we essentially
subtracted 12x squared.

00:03:00.240 --> 00:03:05.620
And the way we got that 12,
we multiplied 3 times 4,

00:03:05.620 --> 00:03:07.100
and then we subtracted.

00:03:07.100 --> 00:03:10.360
Here we're multiplying
3 times negative 4.

00:03:10.360 --> 00:03:14.027
We're essentially multiplying
3 times 4 and then subtracting.

00:03:14.027 --> 00:03:15.610
That's why we put
that negative there,

00:03:15.610 --> 00:03:17.109
so we don't have
to keep remembering

00:03:17.109 --> 00:03:18.000
to subtract this row.

00:03:18.000 --> 00:03:19.540
So we could just
keep adding them.

00:03:19.540 --> 00:03:20.998
But that's essentially
what we did.

00:03:20.998 --> 00:03:22.930
We multiplied 3 times this 4.

00:03:22.930 --> 00:03:24.060
And now we subtract it.

00:03:24.060 --> 00:03:26.700
We get that negative
12x squared.

00:03:26.700 --> 00:03:30.447
And then we subtracted, and
you got negative 8x squared.

00:03:30.447 --> 00:03:32.530
And you might say, oh, is
this the same negative 8

00:03:32.530 --> 00:03:33.980
as this right over here?

00:03:33.980 --> 00:03:37.240
Not quite yet,
because over here,

00:03:37.240 --> 00:03:40.630
this negative 8 literally
represents negative 8x.

00:03:40.630 --> 00:03:42.810
This is actually part
of our simplification.

00:03:42.810 --> 00:03:44.510
When we divide
this into that, we

00:03:44.510 --> 00:03:47.880
got 3x squared minus 8x plus 30.

00:03:47.880 --> 00:03:50.660
So over here in the
algebraic long division,

00:03:50.660 --> 00:03:54.010
we then say, how many
times does x plus 4

00:03:54.010 --> 00:03:56.820
go into negative 8x squared?

00:03:56.820 --> 00:04:03.360
Well x goes into negative 8x
squared negative 8x times.

00:04:03.360 --> 00:04:05.090
So this is the key
right over here. x

00:04:05.090 --> 00:04:10.877
goes into it negative 8x times.

00:04:10.877 --> 00:04:12.710
And once again, the
reason why we could just

00:04:12.710 --> 00:04:14.920
put this negative
8 here is we know

00:04:14.920 --> 00:04:17.740
that we are dividing
just by a 1x.

00:04:17.740 --> 00:04:21.519
So you're going to have
the exact same coefficient,

00:04:21.519 --> 00:04:23.170
just one degree lower.

00:04:23.170 --> 00:04:25.600
So this right over
here is our x term,

00:04:25.600 --> 00:04:28.280
and you see it right over
there, just like that.

00:04:28.280 --> 00:04:30.820
So a lot of the simplification
just comes from the idea

00:04:30.820 --> 00:04:33.280
that we are assuming with
the synthetic division

00:04:33.280 --> 00:04:35.020
that this is a 1x.

00:04:35.020 --> 00:04:36.460
But let's just keep going.

00:04:36.460 --> 00:04:40.590
So you have a negative 8x times
this business right over here

00:04:40.590 --> 00:04:46.840
gives you negative 8x times
x is negative 8x squared.

00:04:46.840 --> 00:04:57.080
And then you have negative 8x
times 4, which is negative 32x.

00:04:57.080 --> 00:04:59.080
And we can bring down all
of this business right

00:04:59.080 --> 00:05:01.205
over here, just so it
becomes a little bit simpler.

00:05:01.205 --> 00:05:03.380
So you have a negative 2x.

00:05:03.380 --> 00:05:07.855
And then over here,
you have a minus 1.

00:05:07.855 --> 00:05:09.230
And once again,
when you're doing

00:05:09.230 --> 00:05:11.010
traditional algebraic
long division,

00:05:11.010 --> 00:05:13.349
you're going to subtract
this from that up there.

00:05:13.349 --> 00:05:14.890
So if we're going
to subtract, that's

00:05:14.890 --> 00:05:16.150
like adding the negative.

00:05:18.810 --> 00:05:20.820
These characters cancel out.

00:05:20.820 --> 00:05:24.080
You have a negative 2x plus 32x.

00:05:24.080 --> 00:05:28.480
That gives us a positive 30x.

00:05:28.480 --> 00:05:32.711
And then we can bring down
that negative 1 if we want.

00:05:32.711 --> 00:05:34.210
I'll do that yellow
color, actually.

00:05:34.210 --> 00:05:37.120
We'll bring down
that negative 1.

00:05:37.120 --> 00:05:39.660
So this 30 has the
same coefficient here.

00:05:39.660 --> 00:05:41.480
But this 30 should be up here.

00:05:41.480 --> 00:05:43.760
This is going to be part
of our final answer.

00:05:43.760 --> 00:05:46.880
And to get that, once again,
it all comes from the fact

00:05:46.880 --> 00:05:48.550
that we know that
we had an x here

00:05:48.550 --> 00:05:50.190
when we did the
synthetic division.

00:05:50.190 --> 00:05:56.740
30x divided by x is
just going to be 30.

00:05:56.740 --> 00:05:59.856
That 30 and this 30 is
the exact same thing.

00:05:59.856 --> 00:06:00.730
And then we multiply.

00:06:00.730 --> 00:06:03.000
30 times x is 30x.

00:06:05.417 --> 00:06:07.500
Actually, let me write the
30x in that white color

00:06:07.500 --> 00:06:10.780
because that's the convention
I've been using-- 30x.

00:06:10.780 --> 00:06:16.260
30 times 4 is 120.

00:06:16.260 --> 00:06:19.140
And then we are going to
subtract this from that.

00:06:19.140 --> 00:06:28.700
So we get negative 1
minus 120 is negative 121,

00:06:28.700 --> 00:06:32.590
which, right over there,
is our remainder-- which

00:06:32.590 --> 00:06:35.760
is exactly what
we got over there.

00:06:35.760 --> 00:06:38.420
So hopefully you
see the connection.

00:06:38.420 --> 00:06:40.480
Because we are assuming
that we are dividing

00:06:40.480 --> 00:06:42.690
by x plus or minus
something, we were

00:06:42.690 --> 00:06:44.710
able to make some
simplifying assumptions.

00:06:44.710 --> 00:06:46.654
Whenever you divide
this by an x,

00:06:46.654 --> 00:06:49.070
you know it's going to have
the same coefficient, just one

00:06:49.070 --> 00:06:51.740
degree lower.

00:06:51.740 --> 00:06:53.325
And we kept doing that.

00:06:53.325 --> 00:06:55.450
And so it allowed us to do
it a little bit simpler,

00:06:55.450 --> 00:06:58.665
a little bit faster,
and using less space.


Synthetic division

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.400
In this expression, we're
dividing this third degree

00:00:03.400 --> 00:00:06.300
polynomial by this
first degree polynomial.

00:00:06.300 --> 00:00:08.360
And we could simplify
this by using

00:00:08.360 --> 00:00:10.229
traditional algebraic
long division.

00:00:10.229 --> 00:00:12.020
But what we're going
to cover in this video

00:00:12.020 --> 00:00:13.436
is a slightly
different technique,

00:00:13.436 --> 00:00:15.990
and we call it
synthetic division.

00:00:15.990 --> 00:00:17.640
And synthetic division
is going to seem

00:00:17.640 --> 00:00:20.470
like a little bit of voodoo
in the context of this video.

00:00:20.470 --> 00:00:21.886
In the next few
videos we're going

00:00:21.886 --> 00:00:24.450
to think about why it actually
makes sense, why you actually

00:00:24.450 --> 00:00:28.690
get the same result as
traditional algebraic long

00:00:28.690 --> 00:00:29.540
division.

00:00:29.540 --> 00:00:32.830
My personal tastes are not
to like synthetic division

00:00:32.830 --> 00:00:35.240
because it is very,
very, very algorithmic.

00:00:35.240 --> 00:00:38.150
I prefer to do traditional
algebraic long division.

00:00:38.150 --> 00:00:40.670
But I think you'll see that
this has some advantages.

00:00:40.670 --> 00:00:41.950
It can be faster.

00:00:41.950 --> 00:00:44.900
And it also uses a lot
less space on your paper.

00:00:44.900 --> 00:00:47.380
So let's actually perform
this synthetic division.

00:00:47.380 --> 00:00:49.750
Let's actually simplify
this expression.

00:00:49.750 --> 00:00:52.640
Before we start, there's
two important things

00:00:52.640 --> 00:00:53.450
to keep in mind.

00:00:53.450 --> 00:00:55.400
We're doing, kind of,
the most basic form

00:00:55.400 --> 00:00:56.730
of synthetic division.

00:00:56.730 --> 00:00:59.940
And to do this most basic
algorithm, this most basic

00:00:59.940 --> 00:01:01.650
process, you have
to look for two

00:01:01.650 --> 00:01:04.370
things in this
bottom expression.

00:01:04.370 --> 00:01:09.970
The first is that it has to
be a polynomial of degree 1.

00:01:09.970 --> 00:01:11.262
So you have just an x here.

00:01:11.262 --> 00:01:13.220
You don't have an x
squared, an x to the third,

00:01:13.220 --> 00:01:15.220
an x to the fourth or
anything like that.

00:01:15.220 --> 00:01:19.470
The other thing is, is that
the coefficient here is a 1.

00:01:19.470 --> 00:01:21.910
There are ways to do it if
the coefficient was different,

00:01:21.910 --> 00:01:23.493
but then our synthetic
division, we'll

00:01:23.493 --> 00:01:26.329
have to add a little bit of
bells and whistles to it.

00:01:26.329 --> 00:01:27.870
So in general, what
I'm going to show

00:01:27.870 --> 00:01:30.170
you now will work if
you have something

00:01:30.170 --> 00:01:33.580
of the form x plus or
minus something else.

00:01:33.580 --> 00:01:35.230
So with that said,
let's actually

00:01:35.230 --> 00:01:38.114
perform the synthetic division.

00:01:38.114 --> 00:01:39.530
So the first thing
I'm going to do

00:01:39.530 --> 00:01:42.260
is write all the coefficients
for this polynomial

00:01:42.260 --> 00:01:43.800
that's in the numerator.

00:01:43.800 --> 00:01:45.230
So let's write all of them.

00:01:45.230 --> 00:01:47.210
So we have a 3.

00:01:47.210 --> 00:01:50.780
We have a 4, that's
a positive 4.

00:01:50.780 --> 00:01:54.460
We have a negative 2.

00:01:54.460 --> 00:01:55.900
And a negative 1.

00:01:59.670 --> 00:02:02.380
And you'll see different people
draw different types of signs

00:02:02.380 --> 00:02:04.220
here depending on how they're
doing synthetic division.

00:02:04.220 --> 00:02:05.800
But this is the
most traditional.

00:02:05.800 --> 00:02:07.350
And you want to leave
some space right here

00:02:07.350 --> 00:02:08.473
for another row of numbers.

00:02:08.473 --> 00:02:11.000
So that's why I've gone
all the way down here.

00:02:11.000 --> 00:02:13.130
And then we look
at the denominator.

00:02:13.130 --> 00:02:15.200
And in particular, we're
going to look whatever

00:02:15.200 --> 00:02:17.340
x plus or minus is,
right over here.

00:02:17.340 --> 00:02:20.570
So we'll look at, right over
here, we have a positive 4.

00:02:20.570 --> 00:02:24.540
Instead of writing a positive 4,
we write the negative of that.

00:02:24.540 --> 00:02:30.090
So we write the negative,
which would be negative 4.

00:02:33.470 --> 00:02:35.250
And now we are
all set up, and we

00:02:35.250 --> 00:02:38.660
are ready to perform
our synthetic division.

00:02:38.660 --> 00:02:40.150
And it's going to
seem like voodoo.

00:02:40.150 --> 00:02:43.350
In future videos, we'll
explain why this works.

00:02:43.350 --> 00:02:45.700
So first, this first
coefficient, we literally just

00:02:45.700 --> 00:02:47.130
bring it straight down.

00:02:47.130 --> 00:02:48.990
And so you put the 3 there.

00:02:48.990 --> 00:02:53.200
Then you multiply what you
have here times the negative 4.

00:02:53.200 --> 00:02:55.820
So you multiply it
times the negative 4.

00:02:55.820 --> 00:02:59.840
3 times negative
4 is negative 12.

00:02:59.840 --> 00:03:02.820
Then you add the 4
to the negative 12.

00:03:02.820 --> 00:03:06.960
4 plus negative
12 is negative 8.

00:03:06.960 --> 00:03:10.500
Then you multiply negative
8 times the negative 4.

00:03:10.500 --> 00:03:12.480
I think you see the pattern.

00:03:12.480 --> 00:03:17.610
Negative 8 times negative
4 is positive 32.

00:03:17.610 --> 00:03:21.180
Now we add negative
2 plus positive 32.

00:03:21.180 --> 00:03:24.400
That gives us positive 30.

00:03:24.400 --> 00:03:28.600
Then you multiply the positive
30 times the negative 4.

00:03:28.600 --> 00:03:34.200
And that gives you negative 120.

00:03:34.200 --> 00:03:38.220
And then you add the negative
1 plus the negative 120.

00:03:38.220 --> 00:03:43.272
And you end up with
a negative 121.

00:03:43.272 --> 00:03:44.980
Now the last thing
you do is say, well, I

00:03:44.980 --> 00:03:45.970
have one term here.

00:03:45.970 --> 00:03:47.924
And in this plain,
vanilla, simple version

00:03:47.924 --> 00:03:50.090
of synthetic division, we're
only dealing, actually,

00:03:50.090 --> 00:03:51.820
when you have x plus
or minus something.

00:03:51.820 --> 00:03:53.760
So you're only going
to have one term there.

00:03:53.760 --> 00:03:57.760
So you separate out one term
from the right, just like that.

00:03:57.760 --> 00:03:59.770
And we essentially
have our answer,

00:03:59.770 --> 00:04:02.250
even though it
seems like voodoo.

00:04:02.250 --> 00:04:07.230
So to simplify this, you get,
and you could have a drum roll

00:04:07.230 --> 00:04:11.040
right over here,
this right over here,

00:04:11.040 --> 00:04:13.750
it's going to be
a constant term.

00:04:13.750 --> 00:04:15.460
You could think of it
as a degree 0 term.

00:04:15.460 --> 00:04:16.769
This is going to be an x term.

00:04:16.769 --> 00:04:18.928
And this is going to
be an x squared term.

00:04:18.928 --> 00:04:20.594
You can kind of just
build up from here,

00:04:20.594 --> 00:04:22.520
saying this first one is
going to be a constant.

00:04:22.520 --> 00:04:24.810
Then this is going to be an
x term, then an x squared.

00:04:24.810 --> 00:04:26.851
If we had more you'd have
an x to the third, an x

00:04:26.851 --> 00:04:28.740
to the fourth, so
on and so forth.

00:04:28.740 --> 00:04:41.750
So this is going to be equal
to 3x squared minus 8x plus 30.

00:04:44.420 --> 00:04:46.060
And this right over
here you can view

00:04:46.060 --> 00:04:53.910
as the remainder, so minus
121 over the x plus 4.

00:04:53.910 --> 00:04:55.740
This didn't divide perfectly.

00:04:55.740 --> 00:05:00.397
So over the x plus 4.

00:05:00.397 --> 00:05:02.730
Another way you could have
done it, you could have said,

00:05:02.730 --> 00:05:03.760
this is the remainder.

00:05:03.760 --> 00:05:07.850
So I'm going to have a
negative 121 over x plus 4.

00:05:07.850 --> 00:05:13.222
And this is going to be plus
30 minus 8x plus 3x squared.

00:05:13.222 --> 00:05:14.680
So hopefully that
makes some sense.

00:05:14.680 --> 00:05:16.510
I'll do another example
in the next video.

00:05:16.510 --> 00:05:19.795
And then we'll think about
why this actually works.


Why we do the same thing to both sides basic systems

WEBVTT
Kind: captions
Language: en

00:00:00.631 --> 00:00:04.771
so now we have a very very very interesting problem

00:00:04.771 --> 00:00:08.628
on the left hand side of the scale I have two different types of unknown masses

00:00:08.628 --> 00:00:15.665
one of these X masses and we know that they have the same identical mass, we call that identical each of them having a mass of X

00:00:15.665 --> 00:00:23.627
But then we have this other blue thing and that has a mass of Y, which isn't necessarily going to be the same as the mass of X.

00:00:23.627 --> 00:00:33.661
We have two of these X's and a Y.It seems like the total mass or it definitely is the case, their total mass balance it out to these 8 kg right over here.

00:00:33.661 --> 00:00:41.037
Each of these is 1kg block and balances them out. So the first question I'm going to ask you is, can you express this Mathematically?

00:00:41.037 --> 00:00:45.826
Can you express what we're seeing here, the fact that this total mass balances out with this total mass.

00:00:45.826 --> 00:00:49.231
Can you express that mathematically?

00:00:51.247 --> 00:00:54.476
Well let's just think about our total mass on this side.

00:00:54.815 --> 00:01:00.867
We have two masses of mass X so those two are gonna total at 2X, and then you have a mass of Y.

00:01:01.082 --> 00:01:07.967
So then you're gonna have another Y. So the total mass on the left hand side. Alright let me re-write a little bit closer to the center.

00:01:08.105 --> 00:01:13.981
So,it doesn't get too spread out. On the left hand side, I got 2X plus a mass of Y. That's the total mass.

00:01:14.058 --> 00:01:24.783
The total mass on the left hand side is 2X plus Y, the total mass on the right hand side is just 8. 1,2,3,4,5,6,7,8. It is equal to 8

00:01:25.168 --> 00:01:30.750
And since we see that the scale is balance, this total mass must be equal to this total mass.

00:01:30.750 --> 00:01:33.421
So, we can write an equal sign there.

00:01:33.421 --> 00:01:45.625
Now my question to you is there anything we can do just based on the information that we have here to solve for either the mass X or for the mass Y.

00:01:45.625 --> 00:01:48.313
Is there anything that we can do.

00:01:50.744 --> 00:01:54.580
Well the simple answer is just with this information here, there's actually very little.

00:01:54.580 --> 00:01:58.245
You might say that "Oh well, let me take the Y from both sides" You might take this Y block up.

00:01:58.245 --> 00:02:03.491
But if you take this Y block up you have to take away Y from this side and you don't know what Y is.

00:02:03.491 --> 00:02:09.756
And if you think about it algebraically you might get rid of the Y here. Subtracting Y and you're gonna subtract Y from this side too.

00:02:09.756 --> 00:02:11.921
So, you're not gonna get rid of the Y.

00:02:11.921 --> 00:02:18.095
Same thing with the X's, you actually don't have enough information. Y depends on what X is,and X depends on what Y is.

00:02:18.419 --> 00:02:25.918
Lucky for us however,we do have some more of these blocks laying around. And what we do is we take one of these X blocks.

00:02:25.918 --> 00:02:32.579
And I stack it over here,and I also take one of the Y block and I stack it right over there.

00:02:32.749 --> 00:02:37.366
And then I keep adding all these ones until I balance these things out.

00:02:37.366 --> 00:02:42.668
So, I keep adding these ones. Obviously if I just place this, this will go down cause there's nothing on that side.

00:02:42.668 --> 00:02:56.744
But I keep adding these blocks until it all balances out and I find that my scale balances once I have 5 kg on the right hand side

00:02:57.005 --> 00:03:03.633
So, once again let me ask you this information having X and Y on the left hand side and a 5kg on the right hand side

00:03:03.633 --> 00:03:07.857
And the fact that they are balance, how can we represent that mathematically?

00:03:09.996 --> 00:03:21.593
Well our total mass on the left hand side is X plus Y. And our total mass, let me right that once again a little bit closer to the center.

00:03:21.808 --> 00:03:31.280
It's X plus Y on the left hand side and the right hand side I have 5 kg. I have 5kg.

00:03:32.895 --> 00:03:39.979
I have 5 kg on the right hand side. And we know that's actually balance the scale. So these total masses must be equal to each other.

00:03:40.564 --> 00:03:46.082
And this information by itself, once again. There's nothing I can do with it. I don't know what X and Y.

00:03:46.082 --> 00:03:51.750
If Y is 4 maybe X is 1 or maybe X is 4, Y is 1. Who knows what these are.

00:03:51.919 --> 00:04:00.998
The interesting thing is we can actually use both of these information to figure out what X and Y actually is.

00:04:01.168 --> 00:04:06.283
And I'm giving you a few seconds to think about how we can approach this situation.

00:04:08.920 --> 00:04:13.218
Well think about it this way, we know that X plus Y is equal to 5.

00:04:13.572 --> 00:04:18.188
So if we were to get rid of an X and a Y on this side,on the left hand side of the equation.

00:04:18.342 --> 00:04:24.486
What would we have to get rid of on the right hand side of the, or if we know if we get rid of X and Y on the left hand side of the scale

00:04:24.640 --> 00:04:28.576
What would we get rid of the right hand side of the scale to take away the same mass?

00:04:29.007 --> 00:04:38.542
Well if we take away the X and Y on the left hand side, we know that an X plus Y is 5kg. So, we'll just have to take 5kg from the right hand side.

00:04:38.758 --> 00:04:45.439
So, lets think about what that would do. Well then I'll just have an X over here, I'll just have some of these masses left over here.Then I would what X is.

00:04:45.763 --> 00:04:48.831
Now lets think about how we can represent that algebraically

00:04:49.169 --> 00:04:54.907
Essentially for taking an X and Y from the left hand side. If I'm taking an X and Y from the left hand side.

00:04:55.138 --> 00:05:09.758
I'm subtracting an X, and I'm subtracting an X. Actually let me think of it this way. I'm subtracting an X plus Y. I'm subtracting an X and Y on the left hand side.

00:05:09.896 --> 00:05:12.276
But then what am I gonna do on the right hand side?

00:05:12.629 --> 00:05:17.689
Well an X and a Y we know has a mass of 5. So we can subtract 5 from the right hand side.

00:05:18.505 --> 00:05:22.428
And the only way I'm gonna be able to do this is because of the information that we got from the second scale.

00:05:22.675 --> 00:05:29.820
So I can take away 5. So this is going to be equal to taking away 5. Taking away X and a Y is equal to taking away 5.

00:05:30.129 --> 00:05:37.390
And we know that because an X and a Y is equal to 5kg. And if we take away an X and a Y on the left hand side, what do we left with?

00:05:37.729 --> 00:05:42.526
Well this is gonna be the same thing. Let me rewrite this part.

00:05:43.342 --> 00:05:54.970
This, taking away an X and a Y is the same thing if you distribute the negative sign as taking away an X and taking away a Y.

00:05:56.718 --> 00:06:05.738
And so on the left hand side, we're left with just 2X and we have taken away one of the X's, we're left just an X.

00:06:06.601 --> 00:06:09.601
And we had a Y and we've took away one of the Y. So we're left with no Y.

00:06:09.970 --> 00:06:12.227
We see that visually, we're left with just an X here.

00:06:12.227 --> 00:06:14.134
And what do we have on the right hand side?

00:06:14.564 --> 00:06:26.099
We had 8 and we know X and Y is equal to 5, so we took away 5. So to keep the scale balance. And so 8 minus 5 is going to be 3.

00:06:26.453 --> 00:06:38.949
8 minus 5 is equal to 3 and just like that using this extra information we're able to figure out that the mass of X is equal to 3.

00:06:39.242 --> 00:06:47.001
Now, one final question. We're able to figure out the mass of X, can you figure out what the mass of Y is.

00:06:48.648 --> 00:06:53.444
Well we can go back to either one of these scales. Probably be simpler to go back to this one.

00:06:53.766 --> 00:07:01.157
We know that the mass of X plus the mass of Y is equal to 5. So we could say, one thing we know that X is now is equal to 3.

00:07:01.480 --> 00:07:13.707
We know that this is now a 3kg mass. We can rewrite this is 3 plus Y is equal to 5

00:07:14.323 --> 00:07:17.554
Well now we say, we could take 3 away from both sides.

00:07:17.554 --> 00:07:24.103
if I take 3 away from the left hand side I just have to take 3 away from the right hand side to keep my scale balanced.

00:07:24.333 --> 00:07:30.284
And I'll be left with the mass of Y is balance with a mass of 2 or Y is equal to 2.

00:07:30.667 --> 00:07:43.488
That's an analogy of taking 3 from both sides of this equation. And on the left hand side, I'm just left with a Y and on the right hand side I'm just left with a 2.

00:07:43.703 --> 00:07:52.200
So, X is equal to 3kg and Y is equal to 2kg and what I encouraged you to do is verify that it made sense right up here.

00:07:52.369 --> 00:08:00.168
Figure out what the total mass on the left hand and the right hand or verify what the total mass right over here really was 8 to begin with.

00:08:00.492 --> 00:08:08.617
And you'll see that 2Xs are gonna be 6kg plus my Y is 2kg that will balance 8kg.

00:08:08.617 --> 00:08:12.617
And 3 plus 2 was equal to 5.


Why we do the same thing to both sides: Multi-step equations

WEBVTT
Kind: captions
Language: en

00:00:00.015 --> 00:00:03.932
Alright, now we have a very interesting situation.

00:00:04.193 --> 00:00:09.951
On both sides of the scale, we have our mystery mass and now I'm calling the mystery mass having a mass of Y.

00:00:09.951 --> 00:00:12.066
Just to show you that it doesn't always have to be X.

00:00:12.127 --> 00:00:14.902
It can be any symbol as long as you can keep track of that symbol.

00:00:14.902 --> 00:00:18.523
But all these have the same mass. That's why I wrote Y on all of them.

00:00:18.616 --> 00:00:23.680
And we also have a little 1 kilogram boxes on both sides of the scale.

00:00:23.726 --> 00:00:28.867
So the first thing I wanna do, we're gonna do step by step and try to figure out what this mystery mass is.

00:00:28.897 --> 00:00:35.025
But the first thing I wanna do is, is, is, have you think about, whether you can represent this algebraically?

00:00:35.055 --> 00:00:40.155
Whether with, with a little bit of mathematic symbolry, you can represent what's going on this scale.

00:00:40.293 --> 00:00:46.638
Over here, I have three Ys and three of these boxes and their total mass is equal to this one Y.

00:00:46.638 --> 00:00:49.751
And I think I have about let's see, I have 7 boxes right over here.

00:00:49.751 --> 00:00:51.504
So I'll give you a few seconds to do that.

00:00:53.647 --> 00:00:58.445
So let's think about the total mass over here. We have 3 boxes and a mass Y.

00:00:58.599 --> 00:01:04.334
So they're going to have a mass of 3Y, and then you have 3 boxes with a mass of 1 kilogram.

00:01:04.380 --> 00:01:11.892
So they're going to have a mass of 3 kilograms. Now over here, I have 1 box with a mass of Y kilograms.

00:01:11.953 --> 00:01:17.429
So that's going to be my 1Y right over there. I could've written 1Y but that's, I don't need to.

00:01:17.429 --> 00:01:23.299
A Y is the same thing as 1Y. So I have the Y kilograms right there. And I have seven of these, right?

00:01:23.422 --> 00:01:30.254
1,2,3,4,5,6,7. Yup, seven of these. So I have Y plus 7 kilograms on the right-hand side.

00:01:30.254 --> 00:01:35.908
And once again, it's balanced. The scale is balanced. This mass, total mass is equal to this total mass.

00:01:36.092 --> 00:01:40.869
So we can write an equal sign, right over there. So that's a good starting point.

00:01:40.869 --> 00:01:45.833
We were able to represent this situation to this real-life simple situation.

00:01:45.833 --> 00:01:50.931
You know, back in the day, when people actually had to figure out the mass of things if you were to go the jewelry store, whatever.

00:01:50.931 --> 00:01:55.646
They actually did had problems like this. We were able to represent it mathematically.

00:01:56.014 --> 00:02:03.498
Now the next thing to do is, what are some reasonable next steps? How can we start to simplify this a little bit?

00:02:03.513 --> 00:02:05.915
You know, once again, I'll give you a few seconds to think about that.

00:02:08.424 --> 00:02:12.680
Well, the neat thing about algebra is there's actually multiple paths that you could go down.

00:02:12.772 --> 00:02:19.154
You might say, well why won't we remove 3 of these what, of these yellow blocks from both sides?

00:02:19.154 --> 00:02:24.110
That would be completely legitimate. You might say, well, why won't we remove 1 of these Ys from both sides?

00:02:24.263 --> 00:02:29.486
That also would be legitimate. And we could do it in either order. So let's just pick one of them.

00:02:29.486 --> 00:02:35.834
Let's say that we've first want to remove, let's say that we first want to remove the, a Y from either sides.

00:02:35.834 --> 00:02:39.579
Just so that we feel a little bit more comfortable with all of our Ys sitting on the 1 side.

00:02:39.579 --> 00:02:46.234
And so the best way, if we don't want all of our Ys to sit on the 1 side, we can remove, we can remove a Y from each side.

00:02:46.234 --> 00:02:52.903
Remember, if you removed a Y from only 1 side, that would unbalance the scale. The scale was already balanced whatever you have to do to the one side after you do the other.

00:02:53.163 --> 00:03:02.178
So I'm gonna remove a Y, I'm gonna remove Y mass from both sides. Now what will that look like algebraically?

00:03:02.178 --> 00:03:09.381
Well. I remove the Y from both sides. So I subtracted Y from the left-hand side and I subtracted Y from the right-hand side.

00:03:09.381 --> 00:03:14.189
That's exactly what I did. The mass, it had a massive Y. I don't know what that is but I did take it away.

00:03:14.189 --> 00:03:20.682
I lifted that block, that little, that little block. And so, on the left-hand side, on the left-hand side, what am I left with?

00:03:20.682 --> 00:03:23.986
That you can pick it up mathematically, you can even look up here, and look up here what you're left with.

00:03:23.986 --> 00:03:29.566
If I had 3 of something, and I take away 1 of them, if I take away 1 of them, I'm left with 2 of that something.

00:03:29.566 --> 00:03:35.088
So I'm left with 2Y, right over here, you see it. I had 3, I got rid of 1, so I'm left with 2.

00:03:35.256 --> 00:03:44.543
And I still have those 3 yellow blocks. So I still have those 3 yellow blocks. On the right-hand side, I had a Y, I took away the Y, and so now I have no Ys left.

00:03:44.543 --> 00:03:51.898
We see it visually right over here. And I still have 7 of the yellow blocks. So I still have 7 of the yellow blocks.

00:03:52.082 --> 00:04:01.332
And since I took the exact same mass from both sides of the scale, the scale is still going to be balanced. It was balanced before, I took away the same thing from both sides.

00:04:01.332 --> 00:04:07.662
And so the scale is still, is still balanced. So this is going to be equal to that.

00:04:07.662 --> 00:04:12.319
Now, now just trying to look a bit similar to what we saw in that last video.

00:04:12.319 --> 00:04:15.311
But I will ask you, what can we do from this point?

00:04:15.325 --> 00:04:24.059
What can we do from this point to simplify it further or so, even better, think of it so we could isolate the, these Ys on the left-hand side.

00:04:24.059 --> 00:04:26.949
And I'll give you a few seconds to think about that.

00:04:28.303 --> 00:04:35.894
Well, if we want to isolate these Ys on the left-hand side, these 2 Ys, the best way is to get rid of this 3, to get rid of these 3 blocks.

00:04:35.894 --> 00:04:41.862
So why won't we do that? Let's take 3 blocks from this side but we can't just take it from that side if we want to keep it balanced.

00:04:41.862 --> 00:04:46.746
We have to do it to this side too. We gotta take away, we gotta take away 3 blocks.

00:04:46.746 --> 00:04:50.761
So we're subtracting 3 from that side, and subtracting 3 from the right side.

00:04:50.761 --> 00:04:58.230
So on the left-hand side, on the left-hand side, we're gonna be left with just these 2 blocks of mass Y.

00:04:58.230 --> 00:05:04.068
So our total mass is now going to be 2Y. These 3 minus 3 is 0, and you see that here.

00:05:04.068 --> 00:05:08.905
We're just left with 2Ys right over here, and on the right-hand side, we got rid of 3 of the blocks.

00:05:08.905 --> 00:05:17.705
So we only have 4 of them left. So you have 4 of them left. So you have 2 of these Y masses is equal to 4 kilograms.

00:05:18.210 --> 00:05:22.084
And because we did the same thing to both sides, the scale is still balanced.

00:05:22.253 --> 00:05:27.626
And now, well how do we solve this? And you might be able to solve this in your head. I have 2 times something is equal to 4.

00:05:27.626 --> 00:05:33.769
And you can kind of think about what that is, but if we want to stay true to what we've been doing before, let's think about it.

00:05:33.769 --> 00:05:39.287
I have 2 of something, is equal to something else. What if I multiply both sides by 2?

00:05:39.287 --> 00:05:45.499
Oh, sorry, what if I multiply both sides by 1/2 or another way is dividing both sides by 2.

00:05:45.499 --> 00:05:52.763
If I multiply this side by 1/2, if I essentially take away half of the mass, or I'll only leave half of the mass,

00:05:52.763 --> 00:06:00.903
Then, I'm only gonna have 1 block here, and if I take away half of the mass over here, I'm gonna have to take away 2 of these blocks right over there.

00:06:00.903 --> 00:06:06.724
And what I just did, you could say I multiplied both sides by 1/2 or just for a sake of a little change,

00:06:06.724 --> 00:06:13.570
You could say I divided both sides by 2. And on the left-hand side, I'm left with a mass of Y.

00:06:13.953 --> 00:06:23.333
And on the right-hand side, I'm left with a mass of 4 divided by 2 is 2. And once again I can still write this equal sign because the scale is balanced.

00:06:23.333 --> 00:06:29.489
I did the exact same thing to both sides. I left half to what was on the left-hand side, and half of it was on the right-hand side.

00:06:29.489 --> 00:06:34.712
It was balanced before, half of each side, so it's going to be balanced again. But there, we've done it.

00:06:34.865 --> 00:06:39.115
We've solved something that's actually not so easy to solve. Or it might not look so easy first.

00:06:39.407 --> 00:06:47.098
We figured out that our mystery mass Y is 2 kilograms. And you can verify this, this is the really fun thing about algebra.

00:06:47.544 --> 00:06:53.739
Is it, once you get to this point, you can go back and think about whether the original, the original problem you saw made sense.

00:06:54.108 --> 00:07:01.334
Let's do that. Let's think about whether the original problem made sense. And to do that, I want you to, I want you to calculate.

00:07:01.518 --> 00:07:06.670
Now that we know what the mass Y is it's 2 kilograms. What was the total mass on each side?

00:07:08.964 --> 00:07:18.423
Well, let's calculate it. We have 2 right, right over here, this is 2 kilograms, I'll do that in purple color,so this is a 2, this is a 2, this is a 2.

00:07:18.499 --> 00:07:29.678
So we had 6 kilograms plus these 3, we had 9 kilograms on the left-hand side. And on the right-hand side, I had these 7 plus 2 here, 7 plus 2 is 9 kilograms.

00:07:29.678 --> 00:07:34.231
That's why it was balanced. Our mystery mass, we had 9 kilograms, total on both sides.