Put-call parity

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:04.050
If we want to get the upside
of owning a stock while still

00:00:04.050 --> 00:00:07.640
mitigating the downside, in
case the stock price goes down,

00:00:07.640 --> 00:00:12.210
we saw that we could buy a stock
and an appropriate put option.

00:00:12.210 --> 00:00:14.940
So that when the stock
goes below some price,

00:00:14.940 --> 00:00:16.640
the put option
starts to have value,

00:00:16.640 --> 00:00:18.300
and so it mitigates
our downside.

00:00:18.300 --> 00:00:21.080
And just as a review,
these payoff diagrams

00:00:21.080 --> 00:00:23.300
are the values of-- or at
least the one on the left,

00:00:23.300 --> 00:00:25.930
is the value of our holdings
at some future date.

00:00:25.930 --> 00:00:28.610
And we're defining that
date to be the maturity

00:00:28.610 --> 00:00:31.940
date of the options
under question.

00:00:31.940 --> 00:00:34.810
Now, and this one over here
is the profit at that maturity

00:00:34.810 --> 00:00:37.110
date, and that's why we're
subtracting the actual costs

00:00:37.110 --> 00:00:39.620
to enter the position on
this one on the right.

00:00:39.620 --> 00:00:42.730
Now the question I want to
answer in this video is how can

00:00:42.730 --> 00:00:46.940
we get the same payoff diagram
without buying either stocks

00:00:46.940 --> 00:00:47.910
or puts?

00:00:47.910 --> 00:00:49.580
And as a bit of a
clue, think about what

00:00:49.580 --> 00:00:52.250
happens if we were to
just to buy a call option.

00:00:52.250 --> 00:00:53.990
Actually let me do it
in that same color.

00:00:53.990 --> 00:00:55.970
So if you were to just
have a call option,

00:00:55.970 --> 00:00:58.230
the payoff diagram
would look like this.

00:00:58.230 --> 00:01:01.682
You would never exercise the
call option at expiration,

00:01:01.682 --> 00:01:03.140
unless-- and we're
assuming this is

00:01:03.140 --> 00:01:05.170
at expiration or at maturity.

00:01:05.170 --> 00:01:07.300
But if the stock
price goes above $50,

00:01:07.300 --> 00:01:12.460
you would then exercise your
option to buy it at $50.

00:01:12.460 --> 00:01:15.090
So then it starts to
have value as the stock

00:01:15.090 --> 00:01:16.440
price goes above $50.

00:01:16.440 --> 00:01:18.420
If the stock price
goes to $60, you

00:01:18.420 --> 00:01:20.540
would exercise your
option to buy at $50,

00:01:20.540 --> 00:01:24.140
and then you could sell at
$60 and you would make $10.

00:01:24.140 --> 00:01:27.230
So you start to get
some of the upside.

00:01:27.230 --> 00:01:29.330
So how can we
shift this graph up

00:01:29.330 --> 00:01:31.790
to get exactly the
same payoff diagram?

00:01:31.790 --> 00:01:33.760
Well, we could
have a call option,

00:01:33.760 --> 00:01:37.330
and we could own something
that would essentially

00:01:37.330 --> 00:01:40.400
shift this entire
graph up by $50.

00:01:40.400 --> 00:01:43.670
So we could have, essentially,
a $50 bond, or a bond

00:01:43.670 --> 00:01:46.100
to that is worth-- let
me write it this way.

00:01:46.100 --> 00:01:57.950
A bond that is worth $50
at option expiration.

00:01:57.950 --> 00:01:59.899
So if there's some
interest we're getting,

00:01:59.899 --> 00:02:01.940
we might be able to buy
it for a little bit less.

00:02:01.940 --> 00:02:04.870
If there's zero interest, then
it's pretty much like cash,

00:02:04.870 --> 00:02:06.420
we would pay $50 for it.

00:02:06.420 --> 00:02:09.479
But the payoff diagram for a
bond that will be worth $50

00:02:09.479 --> 00:02:14.010
at this date, at maturity,
or at expiration, the payoff

00:02:14.010 --> 00:02:16.000
diagram for just the bond
would look like this.

00:02:16.000 --> 00:02:17.374
It would just be
a straight line.

00:02:17.374 --> 00:02:20.430
It's guaranteed to pay you $50.

00:02:20.430 --> 00:02:25.720
So if you own the bond and
the call option, below $50,

00:02:25.720 --> 00:02:27.620
the call option is
worthless, so you're just

00:02:27.620 --> 00:02:30.020
going to have the
bond over here.

00:02:30.020 --> 00:02:33.330
And then above $50, you
still have the bond,

00:02:33.330 --> 00:02:35.259
but now the call option
is worth something.

00:02:35.259 --> 00:02:37.550
So you have the value of the
bond plus the call option.

00:02:37.550 --> 00:02:40.790
So at $60, the call option's
worth $10, the bonds worth $50,

00:02:40.790 --> 00:02:43.007
the combination is worth $60.

00:02:43.007 --> 00:02:45.340
And so the combination of the
call option plus the bond,

00:02:45.340 --> 00:02:47.150
you'll see it here
on the left, it's

00:02:47.150 --> 00:02:49.230
actually going to
have the same payoff

00:02:49.230 --> 00:02:52.410
diagram as the
stock plus the put.

00:02:52.410 --> 00:02:55.460
So you have the situation
here that a stock

00:02:55.460 --> 00:02:58.740
plus an appropriately
priced put or a put

00:02:58.740 --> 00:03:00.470
with a appropriate
strike price is

00:03:00.470 --> 00:03:02.440
going to be the same
thing when it comes

00:03:02.440 --> 00:03:06.480
to payoff, at a future
date, at expiration,

00:03:06.480 --> 00:03:10.260
as a bond plus a call option.

00:03:10.260 --> 00:03:13.310
And this right here is
called put call parity.

00:03:13.310 --> 00:03:15.560
And it shows the
relationship between all

00:03:15.560 --> 00:03:16.870
of these different securities.

00:03:16.870 --> 00:03:20.920
And if any of the prices
start to kind of not

00:03:20.920 --> 00:03:23.370
make this thing hold
true, there might

00:03:23.370 --> 00:03:25.160
be an arbitrage opportunity.

00:03:25.160 --> 00:03:27.991
But we'll cover that
in future videos.


Put writer payoff diagrams

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Language: en

00:00:00.482 --> 00:00:05.092
If I were to buy a put option with a fifty dollar excercise price

00:00:05.092 --> 00:00:09.338
and if I were to buy it for $10.00, then the value of my position

00:00:09.338 --> 00:00:12.277
the payoff for that put option, at the maturity

00:00:12.277 --> 00:00:17.713
or at the expiration I should say. At the expiration of the option.

00:00:17.713 --> 00:00:21.282
Depending on what the stock price is, and expiration would look like this:

00:00:21.282 --> 00:00:24.600
if the stock price is worth, if the stock price goes to zero

00:00:24.600 --> 00:00:26.408
then the put option is worth fifty

00:00:26.408 --> 00:00:28.354
because I could buy the stock at zero

00:00:28.354 --> 00:00:30.164
and excercise my option

00:00:30.164 --> 00:00:34.667
to sell at fifty. At "putting" the stock to someone else

00:00:34.667 --> 00:00:38.333
at fifty dollars. All the way to if the stock becomes worth fifty

00:00:38.333 --> 00:00:41.364
then my put option, I wouldn't need to excercise it

00:00:41.364 --> 00:00:43.800
because why would I?

00:00:43.800 --> 00:00:46.533
It's worthless to have the option to sell something at fifty

00:00:46.533 --> 00:00:49.129
where you can just sell the actual stock in the open market

00:00:49.129 --> 00:00:54.344
or buy the stock at fifty. So then the put option becomes worthless

00:00:54.344 --> 00:00:56.600
for a stock price above that.

00:00:56.600 --> 00:00:58.146
Now, this is the payoff diagram.

00:00:58.146 --> 00:01:00.369
And this is when we just think about the value and expiration.

00:01:00.385 --> 00:01:03.738
If we think about the actual profit and loss at expiration, it would look like this

00:01:03.738 --> 00:01:06.400
It would just be shifted down by ten dollars

00:01:06.400 --> 00:01:09.267
because we have to pay $10 to get this value.

00:01:09.267 --> 00:01:12.929
So if the stock is worth zero, the put option is worth

00:01:12.929 --> 00:01:17.533
$50, but I spent $10 dollars to get it, so the profit is going to be

00:01:17.533 --> 00:01:19.333
$40 dollars.

00:01:19.333 --> 00:01:21.748
And so then at $50, I wouldn't excercise the put option

00:01:21.748 --> 00:01:24.733
so I've lost the $10 dollars I spent on the option

00:01:24.733 --> 00:01:29.267
so my payoff diagram would look like, I'm gonna draw it,

00:01:29.267 --> 00:01:33.005
relatively neatly. My payoff diagram would look like this

00:01:33.005 --> 00:01:36.954
Once again, this payoff diagram just incorporates the price of the option

00:01:36.954 --> 00:01:40.262
So it's the actual profit. This is just the value at expiration,

00:01:40.262 --> 00:01:42.975
depending upon what the value of the stock is at expiration.

00:01:42.975 --> 00:01:46.333
Now this is just a situation if you were to buy an option

00:01:46.333 --> 00:01:48.502
but there has to be someone on the other side of the contract

00:01:48.502 --> 00:01:53.800
someone who's holding, agreeing to buy the option for you

00:01:53.815 --> 00:01:59.667
So you could actually have the writer, you could actually have the writer, of the put

00:01:59.667 --> 00:02:03.533
the payoff diagram we just showed is the person who owns the put,

00:02:03.533 --> 00:02:05.400
but someone else had to have created the put.

00:02:05.400 --> 00:02:08.667
They said, "Oh, you know what, I will give you the right.

00:02:08.667 --> 00:02:16.324
"I will give you the right to sell, to sell me the stock at $50, up to some expiration date."

00:02:16.324 --> 00:02:18.800
So what does their payoff diagram look like?

00:02:18.800 --> 00:02:23.667
Well if this guy is going to be able to make $50, this guy over here

00:02:23.667 --> 00:02:27.267
the writer of the put, the writer of the put

00:02:27.267 --> 00:02:30.933
is going to lose $50. He's going to have to essentially go out

00:02:30.933 --> 00:02:36.415
he's actually going to have to buy that $50 put or buy that $50 stock from this person

00:02:36.415 --> 00:02:39.113
because he has to uphold his side of the transaction

00:02:39.113 --> 00:02:41.754
but he's buying something for $50 that's worthless

00:02:41.754 --> 00:02:46.933
because based over, at this end of the axis, the stock would be worth nothing

00:02:46.933 --> 00:02:51.862
so he's taking a $50 loss, all the way to him not having to do anything

00:02:51.862 --> 00:02:55.538
because the put holder won't actually excercise their options

00:02:55.538 --> 00:02:58.359
if their stock price is $50, so their payoff diagram is going to look like this

00:02:58.359 --> 00:03:01.498
So you can see it's actually the mirror image

00:03:01.498 --> 00:03:05.800
of the payoff diagram of this person on the other side of the contract

00:03:05.800 --> 00:03:09.733
And if you were to add these two payoff diagrams, you would be neutral,

00:03:09.733 --> 00:03:12.667
because all of the money is exchanging hands between the buyer

00:03:12.667 --> 00:03:13.677
and the seller of the put.

00:03:13.677 --> 00:03:16.267
If you look at the actual profit or loss

00:03:16.267 --> 00:03:19.000
if the put is not excercised

00:03:19.000 --> 00:03:24.000
then the writer of the put essentially just got a free $10

00:03:24.000 --> 00:03:26.667
He sold the put, he sold the put to this guy for $10

00:03:26.667 --> 00:03:29.333
He created the put and sold it to that guy for $10

00:03:29.333 --> 00:03:32.733
the put is not excercised, he gets to keep that $10.

00:03:32.733 --> 00:03:36.267
But then if the stock goes down and he's forced to buy the stock

00:03:36.267 --> 00:03:38.600
from the owner of the put

00:03:38.600 --> 00:03:41.400
he has to buy it because that's his side of the deal

00:03:41.400 --> 00:03:43.600
then all of a sudden he loses money

00:03:43.600 --> 00:03:47.000
So he would go, if all the way down if the stock is worth nothing,

00:03:47.000 --> 00:03:50.133
He is forced to buy something for $50 that is worth nothing

00:03:50.133 --> 00:03:56.067
He would take a $50 loss, but he paid the $10 on the actual price of the option

00:03:56.067 --> 00:03:59.400
so it would be a negative $40 profit

00:03:59.400 --> 00:04:01.815
So his profit and loss would look like this

00:04:01.815 --> 99:59:59.999
But once again, these are the mirror images of each other.


Khan Academy on Nightly News

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.314
Brian: We continue our series from Education Nation

00:00:03.314 --> 00:00:06.379
tonight with the work of a remarkable young man who is

00:00:06.379 --> 00:00:10.512
applying his three degrees from MIT and one from

00:00:10.512 --> 00:00:13.928
Harvard to help a lot of young people around the world

00:00:13.928 --> 00:00:15.094
do better in school.

00:00:15.094 --> 00:00:17.756
His classroom, in this case, is the Internet and his

00:00:17.756 --> 00:00:21.420
videos have become a teaching sensation from

00:00:21.420 --> 00:00:24.585
Mountain View, California, NBC's Kristen Welker has

00:00:24.585 --> 00:00:27.398
our story tonight.

00:00:27.398 --> 00:00:29.138
(kids cheering)

00:00:29.138 --> 00:00:31.635
Kristen: What makes 5th graders cheer?

00:00:31.635 --> 00:00:33.506
Would you believe math?

00:00:33.506 --> 00:00:34.376
Boy: Yes.

00:00:34.376 --> 00:00:36.262
Girl: I'm starting to really like math now.

00:00:36.262 --> 00:00:38.840
Kristen: These kids are learning with the help of

00:00:38.840 --> 00:00:41.526
Khan Academy, an online school.

00:00:41.526 --> 00:00:43.656
Adult: You got it right. Good job.

00:00:43.656 --> 00:00:46.179
Videos that are interactive and fun,

00:00:46.179 --> 00:00:49.788
explaining difficult concepts in a conversational way.

00:00:49.788 --> 00:00:51.318
Khan: So if you want the slope of a line.

00:00:51.318 --> 00:00:53.653
Kristen: Khan Academy is the brainchild of

00:00:53.653 --> 00:00:55.827
34-year-old Sal Khan.

00:00:55.827 --> 00:00:58.039
Khan: Everything in my gut says that's the right way

00:00:58.039 --> 00:01:00.333
that education should move in the future.

00:01:00.333 --> 00:01:03.497
In 2004, Khan started tutoring his cousin,

00:01:03.497 --> 00:01:05.128
Nadia Rahman, in math.

00:01:05.128 --> 00:01:06.106
Khan: So, this is it.

00:01:06.106 --> 00:01:08.485
Kristen: She was out of state so he coached her

00:01:08.485 --> 00:01:11.771
online from a closet in his home.

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Now studying Pre-Med at Sarah Lawrence College,

00:01:14.870 --> 00:01:17.703
Nadia says she has her cousin to thank.

00:01:17.703 --> 00:01:20.315
Nadia: So I kind of, was able to learn on my own pace.

00:01:20.315 --> 00:01:23.703
Kristen: Khan posted more lessons online for anyone to use.

00:01:23.703 --> 00:01:28.065
The videos went viral and positive feedback poured in.

00:01:28.065 --> 00:01:29.649
Khan: Your videos will allow us to help ...

00:01:29.649 --> 00:01:34.117
Kristen: Khan eventually made 2,200 videos; basic math,

00:01:34.117 --> 00:01:38.042
algebra, trigonometry, calculus, all for free.

00:01:38.042 --> 00:01:39.165
Speaker: 9 divided by ...

00:01:39.165 --> 00:01:43.253
Khan's lessons have been viewed on YouTube 43 million times,

00:01:43.276 --> 00:01:44.757
Kristen: X equals zero.

00:01:44.757 --> 00:01:46.025
Khan: You got it, very good.
Kristen: Yeah.

00:01:46.025 --> 00:01:48.995
making them the largest educational source on the site

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and he's attracting big attention.

00:01:51.705 --> 00:01:55.693
Bill: I'm actually envious of the amazing work that he's done.

00:01:55.719 --> 00:01:57.679
Kristen: We caught up with Bill Gates promoting

00:01:57.679 --> 00:02:00.430
Khan Academy at this year's Tech Conference,

00:02:00.430 --> 00:02:02.680
which focuses on new ideas.

00:02:02.680 --> 00:02:05.902
Bill: It's phenomenal. It's the cutting edge of

00:02:05.902 --> 00:02:07.014
where education is going.

00:02:07.014 --> 00:02:09.233
Kristin: The Gates Foundation and Google recently

00:02:09.233 --> 00:02:12.969
donated a total of 3.5 million dollars to help turn

00:02:12.969 --> 00:02:16.696
Khan's not-for-profit into a global classroom.

00:02:16.696 --> 00:02:18.335
Khan: Hopefully, Khan Academy will teach people

00:02:18.335 --> 00:02:20.895
either there are no such things as geniuses

00:02:20.895 --> 00:02:23.445
or that to some degree everyone is.

00:02:23.445 --> 00:02:25.747
Kristin: A philosophy that is opening the doors of

00:02:25.747 --> 00:02:28.222
learning all over the world.

00:02:28.222 --> 00:02:32.000
Kristin Welker, NBC News, Mountain View, California.


The substitution method

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.215
Use substitution to
solve for x and y.

00:00:03.215 --> 00:00:04.840
And we have a system
of equations here.

00:00:04.840 --> 00:00:08.370
The first equation is
2y is equal to x plus 7.

00:00:08.370 --> 00:00:13.307
And the second equation here
is x is equal to y minus 4.

00:00:13.307 --> 00:00:15.390
So what we want to do,
when they say substitution,

00:00:15.390 --> 00:00:17.180
what we want to do
is substitute one

00:00:17.180 --> 00:00:19.150
of the variables
with an expression

00:00:19.150 --> 00:00:21.234
so that we have an equation
and only one variable.

00:00:21.234 --> 00:00:22.441
And then we can solve for it.

00:00:22.441 --> 00:00:24.130
Let me show you what
I'm talking about.

00:00:24.130 --> 00:00:25.713
So let me rewrite
this first equation.

00:00:25.713 --> 00:00:29.490
2y is equal to x plus 7.

00:00:29.490 --> 00:00:31.740
And we have the second
equation over here,

00:00:31.740 --> 00:00:36.002
that x is equal to y minus 4.

00:00:36.002 --> 00:00:37.710
So if we're looking
for an x and a y that

00:00:37.710 --> 00:00:40.740
satisfies both constraints,
well we could say, well look,

00:00:40.740 --> 00:00:43.640
at the x and y have to
satisfy both constraints, both

00:00:43.640 --> 00:00:45.840
of these constraints
have to be true.

00:00:45.840 --> 00:00:49.070
So x must be equal to y minus 4.

00:00:49.070 --> 00:00:52.715
So anywhere in this top
equation where we see an x,

00:00:52.715 --> 00:00:54.340
anywhere we see an
x, we say well look,

00:00:54.340 --> 00:00:59.690
that x by the second constraint
has to be equal to y minus 4.

00:00:59.690 --> 00:01:02.310
So everywhere we see an
x, we can substitute it

00:01:02.310 --> 00:01:05.030
with a y minus 4.

00:01:05.030 --> 00:01:06.150
So let's do that.

00:01:06.150 --> 00:01:09.910
So if we substitute y minus
4 for x in this top equation,

00:01:09.910 --> 00:01:14.940
the top equation becomes 2y
is equal to instead of an x,

00:01:14.940 --> 00:01:17.090
the second constraint
tells us that x

00:01:17.090 --> 00:01:18.920
needs to be equal to y minus 4.

00:01:18.920 --> 00:01:23.480
So instead of an x,
we'll write a y minus 4,

00:01:23.480 --> 00:01:27.390
and then we have a plus 7.

00:01:27.390 --> 00:01:32.420
All I did here is I
substituted y minus 4 for x.

00:01:32.420 --> 00:01:35.190
The second constraint tells
us that we need to do it.

00:01:35.190 --> 00:01:37.750
y minus 4 needs to be
equal to x or x needs

00:01:37.750 --> 00:01:39.640
to be equal to y minus 4.

00:01:39.640 --> 00:01:42.470
The value here is now
we have an equation,

00:01:42.470 --> 00:01:44.090
one equation with one variable.

00:01:44.090 --> 00:01:46.250
We can just solve for y.

00:01:46.250 --> 00:01:52.380
So we get 2y is equal to y, and
then we have minus 4 plus 7.

00:01:52.380 --> 00:01:54.630
So y plus 3.

00:01:54.630 --> 00:02:00.550
We can subtract y from both
sides of this equation.

00:02:00.550 --> 00:02:04.010
The left hand side,
2y minus y is just y.

00:02:04.010 --> 00:02:07.804
y is equal to-- these
cancel out. y is equal to 3.

00:02:07.804 --> 00:02:09.470
And then we could go
back and substitute

00:02:09.470 --> 00:02:11.950
into either of these
equations to solve for x.

00:02:11.950 --> 00:02:14.570
This is easier right over
here, so let's substitute

00:02:14.570 --> 00:02:17.600
right over here. x needs
to be equal to y minus 4.

00:02:17.600 --> 00:02:26.220
So we could say that x
is equal to 3 minus 4

00:02:26.220 --> 00:02:28.830
which is equal to negative 1.

00:02:28.830 --> 00:02:34.130
So the solution to this system
is x is equal to negative 1

00:02:34.130 --> 00:02:37.002
and y is equal to 3.

00:02:37.002 --> 00:02:39.210
And you can verify that it
works in this top equation

00:02:39.210 --> 00:02:39.876
right over here.

00:02:39.876 --> 00:02:45.490
2 times 3 is 6 which is indeed
equal to negative 1 plus 7.

00:02:45.490 --> 00:02:49.230
Now I want to show you that
over here we substituted--

00:02:49.230 --> 00:02:51.830
we had an expression that,
or we had an equation,

00:02:51.830 --> 00:02:53.750
that explicitly solved for x.

00:02:53.750 --> 00:02:55.490
So we were able to
substitute the x's.

00:02:55.490 --> 00:02:56.265
What I want to show
you is we could

00:02:56.265 --> 00:02:57.681
have done it the
other way around.

00:02:57.681 --> 00:03:01.350
We could have solved for y and
then substituted for the y's.

00:03:01.350 --> 00:03:02.069
So let's do that.

00:03:02.069 --> 00:03:04.110
And we could have substituted
from one constraint

00:03:04.110 --> 00:03:05.940
into the other
constraint or vice versa.

00:03:05.940 --> 00:03:08.750
Either way, we would have
gotten the same exact answer.

00:03:08.750 --> 00:03:11.500
So instead of saying x
is equal to y minus 4,

00:03:11.500 --> 00:03:14.480
in that second
equation, if we add 4

00:03:14.480 --> 00:03:20.970
to both sides of this equation,
we get x plus 4 is equal to y.

00:03:20.970 --> 00:03:23.860
This and this is the
exact same constraint.

00:03:23.860 --> 00:03:25.720
I just added 4 to
both sides of this

00:03:25.720 --> 00:03:27.610
to get this
constraint over here.

00:03:27.610 --> 00:03:31.490
And now since we've solved
this equation explicitly for y,

00:03:31.490 --> 00:03:35.280
we can use the first
constraint, the first equation.

00:03:35.280 --> 00:03:37.430
And everywhere
where we see a y, we

00:03:37.430 --> 00:03:40.080
can substitute it with x plus 4.

00:03:40.080 --> 00:03:43.200
So it's 2 times--
instead of 2 times y,

00:03:43.200 --> 00:03:47.530
we can write 2 times x plus 4.

00:03:47.530 --> 00:03:51.940
2 times x plus 4 is
equal to x plus 7.

00:03:51.940 --> 00:03:53.540
We can distribute this 2.

00:03:53.540 --> 00:03:59.720
So we get 2x plus 8
is equal to x plus 7.

00:03:59.720 --> 00:04:04.440
We can subtract x from both
sides of this equation.

00:04:04.440 --> 00:04:07.810
And then we can subtract 8 from
both sides of this equation,

00:04:07.810 --> 00:04:09.170
subtract 8.

00:04:09.170 --> 00:04:11.610
The left hand side,
that cancels out.

00:04:11.610 --> 00:04:13.180
We're just left with an x.

00:04:13.180 --> 00:04:14.960
On the right hand
side, that cancels out,

00:04:14.960 --> 00:04:18.170
and we are left
with a negative 1.

00:04:18.170 --> 00:04:20.160
And then we can
substitute back over here

00:04:20.160 --> 00:04:25.830
we have y is equal to x plus 4,
or so y is equal to negative 1

00:04:25.830 --> 00:04:28.580
plus 4 which is equal to 3.

00:04:28.580 --> 00:04:30.520
So once again, we
got the same answer

00:04:30.520 --> 00:04:33.720
even though this time we
substituted for y instead

00:04:33.720 --> 00:04:35.710
of substituting for x.

00:04:35.710 --> 00:04:38.350
Hopefully you found
that interesting.


Systems of equations word problems example 3

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.860
As a birthday gift,
Zoey gave her niece

00:00:02.860 --> 00:00:05.080
an electronic piggy
bank that displays

00:00:05.080 --> 00:00:06.880
the total amount of
money in the bank

00:00:06.880 --> 00:00:09.610
as well as the total
number of coins.

00:00:09.610 --> 00:00:12.820
After depositing some number
of nickels and quarters only--

00:00:12.820 --> 00:00:15.630
so we only have
nickels and quarters--

00:00:15.630 --> 00:00:20.570
the display read money $2.00,
number of coins 16 How many

00:00:20.570 --> 00:00:23.490
nickels and quarters did
Zoey put in the bank?

00:00:23.490 --> 00:00:24.990
So let's define
some variables here.

00:00:24.990 --> 00:00:32.570
Let's let n equal the
number of nickels.

00:00:32.570 --> 00:00:34.120
Maybe I'll write "let" here.

00:00:34.120 --> 00:00:39.486
Let's let q be equal to
the number of quarters.

00:00:42.900 --> 00:00:44.359
So how many total
coins do we have?

00:00:44.359 --> 00:00:46.399
We'', it's going to be
the number of nickels plus

00:00:46.399 --> 00:00:47.370
the number of quarters.

00:00:47.370 --> 00:00:50.500
So we have the nickels
plus the quarters

00:00:50.500 --> 00:00:52.150
need to be equal to--
well, it tells us

00:00:52.150 --> 00:00:53.579
we have 16 total coins.

00:00:53.579 --> 00:00:55.370
So if we add up the
total number of nickels

00:00:55.370 --> 00:00:58.260
plus the number of
quarters, we have 16 coins.

00:00:58.260 --> 00:01:00.180
So that's one
equation right there.

00:01:00.180 --> 00:01:02.700
And then how much
total money do we have?

00:01:02.700 --> 00:01:05.280
Well, however many
nickels we have,

00:01:05.280 --> 00:01:09.690
we can multiply that
times 0.05, and that'll

00:01:09.690 --> 00:01:12.390
tell us how much money
we have in nickels.

00:01:12.390 --> 00:01:17.727
So 0.05 times the nickels
plus the amount of money

00:01:17.727 --> 00:01:18.560
we have in quarters.

00:01:18.560 --> 00:01:25.540
Well, that'll just be $0.25
per quarter, or 0.25 of $1.

00:01:25.540 --> 00:01:29.480
So let me write 0.25 times
the number of quarters.

00:01:29.480 --> 00:01:32.140
For example, if I had 4
quarters and no nickels,

00:01:32.140 --> 00:01:35.300
I'd have 4 times
$0.25 which is $1.

00:01:35.300 --> 00:01:36.490
And no money due to nickels.

00:01:36.490 --> 00:01:39.000
So it's however may
nickels times $0.05

00:01:39.000 --> 00:01:41.370
plus however many
quarters times $0.25.

00:01:41.370 --> 00:01:43.540
That's the total
amount of money I have.

00:01:43.540 --> 00:01:47.440
And her piggy bank
tells me that is $2.00.

00:01:47.440 --> 00:01:49.580
That is equal to $2.00.

00:01:49.580 --> 00:01:51.840
So we have two equations
with two unknowns.

00:01:51.840 --> 00:01:55.280
We can solve for n and q.

00:01:55.280 --> 00:01:57.990
And let's do it by substitution.

00:01:57.990 --> 00:02:02.240
So the easiest thing that
we could do here, let's

00:02:02.240 --> 00:02:04.560
solve for q over here.

00:02:04.560 --> 00:02:06.920
So if n plus q is
equal to 16, we

00:02:06.920 --> 00:02:10.250
could subtract n from both
sides of this equation.

00:02:10.250 --> 00:02:13.940
So if n plus q is equal
to 16, if we subtract n

00:02:13.940 --> 00:02:18.640
from both sides, we get
q is equal to 16 minus n.

00:02:18.640 --> 00:02:21.100
So all I did is I rewrote
this first constraint

00:02:21.100 --> 00:02:23.230
right over there.

00:02:23.230 --> 00:02:25.400
So since this first
constraint is telling us

00:02:25.400 --> 00:02:27.120
that q, the number
of quarters, must

00:02:27.120 --> 00:02:29.560
be 16 minus the
number of nickels,

00:02:29.560 --> 00:02:31.580
in the second constraint,
every place that we

00:02:31.580 --> 00:02:34.360
see a q, every place
we see quarters,

00:02:34.360 --> 00:02:36.770
we can replace it
with 16 minus n.

00:02:36.770 --> 00:02:37.960
So let's do that.

00:02:37.960 --> 00:02:40.740
So the second constraint when
we make the substitution becomes

00:02:40.740 --> 00:02:46.660
0.05n plus 0.25.

00:02:46.660 --> 00:02:49.550
Instead of q, I'm going
to write 16 minus n.

00:02:52.630 --> 00:02:57.500
That's what the first constraint
tells us. q must be 16 minus n.

00:02:57.500 --> 00:03:00.130
That is going to
be equal to $2.00.

00:03:00.130 --> 00:03:02.070
We're solving this
system by substitution.

00:03:02.070 --> 00:03:03.830
Now let's see if I
can simplify this.

00:03:03.830 --> 00:03:09.090
So we have 0.05n plus-- let's
distribute the 0.25 times

00:03:09.090 --> 00:03:12.260
the 16 and the 0.25
times the negative n.

00:03:12.260 --> 00:03:15.560
0.25 times 16, that's the
same thing as 1/4 times 16.

00:03:15.560 --> 00:03:17.630
That's just going to be 4.

00:03:17.630 --> 00:03:24.040
And then 0.25 times
negative n is minus 0.25n.

00:03:24.040 --> 00:03:27.040
And that is going to
be equal to $2.00.

00:03:27.040 --> 00:03:29.960
Let me scroll down a little bit.

00:03:29.960 --> 00:03:32.220
I'll scroll down a little bit.

00:03:32.220 --> 00:03:38.870
See we have 0.05n minus 0.25n.

00:03:38.870 --> 00:03:44.850
So if I have 0.05 minus 0.25,
let me combine these terms.

00:03:44.850 --> 00:03:47.610
So if I have 0.05 of something,
and I'm going to subtract from

00:03:47.610 --> 00:03:53.080
that 0.25 of that something,
that'll give me negative 0.20

00:03:53.080 --> 00:03:54.600
of that something.

00:03:54.600 --> 00:03:58.000
If I combine these two terms,
I get negative 0.20 or negative

00:03:58.000 --> 00:04:00.190
0.2n.

00:04:00.190 --> 00:04:03.130
And then of course,
I have the plus 4.

00:04:03.130 --> 00:04:08.230
Plus 4 is equal to $2.00, or we
could even just write 2 there.

00:04:08.230 --> 00:04:10.640
Now, we can isolate the
n on the left-hand side

00:04:10.640 --> 00:04:12.760
by subtracting 4
from both sides.

00:04:12.760 --> 00:04:15.220
So let's subtract
4 from both sides.

00:04:15.220 --> 00:04:18.339
And we are left with, on the
left-hand side, negative--

00:04:18.339 --> 00:04:23.140
I could just write that is
negative 0.20n is equal to 2

00:04:23.140 --> 00:04:24.940
minus 4 is negative 2.

00:04:27.480 --> 00:04:34.650
And then we could divide
both sides by negative 0.2.

00:04:34.650 --> 00:04:40.300
Or I could write negative
0.20, the same thing.

00:04:40.300 --> 00:04:43.140
We're not going to go too deep
into the significance in all

00:04:43.140 --> 00:04:43.640
that.

00:04:43.640 --> 00:04:45.556
We're assuming that we
have infinite precision

00:04:45.556 --> 00:04:46.420
on everything.

00:04:46.420 --> 00:04:50.920
So negative 2 divided
by negative 0.2,

00:04:50.920 --> 00:04:54.130
these guys cancel out,
and we are left with n

00:04:54.130 --> 00:04:57.550
is equal to-- the
negatives cancel out.

00:04:57.550 --> 00:05:00.670
2 divided by 0.2 is
just going to be 10.

00:05:00.670 --> 00:05:02.230
n is equal to 10.

00:05:02.230 --> 00:05:05.317
And then we know that q
is equal to 16 minus n

00:05:05.317 --> 00:05:06.400
from the first constraint.

00:05:06.400 --> 00:05:12.580
q is equal to 16 minus n, which
is 10, which is going to be 6.

00:05:12.580 --> 00:05:17.465
So Zoey put in 10 nickels.

00:05:17.465 --> 00:05:19.410
I want to do that in
a different color.

00:05:19.410 --> 00:05:25.235
She put in 10 nickels and
6 quarters in the bank.

00:05:25.235 --> 00:05:26.110
And we can verify it.

00:05:26.110 --> 00:05:28.090
So clearly she has 16 coins.

00:05:28.090 --> 00:05:29.350
So that part makes sense.

00:05:29.350 --> 00:05:32.382
10 nickels, 6 quarters,
that's 16 coins.

00:05:32.382 --> 00:05:33.090
That makes sense.

00:05:33.090 --> 00:05:35.631
And we could also verify that
it's the right amount of money.

00:05:35.631 --> 00:05:41.840
10 nickels are going to be
$0.50, 10 times $0.05 each.

00:05:41.840 --> 00:05:44.000
So it's going to be $0.50.

00:05:44.000 --> 00:05:48.510
And then 6 quarters
is going to be $1.50.

00:05:48.510 --> 00:05:50.530
So it's going to be $1.50.

00:05:50.530 --> 00:05:54.090
So the total amount of money she
has is $0.50 plus $1.50 which

00:05:54.090 --> 00:05:55.590
is $2.00.

00:05:55.590 --> 00:05:57.498
So it all works out.


Substitution method 2

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.137
Use substitution to
solve for x and y.

00:00:03.137 --> 00:00:04.720
And they give us a
system of equations

00:00:04.720 --> 00:00:07.980
here. y is equal to
negative 5x plus 8

00:00:07.980 --> 00:00:11.800
and 10x plus 2y is
equal to negative 2.

00:00:11.800 --> 00:00:14.010
So they've set it up
for us pretty well.

00:00:14.010 --> 00:00:16.627
They already have y
explicitly solved for up here.

00:00:16.627 --> 00:00:18.210
So they tell us,
this first constraint

00:00:18.210 --> 00:00:22.787
tells us that y must be
equal to negative 5x plus 8.

00:00:22.787 --> 00:00:24.620
So when we go to the
second constraint here,

00:00:24.620 --> 00:00:28.250
every time we see a y, we say,
well, the first constraint

00:00:28.250 --> 00:00:32.330
tells us that y must be
equal to negative 5x plus 8.

00:00:32.330 --> 00:00:34.810
So everywhere we see a
y, we can substitute it

00:00:34.810 --> 00:00:37.277
with negative 5x plus 8.

00:00:37.277 --> 00:00:39.360
Because that's what the
first constraint tells us.

00:00:39.360 --> 00:00:40.319
y is equal to that.

00:00:40.319 --> 00:00:42.110
I don't want to be
repetitive, but I really

00:00:42.110 --> 00:00:45.187
want you to internalize that's
all it's saying. y is that.

00:00:45.187 --> 00:00:47.270
So every time we see a y
in the second constraint,

00:00:47.270 --> 00:00:48.710
we can substitute it with that.

00:00:48.710 --> 00:00:49.690
So let's do it.

00:00:49.690 --> 00:00:54.970
So the second equation
over here is 10x plus 2.

00:00:54.970 --> 00:00:56.610
And instead of
writing a y there,

00:00:56.610 --> 00:00:58.360
and I've said it
multiple times already,

00:00:58.360 --> 00:01:00.560
we can write a
negative 5x plus 8.

00:01:00.560 --> 00:01:03.280
The first constraint
tells us that's what y is.

00:01:03.280 --> 00:01:10.669
So negative 5x plus 8
is equal to negative 2.

00:01:10.669 --> 00:01:12.460
Now, we have one equation
with one unknown.

00:01:12.460 --> 00:01:14.520
We can just solve for x.

00:01:14.520 --> 00:01:16.974
We have 10x plus.

00:01:16.974 --> 00:01:17.890
So we can multiply it.

00:01:17.890 --> 00:01:21.740
We can distribute this 2
onto both of these terms.

00:01:21.740 --> 00:01:28.170
So we have 2 times negative
5x is negative 10x.

00:01:28.170 --> 00:01:30.690
And then 2 times 8 is 16.

00:01:30.690 --> 00:01:35.740
So plus 16 is equal
to negative 2.

00:01:35.740 --> 00:01:37.280
Now we have 10x minus 10x.

00:01:37.280 --> 00:01:38.350
Those guys cancel out.

00:01:38.350 --> 00:01:41.350
10x minus 10x is equal to 0.

00:01:41.350 --> 00:01:42.750
So these guys cancel out.

00:01:42.750 --> 00:01:47.350
And we're just left with
16 equals negative 2,

00:01:47.350 --> 00:01:48.290
which is crazy.

00:01:48.290 --> 00:01:52.050
We know that 16 does
not equal negative 2.

00:01:52.050 --> 00:01:54.550
This is an inconsistent result.

00:01:54.550 --> 00:01:58.380
And that's because these two
lines actually don't intersect.

00:01:58.380 --> 00:02:00.930
And we could see that by
actually graphing these lines.

00:02:00.930 --> 00:02:03.450
Whenever you get something
like some number equalling

00:02:03.450 --> 00:02:05.890
some other number that
they're clearly not equal to,

00:02:05.890 --> 00:02:07.780
that means it's an
inconsistent result,

00:02:07.780 --> 00:02:11.030
It's an inconsistent system,
and that these lines actually

00:02:11.030 --> 00:02:12.760
don't intersect.

00:02:12.760 --> 00:02:15.280
So let me just graph these
just to make it clear.

00:02:15.280 --> 00:02:19.830
This first equation is already
in slope y-intercept form.

00:02:19.830 --> 00:02:21.960
So it looks something like this.

00:02:21.960 --> 00:02:23.690
That's our x-axis.

00:02:23.690 --> 00:02:26.120
This is our y-axis.

00:02:26.120 --> 00:02:32.170
And it's negative 5x plus 8,
so 1, 2, 3, 4, 5, 6, 7, 8.

00:02:32.170 --> 00:02:35.720
And then it has a very
steep downward slope.

00:02:35.720 --> 00:02:38.157
Every time you move forward
1, you have to go down 5.

00:02:38.157 --> 00:02:39.490
So it looks something like that.

00:02:39.490 --> 00:02:42.350
That's this first
equation right over there.

00:02:42.350 --> 00:02:44.020
The second equation,
let me rewrite it

00:02:44.020 --> 00:02:45.520
in slope y-intercept form.

00:02:45.520 --> 00:02:49.790
So it's 10x plus 2y is
equal to negative 2.

00:02:49.790 --> 00:02:51.870
Let's subtract 10x
from both sides.

00:02:51.870 --> 00:02:56.540
You get 2y is equal to
negative 10x minus 2.

00:02:56.540 --> 00:02:58.070
Let's divide both sides by 2.

00:02:58.070 --> 00:03:03.560
You get y is equal to negative
5x, negative 5x minus 1.

00:03:03.560 --> 00:03:05.200
So it's y-intercept
is negative 1.

00:03:05.200 --> 00:03:06.120
It's right over there.

00:03:06.120 --> 00:03:08.990
And it has the same
slope as this first line.

00:03:08.990 --> 00:03:09.950
So it looks like this.

00:03:09.950 --> 00:03:11.250
It's parallel.

00:03:11.250 --> 00:03:13.180
It's just shifted down a bit.

00:03:13.180 --> 00:03:14.900
So it just looks like that.

00:03:14.900 --> 00:03:16.180
So they're parallel lines.

00:03:16.180 --> 00:03:18.530
They have the same slope,
different y-intercepts.

00:03:18.530 --> 00:03:19.970
We get an inconsistent result.

00:03:23.390 --> 00:03:24.690
They don't intersect.

00:03:24.690 --> 00:03:26.260
And the telltale
sign of that, when

00:03:26.260 --> 00:03:27.760
you're doing it
algebraically, is

00:03:27.760 --> 00:03:29.500
you get something
wacky like this.

00:03:29.500 --> 00:03:31.140
This is why it's
called inconsistent.

00:03:31.140 --> 00:03:35.020
It's not consistent for 16
to be equal to negative 2.

00:03:35.020 --> 00:03:36.290
These don't intersect.

00:03:36.290 --> 00:03:40.310
There's no solution to both of
these constraints, no x and y

00:03:40.310 --> 00:03:43.350
that satisfies both of them.


Addition elimination method 1

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.790
Use elimination to
solve for x and y.

00:00:03.790 --> 00:00:07.020
And they gave us two equations
here-- x plus 2y is equal to 6

00:00:07.020 --> 00:00:09.830
and 4x minus 2y is equal to 14.

00:00:09.830 --> 00:00:12.360
So to solve by
elimination, what we

00:00:12.360 --> 00:00:15.160
do is we're going to add
these two equations together

00:00:15.160 --> 00:00:19.100
so that one of the two variables
essentially gets eliminated,

00:00:19.100 --> 00:00:20.430
gets canceled out.

00:00:20.430 --> 00:00:23.210
And what we could do right here,
we see we have a plus 2y here,

00:00:23.210 --> 00:00:25.970
and we see we have a
negative 2y right over here.

00:00:25.970 --> 00:00:27.856
So clearly if we added
these two together,

00:00:27.856 --> 00:00:29.480
the y's would cancel
out, and so that's

00:00:29.480 --> 00:00:31.190
exactly what we're going to do.

00:00:31.190 --> 00:00:33.740
We're going to add the
left side of this equation

00:00:33.740 --> 00:00:35.320
to the left side
of this equation,

00:00:35.320 --> 00:00:37.070
and the right-hand
side of that equation

00:00:37.070 --> 00:00:39.300
to the right-hand side
of the bottom equation.

00:00:39.300 --> 00:00:41.719
And just to make it clear
that this should make sense

00:00:41.719 --> 00:00:44.010
is we're just using both of
these constraints, whenever

00:00:44.010 --> 00:00:46.120
you learn about any
type of equation.

00:00:46.120 --> 00:00:49.580
So if I have x plus
2y is equal to 6,

00:00:49.580 --> 00:00:52.085
you learned early on
in algebra that you

00:00:52.085 --> 00:00:53.931
can manipulate this
equation in any way

00:00:53.931 --> 00:00:55.930
as long as whatever you
do to the left-hand side

00:00:55.930 --> 00:00:57.890
of the equation, you
do the right-hand side.

00:00:57.890 --> 00:01:00.310
If this is equal to
that, the only way

00:01:00.310 --> 00:01:01.790
that the equality
will still hold

00:01:01.790 --> 00:01:04.140
is whatever you do to this--
whatever you add to this

00:01:04.140 --> 00:01:07.492
or multiply it by-- you also
do to the right-hand side.

00:01:07.492 --> 00:01:09.200
So when we're adding
these two equations,

00:01:09.200 --> 00:01:10.533
that's exactly what we're doing.

00:01:10.533 --> 00:01:13.780
We could say, hey, let's add 14
to both sides of this equation.

00:01:13.780 --> 00:01:15.830
So you could add
14 on this side,

00:01:15.830 --> 00:01:17.490
you could add 14 on that side.

00:01:17.490 --> 00:01:20.440
That wouldn't be anything new,
but the second equation right

00:01:20.440 --> 00:01:25.060
here tells us that 4x minus
2y is the same thing as 14.

00:01:25.060 --> 00:01:27.140
So instead of adding 14
on the left-hand side,

00:01:27.140 --> 00:01:30.957
I could add 4x minus 2y.

00:01:30.957 --> 00:01:32.540
When we're adding
these two equations,

00:01:32.540 --> 00:01:34.669
we're really just
adding the same thing.

00:01:34.669 --> 00:01:36.960
You could view it as we're
starting with this equation,

00:01:36.960 --> 00:01:39.110
and then we're adding the
same thing to both sides.

00:01:39.110 --> 00:01:42.130
On the right-hand side it looks
like we're adding 14 to the 6.

00:01:42.130 --> 00:01:45.300
On the left-hand side it looks
like we're adding 4x minus 2y

00:01:45.300 --> 00:01:46.870
to whatever is on
the left-hand side.

00:01:46.870 --> 00:01:48.620
But the second
constraint tells us

00:01:48.620 --> 00:01:52.510
that 14 and 4x minus
2y are the same thing.

00:01:52.510 --> 00:01:55.150
So we're adding the same
thing to both sides.

00:01:55.150 --> 00:01:56.990
So with that said,
let's just do it.

00:01:56.990 --> 00:02:02.340
So the left-hand side, if we add
it up, we have x plus 4x is 5x,

00:02:02.340 --> 00:02:05.830
and then the 2y cancels
out with the negative 2y.

00:02:05.830 --> 00:02:08.150
And then on the right-hand
side we have 6 plus 14.

00:02:08.150 --> 00:02:10.329
6 plus 14 Is 20.

00:02:10.329 --> 00:02:12.370
So we're left with one
equation with one unknown.

00:02:12.370 --> 00:02:14.030
5x is equal to 20.

00:02:14.030 --> 00:02:17.250
We can divide both
sides by 5, and we

00:02:17.250 --> 00:02:20.790
are left with x is equal to 4.

00:02:20.790 --> 00:02:22.290
Now we can go back
and substitute

00:02:22.290 --> 00:02:26.020
in x equals 4 into either of
these equations to solve for y.

00:02:26.020 --> 00:02:27.620
So let's use this top one.

00:02:27.620 --> 00:02:32.150
So we have 4 plus
2y is equal to 6.

00:02:32.150 --> 00:02:35.660
We can subtract 4
from both sides.

00:02:35.660 --> 00:02:40.210
So then we get 2y is equal to 2.

00:02:40.210 --> 00:02:42.550
Divide both sides by 2.

00:02:42.550 --> 00:02:45.700
We get y is equal to 1.

00:02:45.700 --> 00:02:48.140
So the solution,
the x's and y's that

00:02:48.140 --> 00:02:51.420
satisfy both of these
equations, are x is equal to 4,

00:02:51.420 --> 00:02:53.180
and y is equal to 1.

00:02:53.180 --> 00:02:55.660
So this is the solution
for this system,

00:02:55.660 --> 00:02:57.240
or this coordinate
would be the point

00:02:57.240 --> 00:02:58.850
of intersection of
these two lines.

00:02:58.850 --> 00:03:00.130
And we can verify it.

00:03:00.130 --> 00:03:04.350
Let's verify that when
we put x is equal to 4,

00:03:04.350 --> 00:03:07.070
and y is equal to 1, in this
first equation it satisfies it.

00:03:07.070 --> 00:03:11.300
So we have 4 plus 2 times 1.

00:03:11.300 --> 00:03:12.790
That's 4 plus 2.

00:03:12.790 --> 00:03:14.895
That does, indeed, equal 6.

00:03:14.895 --> 00:03:16.770
And then the second
equation right over here,

00:03:16.770 --> 00:03:22.640
you have 4 times
4 minus 2 times 1.

00:03:25.150 --> 00:03:30.200
This is equal to 16 minus 2,
which does indeed equal 14.

00:03:30.200 --> 00:03:33.067
So it definitely does satisfy
both of these equations.

00:03:33.067 --> 00:03:33.650
So we're done.

00:03:33.650 --> 00:03:37.830
X is equal to 4,
and y is equal to 1.


Systems of three variables

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:01.990
Solve this system.

00:00:01.990 --> 00:00:05.650
And here we have three
equations with three unknowns.

00:00:05.650 --> 00:00:07.970
And just so you have a
way to visualize this,

00:00:07.970 --> 00:00:09.630
each of these equations
would actually

00:00:09.630 --> 00:00:11.039
be a plane in three dimensions.

00:00:11.039 --> 00:00:12.830
And so you're actually
trying to figure out

00:00:12.830 --> 00:00:15.716
where three planes in
three dimensions intersect.

00:00:15.716 --> 00:00:17.090
I won't go into
the details here.

00:00:17.090 --> 00:00:18.790
I'll focus more
on the mechanics.

00:00:18.790 --> 00:00:21.440
But you can imagine if I were
to draw three-dimensional space

00:00:21.440 --> 00:00:22.630
over here.

00:00:22.630 --> 00:00:27.727
Now, all of a sudden, it
will an x, y, and z-axes.

00:00:27.727 --> 00:00:30.060
So you can imagine that maybe
this first plane-- and I'm

00:00:30.060 --> 00:00:32.140
not drawing it the way
it might actually look--

00:00:32.140 --> 00:00:34.320
might look something like that.

00:00:34.320 --> 00:00:35.960
I'm just drawing
part of the plane.

00:00:35.960 --> 00:00:40.540
And maybe this plane over here,
it intersects right over there,

00:00:40.540 --> 00:00:43.370
and it comes popping
out like this.

00:00:43.370 --> 00:00:45.299
And then it goes
behind it like that,

00:00:45.299 --> 00:00:46.590
keeps going in every direction.

00:00:46.590 --> 00:00:48.060
I'm just drawing
part of the plane.

00:00:48.060 --> 00:00:50.860
And maybe this plane
over here, maybe it

00:00:50.860 --> 00:00:51.940
does something like this.

00:00:51.940 --> 00:00:55.510
Maybe it intersects
over here and over here,

00:00:55.510 --> 00:00:57.480
and so it pops out like that.

00:00:57.480 --> 00:01:01.560
And then it goes below it like
that, and it goes like that.

00:01:01.560 --> 00:01:03.860
I'm just doing this for
visualization purposes.

00:01:03.860 --> 00:01:06.270
And so the intersection
of this plane, the x, y,

00:01:06.270 --> 00:01:08.790
and z-coordinates that
would satisfy all three

00:01:08.790 --> 00:01:11.010
of these constraints
the way I drew them,

00:01:11.010 --> 00:01:12.490
would be right over here.

00:01:12.490 --> 00:01:13.960
So that's what
we're looking for.

00:01:13.960 --> 00:01:17.730
And a lot of times, three
equations with three

00:01:17.730 --> 00:01:20.290
unknown systems, they
will be inconsistent.

00:01:20.290 --> 00:01:23.560
You won't have a solution here
because it's very possible

00:01:23.560 --> 00:01:27.137
to have three planes that all
don't intersect in one place.

00:01:27.137 --> 00:01:28.970
A very simple example
of that is, well, one,

00:01:28.970 --> 00:01:30.970
they could all be
parallel to each other.

00:01:30.970 --> 00:01:32.884
Or they could
intersect each other,

00:01:32.884 --> 00:01:35.300
but maybe they intersect each
other in kind of a triangle.

00:01:35.300 --> 00:01:37.670
So maybe one plane
looks like that.

00:01:37.670 --> 00:01:45.480
Then another plane maybe pops
out like that, goes underneath.

00:01:45.480 --> 00:01:48.420
And then maybe the
third plane cuts in.

00:01:48.420 --> 00:01:51.495
It does something like this,
where it goes into that plane

00:01:51.495 --> 00:01:54.220
and keeps going out
like that, but it

00:01:54.220 --> 00:01:56.610
intersects this plane over here.

00:01:56.610 --> 00:01:58.580
So you see, this kind
of forms a triangle,

00:01:58.580 --> 00:02:00.480
and they don't all
intersect in one point.

00:02:00.480 --> 00:02:03.430
So in this situation, you would
have an inconsistent system.

00:02:03.430 --> 00:02:05.013
So with that out of
the way, let's try

00:02:05.013 --> 00:02:06.300
to actually solve this system.

00:02:06.300 --> 00:02:08.770
And the trick here is
to try to eliminate

00:02:08.770 --> 00:02:11.820
one variable at a time
from all of the equations,

00:02:11.820 --> 00:02:15.610
making sure that you have the
information from all three

00:02:15.610 --> 00:02:16.577
equations here.

00:02:16.577 --> 00:02:18.410
So what we're going to
do is we could maybe,

00:02:18.410 --> 00:02:19.940
it looks like the
easiest to eliminate--

00:02:19.940 --> 00:02:21.773
since we have a positive
y and a negative y,

00:02:21.773 --> 00:02:24.900
and then another positive y--
it seems like we can eliminate

00:02:24.900 --> 00:02:25.590
the y's.

00:02:25.590 --> 00:02:28.380
We can add these two
equations and come up

00:02:28.380 --> 00:02:31.990
with another equation that will
only be in terms of x and z.

00:02:31.990 --> 00:02:34.830
And then we could use these
two equations to come up

00:02:34.830 --> 00:02:37.550
with another equation that'll
only be in terms of x and z.

00:02:37.550 --> 00:02:41.410
But it'll have all of the x
and z constraint information

00:02:41.410 --> 00:02:44.530
embedded in it because we're
using all three equations.

00:02:44.530 --> 00:02:45.870
So let's do that.

00:02:45.870 --> 00:02:48.600
So first, let's add these two
equations right over here.

00:02:48.600 --> 00:02:54.370
So we have x plus y minus
3z is equal to negative 10,

00:02:54.370 --> 00:03:00.250
and x minus y plus
2z is equal to 3.

00:03:00.250 --> 00:03:01.990
So over here, if we
want to eliminate y

00:03:01.990 --> 00:03:05.150
we can literally just
add these two equations.

00:03:05.150 --> 00:03:09.600
So on the left-hand side, x
plus x is 2x, y plus negative y

00:03:09.600 --> 00:03:10.760
cancels out.

00:03:10.760 --> 00:03:13.780
And then negative
3z plus 2z, that

00:03:13.780 --> 00:03:16.200
gives us just a negative z.

00:03:16.200 --> 00:03:21.110
And then we have negative 10
plus 3, which is negative 7.

00:03:21.110 --> 00:03:22.950
So using these two
equations, we got

00:03:22.950 --> 00:03:26.220
2x minus z is equal
to negative 7,

00:03:26.220 --> 00:03:28.360
just adding these two equations.

00:03:28.360 --> 00:03:31.115
Now let's do these
two equations.

00:03:31.115 --> 00:03:32.490
And we can we
reuse this equation

00:03:32.490 --> 00:03:34.370
as long as we're using
new information here.

00:03:34.370 --> 00:03:35.870
Now we're using the
extra constraint

00:03:35.870 --> 00:03:37.520
of this bottom equation.

00:03:37.520 --> 00:03:43.090
So we have x minus y
plus 2z is equal to 3,

00:03:43.090 --> 00:03:51.780
and we have 2x plus y minus
z is equal to negative 6.

00:03:51.780 --> 00:03:53.450
And if we want to
eliminate the y's, we

00:03:53.450 --> 00:03:55.280
can just add these
two equations.

00:03:55.280 --> 00:03:58.480
So x plus 2x is 3x.

00:03:58.480 --> 00:04:00.830
Negative y plus y cancels out.

00:04:00.830 --> 00:04:04.370
2z minus z, well,
that is just z.

00:04:04.370 --> 00:04:07.370
And that is going to be
equal to 3 plus negative 6

00:04:07.370 --> 00:04:09.040
is negative 3.

00:04:09.040 --> 00:04:10.830
So if I add these
two equations, I

00:04:10.830 --> 00:04:16.110
get 3x plus z is
equal to negative 3.

00:04:16.110 --> 00:04:18.880
Now I have a system of two
equations with two unknowns.

00:04:18.880 --> 00:04:21.690
This is a little bit more
traditional of a problem.

00:04:21.690 --> 00:04:23.040
So let me write them over here.

00:04:23.040 --> 00:04:27.900
So we have 2x minus z
is equal to negative 7,

00:04:27.900 --> 00:04:32.930
and then we have 3x plus
z is equal to negative 3.

00:04:32.930 --> 00:04:34.900
And the way this
problem is set up,

00:04:34.900 --> 00:04:36.780
it gets pretty
simple pretty fast,

00:04:36.780 --> 00:04:39.490
because if we just add these two
equations, the z's cancel out.

00:04:39.490 --> 00:04:41.440
Otherwise, if it didn't
happen so naturally,

00:04:41.440 --> 00:04:43.660
we would have to multiply
one of these equations

00:04:43.660 --> 00:04:46.080
or maybe both of them
by some scaling factor.

00:04:46.080 --> 00:04:50.100
But we can just add
these two equations up.

00:04:50.100 --> 00:04:52.240
On the left-hand
side, 2x plus 3x

00:04:52.240 --> 00:04:58.070
is 5x, negative z plus z
cancels out, negative 7

00:04:58.070 --> 00:05:01.580
plus negative 3, that
is equal to negative 10.

00:05:01.580 --> 00:05:05.720
Divide both sides of
this equation by 5,

00:05:05.720 --> 00:05:09.680
and we get x is
equal to negative 2.

00:05:09.680 --> 00:05:13.049
Now, we can substitute back
to find the other variables.

00:05:13.049 --> 00:05:14.840
We could substitute
back into this equation

00:05:14.840 --> 00:05:17.180
to figure out what
z must be equal to.

00:05:17.180 --> 00:05:22.810
So we have 2 times x, 2
times negative 2 minus z

00:05:22.810 --> 00:05:26.930
is equal to negative
7, or negative 4 minus

00:05:26.930 --> 00:05:29.220
z is equal to negative 7.

00:05:29.220 --> 00:05:32.570
We can add 4 to both
sides of this equation,

00:05:32.570 --> 00:05:37.310
and then we get negative
z is equal to negative 7

00:05:37.310 --> 00:05:39.350
plus 4 is negative 3.

00:05:39.350 --> 00:05:41.800
Multiply or divide both
sides by negative 1,

00:05:41.800 --> 00:05:45.059
and you get z is equal to 3.

00:05:45.059 --> 00:05:46.600
And now we can go
and substitute back

00:05:46.600 --> 00:05:48.700
into one of these
original equations.

00:05:48.700 --> 00:05:52.060
So we have x-- we know x
is negative 2-- so we have

00:05:52.060 --> 00:05:58.320
negative 2 plus y
minus 3 times z.

00:05:58.320 --> 00:06:00.680
Well, we know z is
3-- minus 3 times

00:06:00.680 --> 00:06:03.220
3 should be equal
to negative 10.

00:06:03.220 --> 00:06:05.200
And now we just solve for y.

00:06:05.200 --> 00:06:09.460
So we get negative
2 plus y minus 9

00:06:09.460 --> 00:06:11.490
is equal to negative 10.

00:06:11.490 --> 00:06:14.820
And so negative 2 minus
9, that's negative 11.

00:06:14.820 --> 00:06:20.820
So we have y minus 11
is equal to negative 10.

00:06:20.820 --> 00:06:24.130
So we can add 11 to both
sides of this equation,

00:06:24.130 --> 00:06:31.370
and we get y is equal to
negative 10 plus 11 is 1.

00:06:31.370 --> 00:06:32.070
So we're done.

00:06:32.070 --> 00:06:35.380
We got x is equal to
negative 2, z is equal to 3,

00:06:35.380 --> 00:06:36.510
and y is equal to 1.

00:06:36.510 --> 00:06:38.420
Now we can actually
go back and check it,

00:06:38.420 --> 00:06:42.230
verify that this x, y,
and z works for all three

00:06:42.230 --> 00:06:44.940
constraints, that this
three-dimensional coordinate

00:06:44.940 --> 00:06:47.530
lies on all three planes.

00:06:47.530 --> 00:06:48.420
So let's try it out.

00:06:48.420 --> 00:06:52.490
We got x is negative
2, z is 3, y is 1.

00:06:52.490 --> 00:06:55.190
So if we substitute it-- let
me do it into each of them.

00:06:55.190 --> 00:06:59.730
So in this first equation,
that means we have negative 2

00:06:59.730 --> 00:07:01.300
plus 1.

00:07:01.300 --> 00:07:02.700
Remember, y was equal to 1.

00:07:02.700 --> 00:07:04.720
Let me write it over
here. y is equal to 1,

00:07:04.720 --> 00:07:08.120
x is equal to negative
2, z is equal to 3.

00:07:08.120 --> 00:07:09.835
That was the result we got.

00:07:09.835 --> 00:07:11.160
Yep, that's the result we got.

00:07:11.160 --> 00:07:13.750
So when we test it in this
first one, you have negative 2

00:07:13.750 --> 00:07:19.450
plus 1 minus 3
times 3, so minus 9.

00:07:19.450 --> 00:07:22.170
This should be equal to
negative 10, and it is.

00:07:22.170 --> 00:07:25.800
Negative 2 plus 1 is negative
1 minus 9 is negative 10.

00:07:25.800 --> 00:07:27.310
So it works for the first one.

00:07:27.310 --> 00:07:30.690
Let's try it for the second
equation right over here.

00:07:30.690 --> 00:07:37.960
So we have negative 2 minus
y, so minus 1 plus 2 times z.

00:07:37.960 --> 00:07:40.970
So z is 3, so 2 times 3.

00:07:40.970 --> 00:07:44.230
So plus 6 needs
to be equal to 3.

00:07:44.230 --> 00:07:47.890
So this is negative 3 plus 6,
which is indeed equal to 3.

00:07:47.890 --> 00:07:50.000
So it satisfies the
second equation.

00:07:50.000 --> 00:07:52.030
And then we have the
last one right over here.

00:07:52.030 --> 00:07:55.530
We have 2 times x, so
2 times negative 2,

00:07:55.530 --> 00:07:58.050
which is negative 4.

00:07:58.050 --> 00:08:07.450
Negative 4 plus y, so
plus 1 minus z, so minus 3

00:08:07.450 --> 00:08:09.840
needs to be equal to negative 6.

00:08:09.840 --> 00:08:15.230
Negative 4 plus 1 is negative 3,
and then you subtract 3 again.

00:08:15.230 --> 00:08:17.010
It equals negative 6.

00:08:17.010 --> 00:08:19.610
So it satisfies all
three equations,

00:08:19.610 --> 00:08:22.840
so we can feel pretty
good about our answer.


Addition elimination method 3

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:01.602
Solve for x and y.

00:00:01.602 --> 00:00:03.560
And we have the system
of equations right here.

00:00:03.560 --> 00:00:06.240
We have 2x minus
y is equal to 14,

00:00:06.240 --> 00:00:10.780
and negative 6x plus 3y
is equal to negative 42.

00:00:10.780 --> 00:00:13.850
So we could try to solve
this by elimination,

00:00:13.850 --> 00:00:18.100
and let's see if we
can eliminate our y

00:00:18.100 --> 00:00:19.040
variables first.

00:00:19.040 --> 00:00:23.530
We have a 3y here, and we
have a negative y up here.

00:00:23.530 --> 00:00:24.920
And they won't eliminate.

00:00:24.920 --> 00:00:28.100
If you just add negative y
plus 3y, that won't eliminate.

00:00:28.100 --> 00:00:31.310
But if we could turn this
negative y into a negative 3y,

00:00:31.310 --> 00:00:33.320
then it would cancel
out with the 3y.

00:00:33.320 --> 00:00:35.840
And the best way to turn a
negative y into a negative 3y

00:00:35.840 --> 00:00:39.330
is to multiply this
entire top equation by 3.

00:00:39.330 --> 00:00:41.310
So let's do that.

00:00:41.310 --> 00:00:43.580
Let me get some space
over to the left

00:00:43.580 --> 00:00:47.080
Let's multiply this
entire top equation by 3.

00:00:47.080 --> 00:00:48.810
So I'm going to
multiply it by 3.

00:00:48.810 --> 00:00:50.620
So I multiply 2x by 3.

00:00:50.620 --> 00:00:52.110
I get 6x.

00:00:52.110 --> 00:00:53.930
I multiply negative y by 3.

00:00:53.930 --> 00:00:56.080
I get negative 3y.

00:00:56.080 --> 00:01:00.880
And then I multiply 14 by 3.

00:01:00.880 --> 00:01:04.330
3 times 14 is 42, right?

00:01:04.330 --> 00:01:08.020
3 times 10 is 30,
plus 12, it's 42.

00:01:08.020 --> 00:01:10.410
And then we can add
both of these equations.

00:01:10.410 --> 00:01:12.330
Something interesting
should already maybe

00:01:12.330 --> 00:01:14.550
be showing up on your radar.

00:01:14.550 --> 00:01:16.205
But let's add both
of these equations.

00:01:16.205 --> 00:01:17.880
Let's add the left-hand side.

00:01:17.880 --> 00:01:19.440
Negative 6x plus 6x?

00:01:19.440 --> 00:01:20.890
Well, those cancel out.

00:01:20.890 --> 00:01:22.080
We get 0.

00:01:22.080 --> 00:01:23.910
Then we have 3y minus 3y.

00:01:23.910 --> 00:01:24.890
Those cancel out.

00:01:24.890 --> 00:01:27.000
We get another 0.

00:01:27.000 --> 00:01:30.160
And then finally, you
get negative 42 plus 42.

00:01:30.160 --> 00:01:31.090
Well, that's a 0.

00:01:31.090 --> 00:01:35.230
So we end up with just
0 is equal to 0, which

00:01:35.230 --> 00:01:37.470
is clearly true, but it's
not putting any constraints

00:01:37.470 --> 00:01:38.810
on the x or y.

00:01:38.810 --> 00:01:41.640
And that's because whenever
you have a situation like this,

00:01:41.640 --> 00:01:44.160
where you just get something
that's obviously true-- 0

00:01:44.160 --> 00:01:46.810
equals 0, or 1
equals 1, or 5 equals

00:01:46.810 --> 00:01:48.970
5-- what we're dealing
in this situation

00:01:48.970 --> 00:01:50.580
is where both of
our constraints,

00:01:50.580 --> 00:01:53.720
both of our equations, are
actually the same equation.

00:01:53.720 --> 00:01:56.195
So this right here is
a dependent system.

00:02:01.429 --> 00:02:02.720
And you see it right over here.

00:02:02.720 --> 00:02:06.160
If you take that first
equation, you multiply it by 3,

00:02:06.160 --> 00:02:08.919
you got 6x minus
3y is equal to 42.

00:02:08.919 --> 00:02:13.620
If we then multiplied
it by negative 1,

00:02:13.620 --> 00:02:16.060
you would get the
exact same equation

00:02:16.060 --> 00:02:17.860
as the second equation.

00:02:17.860 --> 00:02:23.560
You would get negative 6x plus
3y is equal to negative 42.

00:02:23.560 --> 00:02:24.950
Or another way to
think about it,

00:02:24.950 --> 00:02:26.670
if you want to go from
the first equation

00:02:26.670 --> 00:02:28.860
to the second equation,
you just multiply

00:02:28.860 --> 00:02:33.630
both sides of the
equation times negative 3.

00:02:33.630 --> 00:02:36.280
So both of these constraints are
actually the same constraints.

00:02:36.280 --> 00:02:38.960
They're just kind of a scaled-up
multiple of each other.

00:02:38.960 --> 00:02:40.560
So if you were to
graph them-- and I

00:02:40.560 --> 00:02:42.800
might as well graph
them for you right here.

00:02:42.800 --> 00:02:45.840
This first equation right
here, let me do it over here.

00:02:45.840 --> 00:02:49.200
It's 2x minus y is equal to 14.

00:02:49.200 --> 00:02:52.020
You could subtract
2x from both sides,

00:02:52.020 --> 00:02:54.030
and you would get-- let
me just subtract it.

00:02:54.030 --> 00:02:56.770
Subtract 2x, subtract 2x.

00:02:56.770 --> 00:02:59.820
On the left-hand side, you're
left with just negative y.

00:02:59.820 --> 00:03:03.710
On the right-hand side, you
have negative 2x plus 14.

00:03:03.710 --> 00:03:06.170
Multiply both sides
by negative 1,

00:03:06.170 --> 00:03:10.730
and you get y is
equal to 2x minus 14.

00:03:10.730 --> 00:03:12.610
So this first
equation over here,

00:03:12.610 --> 00:03:18.890
if I were to draw my axes--
so that is my y-axis, and then

00:03:18.890 --> 00:03:22.440
that is my x-axis.

00:03:22.440 --> 00:03:25.650
This graph right over here,
its y-intercept is negative 14.

00:03:25.650 --> 00:03:29.460
So this is 0, negative 14.

00:03:29.460 --> 00:03:31.190
And it has a slope of 2.

00:03:31.190 --> 00:03:33.200
So it's going to look
something like this.

00:03:33.200 --> 00:03:35.559
It's going to look
something like that.

00:03:35.559 --> 00:03:37.100
And then this second
equation, if you

00:03:37.100 --> 00:03:39.960
were to graph it, because
this is a dependent system,

00:03:39.960 --> 00:03:41.606
it's the exact same line.

00:03:41.606 --> 00:03:43.980
If you were to put this in
slope-intercept form and graph

00:03:43.980 --> 00:03:46.240
it, you would get
the exact same thing.

00:03:46.240 --> 00:03:47.650
It would go right on top of it.

00:03:47.650 --> 00:03:50.120
So there's actually an
infinite number of solutions.

00:03:50.120 --> 00:03:52.600
These two lines
are the same line.

00:03:52.600 --> 00:03:56.120
So they intersect everywhere on
each of the respective lines.

00:03:56.120 --> 00:03:57.839
They're the same line.

00:03:57.839 --> 00:03:59.380
And when you get
something like this,

00:03:59.380 --> 00:04:04.090
0 equals 0 or 1 equals 1, that's
the telltale sign that you're

00:04:04.090 --> 00:04:05.990
dealing with a dependent system.

00:04:05.990 --> 00:04:09.360
If you got something
like 0 equals 1,

00:04:09.360 --> 00:04:10.990
then you would have no solution.

00:04:10.990 --> 00:04:17.741
So this one right here would
be an inconsistent system.

00:04:17.741 --> 00:04:19.074
And this would be the situation.

00:04:19.074 --> 00:04:20.282
So let me make it very clear.

00:04:20.282 --> 00:04:21.579
This is the situation.

00:04:21.579 --> 00:04:25.470
We're dealing with
the same lines.

00:04:25.470 --> 00:04:26.857
The lines are the same lines.

00:04:26.857 --> 00:04:28.940
This is a situation where
you have parallel lines.

00:04:33.320 --> 00:04:34.790
And so they never intersect.

00:04:34.790 --> 00:04:37.689
And then, obviously, you
have the easiest situation.

00:04:37.689 --> 00:04:39.980
I think this is one most of
us are familiar with, where

00:04:39.980 --> 00:04:43.724
you have something like x is
equal to 1, or y is equal to 2,

00:04:43.724 --> 00:04:44.640
or anything like this.

00:04:44.640 --> 00:04:45.880
It doesn't have to be 1 or 2.

00:04:45.880 --> 00:04:47.600
And then this is
clearly a situation

00:04:47.600 --> 00:04:49.790
where you have an
independent system, where

00:04:49.790 --> 00:04:51.690
you have two
different lines that

00:04:51.690 --> 00:04:54.380
intersect in exactly one place.

00:04:54.380 --> 00:04:57.470
Anyway, this-- there's an
infinite number of solutions.

00:04:57.470 --> 00:04:59.780
Any x and y that satisfied
the first equation

00:04:59.780 --> 00:05:01.700
will also satisfy
the second equation,

00:05:01.700 --> 00:05:04.498
because they're fundamentally
the same constraint.


Addition elimination method 2

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:01.737
Solve for x and y.

00:00:01.737 --> 00:00:03.570
And we have two equations
with two unknowns.

00:00:03.570 --> 00:00:06.390
Negative 3y plus
4x is equal to 11,

00:00:06.390 --> 00:00:10.290
and y plus 2x is equal to 13.

00:00:10.290 --> 00:00:13.780
What we could do is try to
solve this using elimination.

00:00:13.780 --> 00:00:15.780
Maybe we can add these
two equations together

00:00:15.780 --> 00:00:17.620
to get some variables
to cancel out.

00:00:17.620 --> 00:00:19.740
But if we just add
y and negative 3y,

00:00:19.740 --> 00:00:20.750
those won't cancel out.

00:00:20.750 --> 00:00:23.920
And if we just add a 2x and
4x, those won't cancel out.

00:00:23.920 --> 00:00:26.210
But maybe we can scale
one of these equations,

00:00:26.210 --> 00:00:28.210
so that we will get
some cancellation.

00:00:28.210 --> 00:00:31.250
If this y could
somehow become a 3y,

00:00:31.250 --> 00:00:34.250
then 3y and negative
3y would cancel out.

00:00:34.250 --> 00:00:36.780
And so the easiest way
to turn this y into a 3y

00:00:36.780 --> 00:00:39.890
is to just multiply
this entire equation,

00:00:39.890 --> 00:00:45.740
so we don't actually change
the information in it, by 3.

00:00:45.740 --> 00:00:47.490
So let's multiply
both sides by 3.

00:00:47.490 --> 00:00:54.650
If we multiply the left-hand
side by 3, we get 3y plus 6x.

00:00:54.650 --> 00:00:56.500
We have to multiply
every term by 3.

00:00:56.500 --> 00:01:02.440
3y plus 6x is equal
to 3 times 13 is 39.

00:01:02.440 --> 00:01:04.290
So this equation
and this equation

00:01:04.290 --> 00:01:05.540
are the exact same equation.

00:01:05.540 --> 00:01:09.100
I've just multiplied both sides
by 3 to go from here to here.

00:01:09.100 --> 00:01:10.780
So I multiplied by 3.

00:01:10.780 --> 00:01:13.260
And of course, we have this
original equation up here.

00:01:13.260 --> 00:01:19.290
That negative 3y plus
4x is equal to 11.

00:01:19.290 --> 00:01:22.370
And now, if we add the left-hand
side and the right-hand sides

00:01:22.370 --> 00:01:24.960
of this equation, something
interesting happens.

00:01:24.960 --> 00:01:27.860
The negative 3y and the
positive 3y cancel out.

00:01:27.860 --> 00:01:30.710
That was the whole point behind
multiplying this second guy

00:01:30.710 --> 00:01:32.090
over here by 3.

00:01:32.090 --> 00:01:33.110
So let's add them.

00:01:33.110 --> 00:01:35.700
So if we add, these
guys cancel out.

00:01:35.700 --> 00:01:38.500
We have 4x plus 6x is 10x.

00:01:38.500 --> 00:01:43.050
And that is equal to 11
plus 39, which is 50.

00:01:43.050 --> 00:01:45.380
And then we can divide
both sides by 10.

00:01:48.360 --> 00:01:53.200
We get x is equal to
50 divided by 10 is 5.

00:01:53.200 --> 00:01:56.610
And then we could go
substitute back to solve for y.

00:01:56.610 --> 00:01:59.320
If x is 5, we could use the
second equation right over

00:01:59.320 --> 00:01:59.820
here.

00:01:59.820 --> 00:02:03.590
We get y plus 2 times 5.

00:02:03.590 --> 00:02:05.280
I'll do it in that same color.

00:02:05.280 --> 00:02:09.570
2 times 5 is equal to 13.

00:02:09.570 --> 00:02:13.350
Or y plus 10 is equal to 13.

00:02:13.350 --> 00:02:16.460
We can subtract 10
from both sides.

00:02:16.460 --> 00:02:18.560
And we will get y.

00:02:18.560 --> 00:02:21.640
These cancel out.
y is equal to 3.

00:02:21.640 --> 00:02:23.660
So our solution that
we get for x and y

00:02:23.660 --> 00:02:25.620
is that x is equal to 5.

00:02:25.620 --> 00:02:27.210
And y is equal to 3.

00:02:27.210 --> 00:02:29.610
And we can verify that it
works in both equations.

00:02:29.610 --> 00:02:32.300
In this top equation-- let
me do it in a new color--

00:02:32.300 --> 00:02:40.290
negative 3 times 3 plus 4
times 5 is-- this is negative

00:02:40.290 --> 00:02:44.130
9 plus 20, which is,
indeed, equal to 11.

00:02:44.130 --> 00:02:47.050
So both of these satisfy
the first equation.

00:02:47.050 --> 00:02:53.500
And then if we take the second
equation, 3 plus 2 times 5,

00:02:53.500 --> 00:02:57.550
that's 3 plus 10, that
does, indeed, equal 13.

00:02:57.550 --> 00:02:59.182
So it definitely satisfies both.

00:02:59.182 --> 00:03:00.140
And I want to be clear.

00:03:00.140 --> 00:03:02.040
You could have solved
this with substitution.

00:03:02.040 --> 00:03:05.600
Or you could have changed which
equation you scale so that we

00:03:05.600 --> 00:03:07.820
get different
cancellations when we add

00:03:07.820 --> 00:03:08.904
or different eliminations.

00:03:08.904 --> 00:03:10.736
So another way that we
could have done this,

00:03:10.736 --> 00:03:12.830
we could've kept the
first equation the same.

00:03:12.830 --> 00:03:17.670
Negative 3y plus
4x is equal to 11.

00:03:17.670 --> 00:03:20.050
But maybe we want this 2x
to cancel with this 4x.

00:03:20.050 --> 00:03:21.424
And the only way
we could do that

00:03:21.424 --> 00:03:24.960
is if this 2x could
become a negative 4x.

00:03:24.960 --> 00:03:27.820
And the way to get from
a 2x to a negative 4x

00:03:27.820 --> 00:03:29.900
is to multiply it by negative 2.

00:03:29.900 --> 00:03:32.770
So let's multiply this entire
equation by negative 2.

00:03:32.770 --> 00:03:36.030
So it would become
negative 2y minus

00:03:36.030 --> 00:03:40.310
4x is equal to negative 26.

00:03:40.310 --> 00:03:42.660
And if we then add
the two equations,

00:03:42.660 --> 00:03:44.130
the x's will cancel out.

00:03:44.130 --> 00:03:46.220
You have negative
3y minus 2y, which

00:03:46.220 --> 00:03:51.100
is negative 5y is
equal to 11 minus 26.

00:03:51.100 --> 00:03:52.950
That's negative 15.

00:03:52.950 --> 00:03:56.800
Now, you can divide both
sides by negative 5.

00:03:56.800 --> 00:04:01.790
And you get-- the negatives
cancel out-- y is equal to 3.

00:04:01.790 --> 00:04:03.870
So in this situation,
we eliminated

00:04:03.870 --> 00:04:05.940
the x's first to solve for y.

00:04:05.940 --> 00:04:08.900
And then you could go back,
substitute, and solve for x.

00:04:08.900 --> 00:04:10.120
You'll get x is equal to 5.

00:04:10.120 --> 00:04:12.860
The first time we did it,
we eliminated the y's.

00:04:12.860 --> 00:04:14.940
We scaled this
equation right here so

00:04:14.940 --> 00:04:16.730
that the y's would cancel out.

00:04:16.730 --> 00:04:19.980
And so we solved for the x
first and then substituted back.

00:04:19.980 --> 00:04:20.880
Either way works.

00:04:20.880 --> 00:04:22.580
You get x is equal to 5.

00:04:22.580 --> 00:04:24.567
y is equal to 3.


Systems of three variables 2

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.060
Solve this system.

00:00:02.060 --> 00:00:04.839
And once again, we have three
equations with three unknowns.

00:00:04.839 --> 00:00:06.630
So this is essentially
trying to figure out

00:00:06.630 --> 00:00:08.720
where three different
planes would

00:00:08.720 --> 00:00:11.120
intersect in three dimensions.

00:00:11.120 --> 00:00:13.884
And to do this, if we want to
do it by elimination, if we want

00:00:13.884 --> 00:00:15.300
to be able to
eliminate variables,

00:00:15.300 --> 00:00:18.750
it looks like, well, it looks
like we have a negative z here.

00:00:18.750 --> 00:00:20.050
We have a plus 2z.

00:00:20.050 --> 00:00:21.760
We have a 5z over here.

00:00:21.760 --> 00:00:26.500
If we were to scale up this
third equation by positive 2,

00:00:26.500 --> 00:00:28.180
then you would have
a negative 2z here,

00:00:28.180 --> 00:00:30.350
and it would cancel
out with this 2z there.

00:00:30.350 --> 00:00:32.670
And then if you were
to scale it up by 5,

00:00:32.670 --> 00:00:34.830
you'd have a negative
5z here, and then that

00:00:34.830 --> 00:00:36.976
could cancel out with
that 5z over there.

00:00:36.976 --> 00:00:38.100
So let's try to cancel out.

00:00:38.100 --> 00:00:40.339
Let's try to eliminate
the z's first.

00:00:40.339 --> 00:00:42.130
So let me start with
this equation up here.

00:00:42.130 --> 00:00:43.480
I'll just rewrite it.

00:00:43.480 --> 00:00:45.580
So we have-- I'll draw
an arrow over here--

00:00:45.580 --> 00:00:53.770
we have x plus 2y plus 5z
is equal to negative 17.

00:00:53.770 --> 00:00:56.170
And then to cancel out or
to eliminate the z's, I'll

00:00:56.170 --> 00:00:59.160
multiply this
equation here times 5.

00:00:59.160 --> 00:01:01.980
So I'm going to multiply
this equation times 5.

00:01:01.980 --> 00:01:04.470
So we have to multiply
both sides by 5.

00:01:04.470 --> 00:01:11.790
So 3x times 5 is 15x, y times
5 is plus 5y, and then negative

00:01:11.790 --> 00:01:14.645
z times 5 is negative 5z--
that's the whole point

00:01:14.645 --> 00:01:18.610
and why we're multiplying it
by 5-- is equal to 3 times

00:01:18.610 --> 00:01:21.580
5, which is equal to 15.

00:01:21.580 --> 00:01:26.330
And so if we add these two
equations, we get x plus 15x

00:01:26.330 --> 00:01:34.980
is 16x, 2y plus 5y is 7y, and
5z minus 5z or plus negative 5z,

00:01:34.980 --> 00:01:36.810
those are going to cancel out.

00:01:36.810 --> 00:01:39.560
And that is going to
be equal to negative 17

00:01:39.560 --> 00:01:42.430
plus 15 is negative 2.

00:01:42.430 --> 00:01:44.580
So we were able to
use the constraints

00:01:44.580 --> 00:01:46.690
in that equation
and that equation,

00:01:46.690 --> 00:01:49.134
and now we have an
equation in just x and y.

00:01:49.134 --> 00:01:50.550
So let's try to
do the same thing.

00:01:50.550 --> 00:01:52.270
Let's trying to
eliminate the z's.

00:01:52.270 --> 00:01:55.750
But now I'll use this
equation and this equation.

00:01:55.750 --> 00:01:58.130
So this equation-- let me
just rewrite it over here.

00:01:58.130 --> 00:02:04.380
We have 2x minus 3y plus
2z is equal to negative 16.

00:02:04.380 --> 00:02:05.750
I just rewrote it.

00:02:05.750 --> 00:02:08.440
And now, so that this
2z gets eliminated,

00:02:08.440 --> 00:02:11.900
let's multiply this
equation times 2.

00:02:11.900 --> 00:02:15.560
So let's multiply it times 2,
so we'll have a negative 2z here

00:02:15.560 --> 00:02:17.650
to eliminate with
the positive 2z.

00:02:17.650 --> 00:02:23.920
So 2 times 3x is 6x,
2 times y is plus 2y,

00:02:23.920 --> 00:02:29.750
and then 2 times negative z is
negative 2z is equal to 2 times

00:02:29.750 --> 00:02:31.420
3 is equal to 6.

00:02:31.420 --> 00:02:34.790
And now we can add
these two equations.

00:02:34.790 --> 00:02:41.050
2x plus 6x is 8x, negative
3y plus 2y is negative y,

00:02:41.050 --> 00:02:43.570
and then these two
guys get canceled out.

00:02:43.570 --> 00:02:45.690
And then that is
equal to negative 16

00:02:45.690 --> 00:02:50.510
plus 6 is negative 10.

00:02:50.510 --> 00:02:53.100
So now we have two
equations with two unknowns.

00:02:53.100 --> 00:02:55.080
We've eliminated the z's.

00:02:55.080 --> 00:02:57.580
And let's see, if we
want to eliminate again,

00:02:57.580 --> 00:02:59.220
we have a negative y over here.

00:02:59.220 --> 00:03:01.190
We have a positive 7y.

00:03:01.190 --> 00:03:04.980
Well, we could eliminate the y's
if we multiplied this times 7

00:03:04.980 --> 00:03:06.250
and add the two equations.

00:03:06.250 --> 00:03:07.470
So let's do that.

00:03:07.470 --> 00:03:11.290
So let's multiply this times 7.

00:03:11.290 --> 00:03:22.460
7 times 8 is 56, so
it's 56x minus 7y

00:03:22.460 --> 00:03:28.440
is equal to 7 times negative
10, is equal to negative 70.

00:03:28.440 --> 00:03:31.530
And now we can add
these two equations.

00:03:31.530 --> 00:03:33.990
I'm now trying to
eliminate the y's.

00:03:33.990 --> 00:03:37.230
So we have 16x plus 56x.

00:03:37.230 --> 00:03:39.660
That is 72x.

00:03:39.660 --> 00:03:43.750
So we have 72x, these
guys eliminate, equal

00:03:43.750 --> 00:03:50.290
to negative 72, negative
2 plus negative 70.

00:03:50.290 --> 00:03:57.155
Divide both sides by 72, and we
get x is equal to negative 1.

00:03:57.155 --> 00:03:58.780
And now we just have
to substitute back

00:03:58.780 --> 00:04:01.110
to figure out what y
and z are equal to.

00:04:01.110 --> 00:04:04.210
So let's go back to this
equation right over here.

00:04:04.210 --> 00:04:07.470
So we have 8x minus y
is equal to negative 10.

00:04:07.470 --> 00:04:10.690
If x is equal to negative 1,
that means 8 times negative 1,

00:04:10.690 --> 00:04:15.540
or negative 8 minus y
is equal to negative 10.

00:04:15.540 --> 00:04:18.370
We can add 8 to both sides.

00:04:18.370 --> 00:04:24.010
And so we have negative
y is equal to negative 2,

00:04:24.010 --> 00:04:28.650
or multiplying both sides by
negative 1, y is equal to 2.

00:04:28.650 --> 00:04:30.380
let me square that off.

00:04:30.380 --> 00:04:33.270
So x is equal to negative
1, y is equal to 2.

00:04:33.270 --> 00:04:35.280
We now just have
to worry about z,

00:04:35.280 --> 00:04:37.570
and we can go back to
any of these up here.

00:04:37.570 --> 00:04:39.280
So I'll just use this last one.

00:04:39.280 --> 00:04:41.230
The numbers seem lower.

00:04:41.230 --> 00:04:43.770
So if we substitute back
into this last equation

00:04:43.770 --> 00:04:46.670
right over here,
we have 3 times x,

00:04:46.670 --> 00:04:53.930
which is 3 times negative 1
plus y, which is 2, minus z

00:04:53.930 --> 00:04:55.420
is equal to 3.

00:04:55.420 --> 00:04:59.920
So it's negative 3 plus
2 minus z is equal to 3.

00:04:59.920 --> 00:05:06.120
And this is negative 1
minus z is equal to 3.

00:05:06.120 --> 00:05:13.050
And then add 1 to both sides,
we get-- these cancel out--

00:05:13.050 --> 00:05:16.310
negative z is equal to 4.

00:05:16.310 --> 00:05:18.250
Multiplying both
sides by negative 1,

00:05:18.250 --> 00:05:23.600
you get z is equal
to negative 4.

00:05:23.600 --> 00:05:24.290
So we're done.

00:05:24.290 --> 00:05:27.500
Let's verify that these
solutions, or the solution of x

00:05:27.500 --> 00:05:30.540
is negative 1, y is equal to
2, z is equal to negative 4,

00:05:30.540 --> 00:05:33.020
actually satisfies all
three of these constraints.

00:05:33.020 --> 00:05:35.100
So let's substitute
into this first one.

00:05:35.100 --> 00:05:37.190
So you have x plus 2y plus 5z.

00:05:37.190 --> 00:05:45.120
So x is negative 1 plus 2
times y, so plus 4, plus 5z.

00:05:45.120 --> 00:05:49.580
So minus 20 has to be
equal to negative 17.

00:05:49.580 --> 00:05:50.540
And this is negative.

00:05:50.540 --> 00:05:52.520
This right here is
positive 3 minus 20

00:05:52.520 --> 00:05:54.730
is indeed equal to negative 17.

00:05:54.730 --> 00:05:56.552
So it satisfies the
first constraint.

00:05:56.552 --> 00:05:57.760
Let's look at the second one.

00:05:57.760 --> 00:06:00.270
2 times x, 2 times negative 1.

00:06:00.270 --> 00:06:03.560
That's negative 2
minus 3 times y.

00:06:03.560 --> 00:06:09.512
That's minus 6-- 3 times
2 is 6-- plus 2 times z.

00:06:09.512 --> 00:06:13.230
z is negative 4, so that's 2
times negative 4 is negative 8.

00:06:13.230 --> 00:06:15.590
That needs to equal negative 16.

00:06:15.590 --> 00:06:18.430
Negative 2 minus negative
8 is-- sorry, negative 2

00:06:18.430 --> 00:06:20.030
minus negative 6 is negative 8.

00:06:20.030 --> 00:06:22.560
Subtract another 8,
you get negative 16.

00:06:22.560 --> 00:06:24.470
So it meets the
second constraint.

00:06:24.470 --> 00:06:27.100
And then, finally, let's
look at the last constraint.

00:06:27.100 --> 00:06:35.900
We have 3 times x, so negative
3 plus y, so plus 2, minus z.

00:06:35.900 --> 00:06:39.760
So minus negative 4 is
the same thing as plus 4.

00:06:39.760 --> 00:06:41.550
That needs to be equal to 3.

00:06:41.550 --> 00:06:46.340
So negative 3 plus 2 is negative
1 plus 4 is indeed equal to 3.

00:06:46.340 --> 00:06:49.330
So we found the point
of intersection in three

00:06:49.330 --> 00:06:51.670
dimensions of
these three planes.

00:06:51.670 --> 00:06:54.930
x is negative 1, z is
negative 4, y is 2.

00:06:54.930 --> 00:06:57.450
And we were able to
verify that it does indeed

00:06:57.450 --> 00:06:59.524
meet all of the constraints.


Three equation application problem

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.940
Solve the following application
problem using three equations

00:00:03.940 --> 00:00:05.510
with three unknowns.

00:00:05.510 --> 00:00:07.540
And they tell us the
second angle of a triangle

00:00:07.540 --> 00:00:11.910
is 50 degrees less than
four times the first angle.

00:00:11.910 --> 00:00:15.200
The third angle is 40
degrees less than the first.

00:00:15.200 --> 00:00:18.090
Find the measures
of the three angles.

00:00:18.090 --> 00:00:21.280
So let's draw ourselves
a triangle here.

00:00:21.280 --> 00:00:26.300
And let's call the first angle
a, the second angle b, and then

00:00:26.300 --> 00:00:28.076
the third angle c.

00:00:28.076 --> 00:00:29.950
And before we even look
at these constraints,

00:00:29.950 --> 00:00:31.990
one property we
know of triangles

00:00:31.990 --> 00:00:36.200
is that the sum of their
angles must be 180 degrees.

00:00:36.200 --> 00:00:42.472
So we know that a plus b plus
c must be equal to 180 degrees.

00:00:42.472 --> 00:00:44.180
Now, with that out of
the way, let's look

00:00:44.180 --> 00:00:46.400
at these other constraints.

00:00:46.400 --> 00:00:50.530
So they tell us the second
angle of a triangle--

00:00:50.530 --> 00:00:52.650
let me do that in
another color--

00:00:52.650 --> 00:00:55.750
they tell us the second
angle of a triangle

00:00:55.750 --> 00:01:00.370
is 50 degrees less than
four times the first angle.

00:01:00.370 --> 00:01:02.440
So we're saying b
is the second angle.

00:01:02.440 --> 00:01:04.840
So they're saying, the
second angle of a triangle

00:01:04.840 --> 00:01:08.730
is 50 degrees less than
four times the first angle.

00:01:08.730 --> 00:01:11.850
So 4 times the first
angle would be 4a,

00:01:11.850 --> 00:01:14.040
we're calling a the first angle.

00:01:14.040 --> 00:01:15.780
So 4 times the
first angle is 4a.

00:01:15.780 --> 00:01:20.770
But it's 50 degrees less
than that, so minus 50.

00:01:20.770 --> 00:01:23.670
Now the next constraint they
give us, the third angle

00:01:23.670 --> 00:01:26.250
is 40 degrees less
than the first.

00:01:26.250 --> 00:01:30.970
So the third angle is 40
degrees less than the first.

00:01:30.970 --> 00:01:34.070
So the first angle, a, is
going to be 40 degrees less

00:01:34.070 --> 00:01:35.590
than that.

00:01:35.590 --> 00:01:38.930
So we have three equations
with three unknowns.

00:01:38.930 --> 00:01:40.830
And so we just have
to solve for it.

00:01:40.830 --> 00:01:43.320
And let's see, what's a
good first variable to try

00:01:43.320 --> 00:01:44.270
to eliminate?

00:01:44.270 --> 00:01:47.940
And just to try to visualize
that a little bit better,

00:01:47.940 --> 00:01:50.990
I'm going to bring these a's
onto the left hand side of each

00:01:50.990 --> 00:01:52.820
of these equations over here.

00:01:52.820 --> 00:01:54.840
So I'm going to rewrite
the first equation.

00:01:54.840 --> 00:02:00.950
We have a plus b plus
c is equal to 180.

00:02:00.950 --> 00:02:03.340
And then this equation,
if we subtract 4a

00:02:03.340 --> 00:02:06.920
from both sides of this
equation, if we subtract 4a

00:02:06.920 --> 00:02:16.670
we have negative 4a plus
b is equal to negative 50.

00:02:16.670 --> 00:02:18.630
And then this equation
right over here,

00:02:18.630 --> 00:02:21.400
if we subtract a
from both sides we

00:02:21.400 --> 00:02:33.620
get negative a plus c
is equal to negative 40.

00:02:33.620 --> 00:02:36.280
I just subtract a
from both sides.

00:02:36.280 --> 00:02:38.600
So we now want to
eliminate variables.

00:02:38.600 --> 00:02:40.340
And we already have--
this third equation

00:02:40.340 --> 00:02:41.750
here is only in
terms of a and c.

00:02:41.750 --> 00:02:43.480
This is only in
terms of a and b.

00:02:43.480 --> 00:02:45.850
And this first one is
in terms of a, b, and c.

00:02:45.850 --> 00:02:49.240
So if we could-- let's see, this
is already in terms of a and c.

00:02:49.240 --> 00:02:51.240
If we could turn these
first two equations,

00:02:51.240 --> 00:02:54.580
if we can use the information
in these first two equations

00:02:54.580 --> 00:02:56.250
to end up with an
equation that's

00:02:56.250 --> 00:02:58.380
only in terms of
a and c, then we

00:02:58.380 --> 00:03:00.510
could use whatever
we end up with along

00:03:00.510 --> 00:03:02.770
with this third equation
right over here,

00:03:02.770 --> 00:03:05.690
and we'll have a system of two
equations with two unknowns.

00:03:05.690 --> 00:03:06.519
So let's do that.

00:03:06.519 --> 00:03:09.060
So if we wanted to just end up
with an equation only in terms

00:03:09.060 --> 00:03:11.180
of a and c using
these first two,

00:03:11.180 --> 00:03:14.009
we would want to
eliminate the b's.

00:03:14.009 --> 00:03:15.800
So we could multiply
one of these equations

00:03:15.800 --> 00:03:17.966
times negative 1, and one
of these positive b's will

00:03:17.966 --> 00:03:19.110
turn into a negative b.

00:03:19.110 --> 00:03:19.950
So let's do that.

00:03:19.950 --> 00:03:22.560
Let's multiply this first
equation over here, let's

00:03:22.560 --> 00:03:26.350
multiply it times negative 1.

00:03:26.350 --> 00:03:29.020
So it'll become
negative a minus b

00:03:29.020 --> 00:03:33.280
minus c is equal
to negative 180.

00:03:33.280 --> 00:03:35.100
And then we have this
green equation right

00:03:35.100 --> 00:03:38.990
over here, which is really just
this equation, just rearranged.

00:03:38.990 --> 00:03:45.600
So we have negative 4a plus
b is equal to negative 50.

00:03:45.600 --> 00:03:48.927
And now we can add
these two equations.

00:03:48.927 --> 00:03:50.760
Actually, let me do
that in the other color,

00:03:50.760 --> 00:03:53.056
just so you see where
that's coming from.

00:03:53.056 --> 00:03:55.060
I'll do it in that green color.

00:03:55.060 --> 00:04:01.160
So this is negative 4a plus
b is equal to negative 50.

00:04:01.160 --> 00:04:03.620
We can add these two up now.

00:04:03.620 --> 00:04:10.370
And we get negative a
minus 4a is negative 5a.

00:04:10.370 --> 00:04:12.030
The b's cancel out.

00:04:12.030 --> 00:04:16.279
We have a minus c is equal
to negative 180 minus 50

00:04:16.279 --> 00:04:18.600
is negative 230.

00:04:18.600 --> 00:04:20.260
So now using these
top two equations

00:04:20.260 --> 00:04:22.357
we have an equation only
in terms of a and c.

00:04:22.357 --> 00:04:24.440
We have another equation
only in terms of a and c.

00:04:24.440 --> 00:04:26.210
And it looks like if we
add them together their c's

00:04:26.210 --> 00:04:27.020
will cancel out.

00:04:27.020 --> 00:04:29.007
So let me just rewrite
this equation over here.

00:04:29.007 --> 00:04:30.590
And you have to be
careful that you're

00:04:30.590 --> 00:04:31.720
using all of the equations.

00:04:31.720 --> 00:04:33.950
Otherwise you'll do
a circular argument.

00:04:33.950 --> 00:04:37.460
You have to be careful that over
here this first equation came

00:04:37.460 --> 00:04:39.430
from these two over here.

00:04:39.430 --> 00:04:42.250
Now I want to combine that
with this third constraint,

00:04:42.250 --> 00:04:45.270
s constraint that's not already
baked into this equation

00:04:45.270 --> 00:04:45.980
right over here.

00:04:45.980 --> 00:04:52.320
So we have negative a plus
c is equal to negative 40.

00:04:52.320 --> 00:04:54.800
And we add these two equations.

00:04:54.800 --> 00:04:59.040
Negative 5a minus
a is negative 6a.

00:04:59.040 --> 00:05:00.740
The c's cancel out.

00:05:00.740 --> 00:05:03.530
And then you have
negative 230 minus 40.

00:05:03.530 --> 00:05:06.190
This is equal to negative 270.

00:05:06.190 --> 00:05:12.080
We can divide both
sides by negative 6.

00:05:12.080 --> 00:05:16.550
And we get a is equal
to negative 270 over 6.

00:05:16.550 --> 00:05:20.630
Let me see how many times--
let me see something.

00:05:20.630 --> 00:05:24.110
270 is divisible
by both 3 and 2.

00:05:24.110 --> 00:05:25.870
So it should be divisible by 6.

00:05:25.870 --> 00:05:27.350
So let me just divide it.

00:05:27.350 --> 00:05:29.280
The negative signs
obviously will-- a negative

00:05:29.280 --> 00:05:31.280
divided by a negative is
going to be a positive.

00:05:31.280 --> 00:05:36.210
And if we take 6 into 270,
6 goes into 27 4 times.

00:05:36.210 --> 00:05:38.420
4 times 6 is 24.

00:05:38.420 --> 00:05:39.670
We subtract.

00:05:39.670 --> 00:05:41.670
We get 3, bring down the 0.

00:05:41.670 --> 00:05:44.970
6 goes into 30 five times.

00:05:44.970 --> 00:05:51.725
So we get a is equal to 45.

00:05:51.725 --> 00:05:53.100
Now let's look at
the other ones.

00:05:53.100 --> 00:05:55.316
We can substitute back
in to solve for c.

00:05:55.316 --> 00:05:58.930
c is equal to a
minus 40 degrees.

00:06:01.640 --> 00:06:04.270
So that is equal to-- let
me write it right over

00:06:04.270 --> 00:06:10.320
here in yellow-- so c
is equal to 45 minus 40,

00:06:10.320 --> 00:06:12.020
which is equal to 5 degrees.

00:06:12.020 --> 00:06:16.490
So, so far we have a
is equal to 45 degrees,

00:06:16.490 --> 00:06:19.740
c is equal to 5 degrees.

00:06:22.826 --> 00:06:24.700
And then you could
substitute into either one

00:06:24.700 --> 00:06:26.470
of these other ones
to figure out b.

00:06:26.470 --> 00:06:28.730
We could use this one
right over here in green.

00:06:28.730 --> 00:06:31.170
B is equal to 4a minus 50.

00:06:31.170 --> 00:06:36.310
So b is going to be equal to
4 times 45 is-- let's see,

00:06:36.310 --> 00:06:37.580
2 times 45 is 90.

00:06:37.580 --> 00:06:39.110
So 4 times 45 is 180.

00:06:39.110 --> 00:06:42.680
So it's going to
be 180 minus 50,

00:06:42.680 --> 00:06:44.840
by this equation
right over here, which

00:06:44.840 --> 00:06:48.110
is equal to 130 degrees.

00:06:48.110 --> 00:06:52.342
So we get b is equal
to 130 degrees.

00:06:52.342 --> 00:06:54.050
And then we can-- let
me write over here.

00:06:54.050 --> 00:06:55.330
So a is equal to 45.

00:06:55.330 --> 00:06:57.780
So if I wanted to
draw this triangle

00:06:57.780 --> 00:06:59.870
it would actually look
something like this.

00:06:59.870 --> 00:07:07.150
a is a 45 degree angle, b is a
130 degree angle, and c is 5.

00:07:07.150 --> 00:07:11.150
So it'll look
something like this.

00:07:11.150 --> 00:07:12.570
It will look
something like this,

00:07:12.570 --> 00:07:17.370
where this is a at 45
degrees, b is 135 degrees,

00:07:17.370 --> 00:07:19.030
and then c is 5 degrees.

00:07:19.030 --> 00:07:20.680
And you can verify
that it works.

00:07:20.680 --> 00:07:22.210
One, you can just
add up the angles.

00:07:22.210 --> 00:07:25.310
45 plus 5 is 50.

00:07:25.310 --> 00:07:26.780
Oh, sorry, this isn't 135.

00:07:26.780 --> 00:07:28.382
It's 130.

00:07:28.382 --> 00:07:29.590
We solved it right over here.

00:07:29.590 --> 00:07:31.630
It's 130 and this is 5.

00:07:31.630 --> 00:07:36.530
So when you add them
all up-- 45 plus 130

00:07:36.530 --> 00:07:40.830
plus 5-- that does
indeed equal 180 degrees.

00:07:40.830 --> 00:07:43.440
45 plus 5 is 50 plus 130.

00:07:43.440 --> 00:07:46.150
So this does
definitely equal 180.

00:07:46.150 --> 00:07:48.010
So it meets our
first constraint,

00:07:48.010 --> 00:07:50.050
Then on our second
constraint, b needs

00:07:50.050 --> 00:07:52.040
to be equal to 4a minus 50.

00:07:52.040 --> 00:07:58.880
Well, 4 times a is 180
minus 50 is 130 degrees.

00:07:58.880 --> 00:08:00.860
So it meets our
second constraint.

00:08:00.860 --> 00:08:04.780
And then our third constraint,
c is a minus 40 degrees.

00:08:04.780 --> 00:08:07.200
Well, a is 45, c is 5.

00:08:07.200 --> 00:08:10.890
So if you subtract 40 from
45 you get 5, which is c.

00:08:10.890 --> 00:08:15.140
So it meets all of our
constraints and we are done.


Addition elimination method 4

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.620
Solve using the
elimination method.

00:00:02.620 --> 00:00:05.290
And they tell us the sum
of two numbers is 70.

00:00:05.290 --> 00:00:06.940
Their difference is 24.

00:00:06.940 --> 00:00:08.780
What are the two numbers?

00:00:08.780 --> 00:00:10.212
So let's use this
first sentence.

00:00:10.212 --> 00:00:12.420
Let's construct an equation
from this first sentence.

00:00:12.420 --> 00:00:13.860
Let's construct a constraint.

00:00:13.860 --> 00:00:14.920
The sum of two numbers.

00:00:14.920 --> 00:00:16.660
Let's call those
numbers x and y.

00:00:16.660 --> 00:00:20.890
So their sum, x plus
y, is equal to 70.

00:00:20.890 --> 00:00:22.810
That's what this first
sentence tells us.

00:00:22.810 --> 00:00:26.430
The second sentence says
their difference is 24.

00:00:26.430 --> 00:00:30.334
So that means that x
minus y is equal to 24.

00:00:30.334 --> 00:00:32.750
We're just going to assume
that x is the larger of the two

00:00:32.750 --> 00:00:34.640
numbers, and y is
the smaller one.

00:00:34.640 --> 00:00:36.473
So when you take their
difference like this,

00:00:36.473 --> 00:00:37.610
you get positive 24.

00:00:37.610 --> 00:00:40.730
So we have a system of two
equations with two unknowns.

00:00:40.730 --> 00:00:43.750
And they want us to solve it
using the elimination method.

00:00:43.750 --> 00:00:45.140
So let's do that.

00:00:45.140 --> 00:00:48.677
So if we were to just
add these two equations,

00:00:48.677 --> 00:00:50.760
on the left-hand side, we
would have a positive y,

00:00:50.760 --> 00:00:52.557
and we would have a
negative y over here.

00:00:52.557 --> 00:00:53.640
And they would cancel out.

00:00:53.640 --> 00:00:55.610
So if we were to just
add these two equations,

00:00:55.610 --> 00:00:57.650
we would be able to
eliminate the y's.

00:00:57.650 --> 00:00:58.820
So let's do that.

00:00:58.820 --> 00:01:01.540
So x plus y plus x minus y.

00:01:01.540 --> 00:01:03.670
Well, the plus y and
the minus y cancel out.

00:01:03.670 --> 00:01:07.210
And you're just left with
an x plus an x, which is 2x.

00:01:07.210 --> 00:01:11.330
And then that is going to
be equal to 70 plus 24.

00:01:11.330 --> 00:01:13.944
70 plus 24 is 94.

00:01:13.944 --> 00:01:15.360
And I want to make
it very clear--

00:01:15.360 --> 00:01:17.120
and I mentioned it
in previous videos--

00:01:17.120 --> 00:01:19.970
that this process of adding the
equations to each other, this

00:01:19.970 --> 00:01:20.750
is nothing new.

00:01:20.750 --> 00:01:23.440
We're really just adding the
same thing to both sides.

00:01:23.440 --> 00:01:25.720
We could do it as to both
sides of this equation.

00:01:25.720 --> 00:01:28.570
You could say we're adding 24
to both sides of this equation.

00:01:28.570 --> 00:01:31.330
Over here were explicitly
adding 24 to the 70.

00:01:31.330 --> 00:01:34.740
And over here you could say
we could add 24 to x plus y,

00:01:34.740 --> 00:01:37.080
but the second constraint
tells us that x minus y

00:01:37.080 --> 00:01:38.920
is the same thing as 24.

00:01:38.920 --> 00:01:40.950
So we're adding the same
thing to both sides.

00:01:40.950 --> 00:01:42.530
Here we're calling it 24.

00:01:42.530 --> 00:01:44.410
Here we're calling it x minus y.

00:01:44.410 --> 00:01:46.500
And we were able
to eliminate the y.

00:01:46.500 --> 00:01:48.500
So we get 2x is equal to 94.

00:01:48.500 --> 00:01:52.250
Now we can divide both
sides of this equation by 2.

00:01:52.250 --> 00:01:56.750
And we are left with x is
equal to-- what is that?

00:01:56.750 --> 00:01:58.090
47.

00:01:58.090 --> 00:02:00.010
And now we can substitute
back into either one

00:02:00.010 --> 00:02:02.100
of these equations
to solve for y.

00:02:02.100 --> 00:02:04.200
So let's try this
first one over here.

00:02:04.200 --> 00:02:08.630
So we have 47 plus
y is equal to 70.

00:02:08.630 --> 00:02:13.270
We can subtract 47 from
both sides of this equation.

00:02:13.270 --> 00:02:15.020
So we subtract 47.

00:02:15.020 --> 00:02:18.550
And we are left with y is
equal to-- what is this?

00:02:18.550 --> 00:02:20.020
23.

00:02:20.020 --> 00:02:22.500
y is equal to 23.

00:02:22.500 --> 00:02:23.980
And you can verify
that it works.

00:02:23.980 --> 00:02:28.230
If you add the two numbers, 47
plus 23, you definitely get 70.

00:02:28.230 --> 00:02:33.190
And then if you take 47 minus
23, you definitely get 24.

00:02:33.190 --> 00:02:36.767
So it definitely meets
both constraints.