Solutions to three variable system

WEBVTT
Kind: captions
Language: en

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Determine whether the
system has no solutions

00:00:03.000 --> 00:00:04.344
or infinite solutions.

00:00:04.344 --> 00:00:06.760
So they're telling us right
from the get go that we're not

00:00:06.760 --> 00:00:09.340
going to have one nice,
clean solution here.

00:00:09.340 --> 00:00:11.374
So let's just try to
solve this system.

00:00:11.374 --> 00:00:12.790
And so to solve
the system we want

00:00:12.790 --> 00:00:16.040
to eliminate one variable first.

00:00:16.040 --> 00:00:19.710
So let's say we want to
focus on eliminating,

00:00:19.710 --> 00:00:21.600
let's eliminate the x's.

00:00:21.600 --> 00:00:23.160
So if we want to
eliminate the x's,

00:00:23.160 --> 00:00:26.580
if I multiply this
first equation over here

00:00:26.580 --> 00:00:29.650
by negative 4, then this
will become a negative 8x

00:00:29.650 --> 00:00:32.270
which should be able to cancel
out with this 8x over here.

00:00:32.270 --> 00:00:34.560
So let me multiply
it by negative 4.

00:00:34.560 --> 00:00:36.580
So times negative 4.

00:00:36.580 --> 00:00:40.000
So 2 times negative
4 is negative 8x.

00:00:40.000 --> 00:00:44.920
Negative 4y times
negative 4 is plus 16y.

00:00:44.920 --> 00:00:49.150
Negative 4 times
z is negative 4z.

00:00:49.150 --> 00:00:52.060
And then that is going to be
equal to negative 4 times 3

00:00:52.060 --> 00:00:54.050
is negative 12.

00:00:54.050 --> 00:00:56.190
And then let me just
rewrite the second equation

00:00:56.190 --> 00:00:57.410
right over here.

00:00:57.410 --> 00:01:05.190
We have 8x minus 2y
plus 4z is equal to 7.

00:01:05.190 --> 00:01:07.540
And now we can add
these two equations

00:01:07.540 --> 00:01:09.250
and hopefully
eliminate the x terms.

00:01:09.250 --> 00:01:12.100
Negative 8x plus
8x, they eliminate.

00:01:12.100 --> 00:01:15.600
16y minus 2y is 14y.

00:01:15.600 --> 00:01:18.580
And then negative 4z plus
4z, those also eliminate.

00:01:18.580 --> 00:01:21.630
We were able to eliminate and
x and a z in one fell swoop.

00:01:21.630 --> 00:01:28.190
And so we get 14y
is equal to-- we

00:01:28.190 --> 00:01:31.990
have negative 12
plus 7 is negative 5.

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We have negative
5, just like that.

00:01:35.544 --> 00:01:36.960
And actually, we
could solve this.

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This is one equation
and one unknown.

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But we haven't used the
third constraint yet.

00:01:40.630 --> 00:01:43.080
We haven't used this
constraint over here.

00:01:43.080 --> 00:01:44.770
So we have-- we could
actually solve it.

00:01:44.770 --> 00:01:57.180
Just divide both sides by 14 and
we get y is equal to negative 5

00:01:57.180 --> 00:01:59.100
over 14.

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So that's interesting,
but it doesn't necessarily

00:02:01.910 --> 00:02:04.640
tell us much about
the solution just yet.

00:02:04.640 --> 00:02:07.162
You want to be very careful
about back substituting

00:02:07.162 --> 00:02:08.620
and all of that,
because we haven't

00:02:08.620 --> 00:02:10.180
used all of the constraints yet.

00:02:10.180 --> 00:02:12.780
We still haven't used
this third constraint.

00:02:12.780 --> 00:02:14.930
So our goal was to eliminate
x's from the get go.

00:02:14.930 --> 00:02:17.430
So let's still try to eliminate
x's using these second two

00:02:17.430 --> 00:02:20.047
constraints and see if we
get anything interesting.

00:02:20.047 --> 00:02:21.880
And the reason why I'm
being especially wary

00:02:21.880 --> 00:02:24.046
is because they're telling
us that this thing either

00:02:24.046 --> 00:02:26.770
has no solutions or an
infinite number of solutions.

00:02:26.770 --> 00:02:28.839
So let's use these
second two constraints.

00:02:28.839 --> 00:02:30.380
And we can eliminate
the x here if we

00:02:30.380 --> 00:02:31.890
keep the first
equation the same.

00:02:31.890 --> 00:02:34.077
And we multiply the
second equation by 2,

00:02:34.077 --> 00:02:35.910
because this will become
a negative 8x which

00:02:35.910 --> 00:02:37.180
will eliminate.

00:02:37.180 --> 00:02:43.260
So our first equation is just 8x
minus 2y plus 4z is equal to 7,

00:02:43.260 --> 00:02:44.345
just rewrote it.

00:02:44.345 --> 00:02:45.970
And then the second
equation over here.

00:02:45.970 --> 00:02:49.980
I'm going to multiply it by 2.

00:02:49.980 --> 00:02:53.586
Negative 4x times
2 is negative 8x.

00:02:53.586 --> 00:02:57.550
y times 2 is plus
2y, positive 2y.

00:02:57.550 --> 00:03:00.830
Negative 2z times
2 is negative 4z.

00:03:00.830 --> 00:03:05.910
And then negative 14
times 2 is negative 28.

00:03:05.910 --> 00:03:08.180
And something very interesting
is about to happen.

00:03:08.180 --> 00:03:11.644
When you use these
two constraints, when

00:03:11.644 --> 00:03:13.560
you try to find the
intersection, essentially,

00:03:13.560 --> 00:03:16.870
between these two planes, you
get something very bizarre.

00:03:16.870 --> 00:03:19.430
8x minus 8x is 0.

00:03:19.430 --> 00:03:22.170
Negative 2y plus
2y is another 0.

00:03:22.170 --> 00:03:25.030
4z minus 4z, another 0.

00:03:25.030 --> 00:03:29.712
And 7 minus 28 is negative 21.

00:03:29.712 --> 00:03:31.170
So using these two
constraints, you

00:03:31.170 --> 00:03:34.430
get this kind of nonsensical
statement or inconsistent

00:03:34.430 --> 00:03:37.780
statement that 0 is
equal to negative 21.

00:03:37.780 --> 00:03:40.150
This obviously can never happen.

00:03:40.150 --> 00:03:45.510
So this is, we end
up with no solution.

00:03:45.510 --> 00:03:49.440
If we got something like
0 is equal to 0, or 21

00:03:49.440 --> 00:03:52.080
is equal to 21, or
negative 21 is equal to 21,

00:03:52.080 --> 00:03:54.860
then it would mean that we have
the same exact planes which

00:03:54.860 --> 00:03:57.160
would mean we have an
infinite number solutions.

00:03:57.160 --> 00:04:00.570
In this circumstance right here,
we have the same coefficients.

00:04:00.570 --> 00:04:02.830
But when you add them
up in this weighted way

00:04:02.830 --> 00:04:04.520
you get a different result.

00:04:04.520 --> 00:04:07.150
So these are actually going
to be parallel planes.

00:04:07.150 --> 00:04:09.390
So if you were to graph
this plane and this plane,

00:04:09.390 --> 00:04:11.480
one would look
like that and then

00:04:11.480 --> 00:04:14.126
the other would look like this.

00:04:14.126 --> 00:04:16.209
And so that the only way
that they could intersect

00:04:16.209 --> 00:04:17.800
is if 0 is equal to negative 21.

00:04:17.800 --> 00:04:19.290
And we know that's impossible.

00:04:19.290 --> 00:04:22.642
They never, ever intersect
these two planes.

00:04:22.642 --> 00:04:25.100
So regardless of this first
constraint and all of the stuff

00:04:25.100 --> 00:04:28.460
we did here, these two
planes will never intersect,

00:04:28.460 --> 00:04:29.830
these two right over here.

00:04:29.830 --> 00:04:32.938
And so we have no solutions.


Salman Khan TED Talk 2011 (from ted.com)

WEBVTT
Kind: captions
Language: en

00:00:15.969 --> 00:00:17.760
SALMAN KHAN: The Khan
Academy is most known

00:00:17.760 --> 00:00:19.590
for its collection of videos.

00:00:19.590 --> 00:00:22.170
So before I go any
farther, let me

00:00:22.170 --> 00:00:24.474
show you a little
bit of a montage.

00:00:24.474 --> 00:00:25.140
[VIDEO PLAYBACK]

00:00:25.140 --> 00:00:27.630
-So the hypotenuse is
now going to be five.

00:00:27.630 --> 00:00:29.500
This animal's fossils
are only found

00:00:29.500 --> 00:00:32.310
in this area of South
America, nice, clean band

00:00:32.310 --> 00:00:35.130
here, and in this
part of Africa.

00:00:35.130 --> 00:00:37.160
We could integrate
over the surface,

00:00:37.160 --> 00:00:40.080
and the notation usually
is a capital sigma.

00:00:40.080 --> 00:00:42.220
National Assembly, they
create the committee

00:00:42.220 --> 00:00:44.900
of public safety, which sounds
like a very nice committee.

00:00:44.900 --> 00:00:48.990
Notice this is an aldehyde,
and it's an alcohol.

00:00:48.990 --> 00:00:52.350
Start differentiating into
effector and memory cells.

00:00:52.350 --> 00:00:54.730
A galaxy, hey, there's
another galaxy.

00:00:54.730 --> 00:00:56.090
Oh look, there's another galaxy.

00:00:56.090 --> 00:00:59.720
And for dollars is their 30
million plus the $20 million

00:00:59.720 --> 00:01:01.460
from the American manufacturer.

00:01:01.460 --> 00:01:06.275
If this does not blow your
mind, then you have no emotion.

00:01:06.275 --> 00:01:13.980
[END VIDEO PLAYBACK]

00:01:13.980 --> 00:01:17.960
SALMAN KHAN: We now have on
the order of 2,200 videos

00:01:17.960 --> 00:01:20.400
covering everything from
basic arithmetic all

00:01:20.400 --> 00:01:22.720
the way to vector calculus
and some of the stuff

00:01:22.720 --> 00:01:24.490
that you saw up there.

00:01:24.490 --> 00:01:27.820
We have a million students
a month using the site,

00:01:27.820 --> 00:01:31.340
watching on the order of
100,000 to 200,000 videos a day.

00:01:31.340 --> 00:01:33.090
But what we're going
to talk about in this

00:01:33.090 --> 00:01:36.150
is how we're going
to the next level.

00:01:36.150 --> 00:01:39.810
But before I do that, I want
to talk a little bit about how

00:01:39.810 --> 00:01:41.330
I got started.

00:01:41.330 --> 00:01:44.950
And some of you all might
know, about five years ago,

00:01:44.950 --> 00:01:46.730
I was an analyst
at a hedge fund.

00:01:46.730 --> 00:01:49.230
And I was in Boston.

00:01:49.230 --> 00:01:52.260
And I was tutoring my cousins
in New Orleans remotely.

00:01:52.260 --> 00:01:55.260
And I started putting the first
YouTube videos up, really just

00:01:55.260 --> 00:01:58.650
as a nice to have, just kind of
a supplement, for my cousins,

00:01:58.650 --> 00:02:01.820
something that might give them
a refresher, or something.

00:02:01.820 --> 00:02:04.279
And as soon as I put those
first YouTube videos up,

00:02:04.279 --> 00:02:05.570
something interesting happened.

00:02:05.570 --> 00:02:08.060
Actually, a bunch of
interesting things happened.

00:02:08.060 --> 00:02:11.180
The first was the
feedback from my cousins.

00:02:11.180 --> 00:02:16.584
They told me that they preferred
me on YouTube than in person.

00:02:16.584 --> 00:02:22.720
[LAUGHTER]

00:02:22.720 --> 00:02:26.949
And once you get over the
backhanded nature of that,

00:02:26.949 --> 00:02:28.990
there was actually something
very profound there.

00:02:28.990 --> 00:02:32.370
They were saying that they
preferred the automated version

00:02:32.370 --> 00:02:34.732
of their cousin to their cousin.

00:02:34.732 --> 00:02:36.940
At first it's very unintuitive,
but when you actually

00:02:36.940 --> 00:02:38.606
think about it from
their point of view,

00:02:38.606 --> 00:02:40.220
it makes a ton of sense.

00:02:40.220 --> 00:02:43.260
You have this situation where
now they can pause and repeat

00:02:43.260 --> 00:02:47.770
their cousin without feeling
like they're wasting my time.

00:02:47.770 --> 00:02:50.590
If they have to review something
that they should have learned

00:02:50.590 --> 00:02:54.570
a couple of weeks ago, or
maybe a couple of years ago,

00:02:54.570 --> 00:02:56.289
they don't have to be
embarrassed and ask

00:02:56.289 --> 00:02:56.830
their cousin.

00:02:56.830 --> 00:02:58.050
They can just
watch those videos.

00:02:58.050 --> 00:02:59.160
If they're bored,
they can go ahead.

00:02:59.160 --> 00:03:01.730
They can watch it at their
own time, at their own pace.

00:03:01.730 --> 00:03:07.280
And probably the least
appreciated aspect of this

00:03:07.280 --> 00:03:10.100
is the notion that the very
first time, the very first time

00:03:10.100 --> 00:03:13.790
that you're trying to get your
brain around a new concept,

00:03:13.790 --> 00:03:16.670
the very last thing you need
is another human being saying

00:03:16.670 --> 00:03:18.327
do you understand this.

00:03:18.327 --> 00:03:20.410
And that's what was happening
with the interaction

00:03:20.410 --> 00:03:21.730
with my cousins before.

00:03:21.730 --> 00:03:28.820
And now they could just do it in
the intimacy of their own room.

00:03:28.820 --> 00:03:31.420
The other thing that happened
is I put them on YouTube

00:03:31.420 --> 00:03:36.050
just for the-- I saw no
reason to make it private.

00:03:36.050 --> 00:03:37.530
So I let other people watch it.

00:03:37.530 --> 00:03:40.230
And then people started
stumbling on it.

00:03:40.230 --> 00:03:42.570
And I started getting some
comments, and some letters,

00:03:42.570 --> 00:03:46.870
and all sorts of feedback from
random people around the world.

00:03:46.870 --> 00:03:50.130
And these are just a few.

00:03:50.130 --> 00:03:53.600
This is actually from one of
the original calculus videos.

00:03:53.600 --> 00:03:56.570
And someone wrote on YouTube,
it was a YouTube comment,

00:03:56.570 --> 00:04:02.650
"First time I smiled
doing a derivative."

00:04:02.650 --> 00:04:04.170
Let's pause here.

00:04:04.170 --> 00:04:09.290
This person did a derivative,
and then they smiled.

00:04:09.290 --> 00:04:11.040
And then in response
to that same comment,

00:04:11.040 --> 00:04:11.956
this is on the thread.

00:04:11.956 --> 00:04:14.090
You could go on YouTube
and look at these comments.

00:04:14.090 --> 00:04:15.830
Someone else wrote,
"Same thing here.

00:04:15.830 --> 00:04:17.820
I actually got a natural
high and a good mood

00:04:17.820 --> 00:04:20.170
for the entire day,
since I remember

00:04:20.170 --> 00:04:23.780
seeing all of this
'matrix text' in class.

00:04:23.780 --> 00:04:26.966
And here I'm all
like, I know Kung Fu."

00:04:26.966 --> 00:04:30.790
[LAUGHTER]

00:04:30.790 --> 00:04:32.900
And we got a lot of
feedback along those lines.

00:04:32.900 --> 00:04:34.820
This clearly was helping people.

00:04:34.820 --> 00:04:38.400
But then, as the viewership
kept growing, and kept growing,

00:04:38.400 --> 00:04:41.032
I started getting
letters from people.

00:04:41.032 --> 00:04:42.490
And it was starting
to become clear

00:04:42.490 --> 00:04:44.780
that it was actually more
than just a nice to have.

00:04:44.780 --> 00:04:48.520
This is just an excerpt
from one of those letters.

00:04:48.520 --> 00:04:51.250
"My 12-year-old son
has autism, and has

00:04:51.250 --> 00:04:53.150
had a terrible time with math.

00:04:53.150 --> 00:04:56.440
We have tried everything, viewed
everything, bought everything.

00:04:56.440 --> 00:05:00.122
We stumbled on your video on
decimals, and it got through.

00:05:00.122 --> 00:05:01.830
Then we went on to
the dreaded fractions.

00:05:01.830 --> 00:05:03.220
Again, he got it.

00:05:03.220 --> 00:05:04.500
We could not believe it.

00:05:04.500 --> 00:05:06.610
He is so excited."

00:05:06.610 --> 00:05:09.980
And so you can
imagine, here I was,

00:05:09.980 --> 00:05:12.140
an analyst at a hedge fund.

00:05:12.140 --> 00:05:15.964
It was very strange for me to
do something of social value.

00:05:15.964 --> 00:05:24.080
[LAUGHTER AND APPLAUSE]

00:05:24.080 --> 00:05:26.460
But I was excited.

00:05:26.460 --> 00:05:27.510
So I kept going.

00:05:27.510 --> 00:05:30.130
And then a few other things
started to dawn on me.

00:05:30.130 --> 00:05:32.780
That not only would it
help my cousins right now,

00:05:32.780 --> 00:05:34.760
or these people who
were sending letters.

00:05:34.760 --> 00:05:37.640
But maybe that this
content will never go old.

00:05:37.640 --> 00:05:40.910
That it could help their
kids or their grandkids.

00:05:40.910 --> 00:05:45.700
If Isaac Newton had done
YouTube videos on calculus,

00:05:45.700 --> 00:05:50.540
I wouldn't have to,
assuming he was good.

00:05:50.540 --> 00:05:51.911
We don't know.

00:05:51.911 --> 00:05:54.200
[LAUGHTER]

00:05:54.200 --> 00:05:56.784
The other thing that happened,
and even at this point, I said,

00:05:56.784 --> 00:05:58.158
OK, maybe it's a
good supplement.

00:05:58.158 --> 00:05:59.590
It's good for
motivated students.

00:05:59.590 --> 00:06:01.792
It's good for maybe
homeschoolers.

00:06:01.792 --> 00:06:04.250
But I didn't think it would be
something that would somehow

00:06:04.250 --> 00:06:05.469
penetrate the classroom.

00:06:05.469 --> 00:06:07.510
But then I started getting
letters from teachers.

00:06:07.510 --> 00:06:09.440
And the teachers
would write saying,

00:06:09.440 --> 00:06:11.860
we've used your videos
to flip the classroom.

00:06:11.860 --> 00:06:13.230
You've given the lectures.

00:06:13.230 --> 00:06:15.021
So now what we do--
and this could actually

00:06:15.021 --> 00:06:18.500
happen in every classroom in
America tomorrow-- what I do

00:06:18.500 --> 00:06:21.350
is I assign the
lectures for homework.

00:06:21.350 --> 00:06:23.670
And what used to
be homework, I now

00:06:23.670 --> 00:06:26.580
have the students
doing in the classroom.

00:06:26.580 --> 00:06:33.290
[APPLAUSE]

00:06:33.290 --> 00:06:34.830
I want to pause
here for a second

00:06:34.830 --> 00:06:36.788
because there's a couple
of interesting things.

00:06:36.788 --> 00:06:38.840
One, when those
teachers are doing that,

00:06:38.840 --> 00:06:40.400
there's the obvious benefit.

00:06:40.400 --> 00:06:42.970
There's the benefit
that now their students

00:06:42.970 --> 00:06:45.250
can enjoy the videos in the
way that my cousins did.

00:06:45.250 --> 00:06:48.562
They can pause, repeat at their
own pace, at their own time.

00:06:48.562 --> 00:06:50.270
But the more interesting
thing-- and this

00:06:50.270 --> 00:06:52.561
is the unintuitive thing when
you talk about technology

00:06:52.561 --> 00:06:55.650
in the classroom-- by removing
the "one size fits all"

00:06:55.650 --> 00:06:58.250
lecture from the classroom
and letting students have

00:06:58.250 --> 00:07:00.684
a self-paced lecture
at home, and then when

00:07:00.684 --> 00:07:02.600
you go to the classroom,
letting them do work,

00:07:02.600 --> 00:07:05.750
having the teacher walk around,
having the peers actually

00:07:05.750 --> 00:07:07.870
be able to interact
with each other,

00:07:07.870 --> 00:07:11.290
these teachers have
used technology

00:07:11.290 --> 00:07:13.460
to humanize the classroom.

00:07:13.460 --> 00:07:15.870
They took a fundamentally
dehumanizing experience,

00:07:15.870 --> 00:07:18.450
a bunch of 30 kids with
their fingers on their lips,

00:07:18.450 --> 00:07:20.160
not allowed to interact
with each other.

00:07:20.160 --> 00:07:21.970
A teacher, no
matter how good, has

00:07:21.970 --> 00:07:23.920
to give this kind of
"one size fits all"

00:07:23.920 --> 00:07:27.640
lecture to 30 students-- blank
faces, slightly antagonistic.

00:07:27.640 --> 00:07:29.110
And now it's a human experience.

00:07:29.110 --> 00:07:31.620
Now they're actually
interacting with each other.

00:07:31.620 --> 00:07:35.880
So once the Khan
Academy-- I quit my job.

00:07:35.880 --> 00:07:37.820
And we turned into
a real organization,

00:07:37.820 --> 00:07:39.940
or a not-for-profit.

00:07:39.940 --> 00:07:42.785
The question is, how do we
take this to the next level?

00:07:42.785 --> 00:07:44.410
How do we take what
those teachers were

00:07:44.410 --> 00:07:46.490
doing to their
natural conclusion?

00:07:46.490 --> 00:07:48.370
And so what I'm
showing over here,

00:07:48.370 --> 00:07:51.330
these are actual
exercises that I

00:07:51.330 --> 00:07:53.717
started writing for my cousins.

00:07:53.717 --> 00:07:55.550
The ones I started were
much more primitive.

00:07:55.550 --> 00:07:59.030
This is a more
competent version of it.

00:07:59.030 --> 00:08:01.490
But the paradigm here is we'll
generate as many questions

00:08:01.490 --> 00:08:03.950
as you need until
you get that concept,

00:08:03.950 --> 00:08:05.370
until you get 10 in a row.

00:08:05.370 --> 00:08:07.610
And the Khan Academy
videos are there.

00:08:07.610 --> 00:08:09.870
You get hints, the actual
steps for that problem,

00:08:09.870 --> 00:08:11.624
if you don't know how to do it.

00:08:11.624 --> 00:08:13.790
But the paradigm, it seems
like a very simple thing.

00:08:13.790 --> 00:08:15.040
10 in a row, you move on.

00:08:15.040 --> 00:08:16.873
But it's fundamentally
different than what's

00:08:16.873 --> 00:08:18.690
happening in
classrooms right now.

00:08:18.690 --> 00:08:21.370
In a traditional
classroom, you have

00:08:21.370 --> 00:08:24.295
a couple of-- homework,
lecture, homework, lecture,

00:08:24.295 --> 00:08:26.360
and then you have
a snapshot exam.

00:08:26.360 --> 00:08:31.470
And that exam, whether you get
a 70%, an 80%, a 90% or a 95%,

00:08:31.470 --> 00:08:33.960
the class moves on
to the next topic.

00:08:33.960 --> 00:08:37.711
And even that 95% student, what
was the 5% they didn't know?

00:08:37.711 --> 00:08:39.169
Maybe they didn't
know what happens

00:08:39.169 --> 00:08:41.297
when you raise something
to the 0-th power.

00:08:41.297 --> 00:08:43.380
And then you go build on
that in the next concept.

00:08:43.380 --> 00:08:47.159
That's analogous to-- imagine
learning to ride a bicycle.

00:08:47.159 --> 00:08:48.200
And I give you a bicycle.

00:08:48.200 --> 00:08:50.070
Maybe I give you a
lecture ahead of time.

00:08:50.070 --> 00:08:51.980
And I give you that
bicycle for two weeks.

00:08:51.980 --> 00:08:53.945
And then I come back
after two weeks.

00:08:53.945 --> 00:08:55.070
And I say, well, let's see.

00:08:55.070 --> 00:08:56.736
You're having trouble
taking left turns.

00:08:56.736 --> 00:08:58.370
You can't quite stop.

00:08:58.370 --> 00:08:59.945
You're an 80% bicyclist.

00:08:59.945 --> 00:09:04.060
So I put a big C stamp
on your forehead.

00:09:04.060 --> 00:09:06.610
And then I say
here's a unicycle.

00:09:06.610 --> 00:09:08.940
But as ridiculous
as that sounds,

00:09:08.940 --> 00:09:12.620
that's exactly what's happening
in our classrooms right now.

00:09:12.620 --> 00:09:14.880
And the idea is
you fast forward.

00:09:14.880 --> 00:09:18.130
And good students start failing
algebra all of a sudden,

00:09:18.130 --> 00:09:20.490
and start failing
calculus all of a sudden,

00:09:20.490 --> 00:09:22.840
despite being smart, despite
having good teachers.

00:09:22.840 --> 00:09:25.006
And it's usually because
they had these Swiss cheese

00:09:25.006 --> 00:09:27.400
gaps that kept building
throughout their foundations.

00:09:27.400 --> 00:09:31.255
So our model is learn math the
way you would learn anything.

00:09:31.255 --> 00:09:32.880
Like the way you
would learn a bicycle.

00:09:32.880 --> 00:09:34.420
Stay on that bicycle.

00:09:34.420 --> 00:09:35.770
Fall off that bicycle.

00:09:35.770 --> 00:09:38.890
Do it as long as necessary
until you have mastery.

00:09:38.890 --> 00:09:41.370
The traditional model,
it penalizes you

00:09:41.370 --> 00:09:43.010
for experimentation and failure.

00:09:43.010 --> 00:09:45.080
But it does not expect mastery.

00:09:45.080 --> 00:09:46.710
We encourage you to experiment.

00:09:46.710 --> 00:09:48.070
We encourage you to failure.

00:09:48.070 --> 00:09:51.364
But we do expect mastery.

00:09:51.364 --> 00:09:53.030
This is just another
one of the modules.

00:09:53.030 --> 00:09:54.730
This is trigonometry.

00:09:54.730 --> 00:09:57.910
This is shifting and
reflecting functions.

00:09:57.910 --> 00:09:59.700
And they all fit together.

00:09:59.700 --> 00:10:01.330
We have about 90
of these right now.

00:10:01.330 --> 00:10:02.600
And you could go to
the site right now.

00:10:02.600 --> 00:10:03.290
It's all free.

00:10:03.290 --> 00:10:04.719
Not trying to sell anything.

00:10:04.719 --> 00:10:06.260
But the general idea
is that they all

00:10:06.260 --> 00:10:07.930
fit into this knowledge map.

00:10:07.930 --> 00:10:09.380
That top node
right there, that's

00:10:09.380 --> 00:10:11.180
literally single-digit addition.

00:10:11.180 --> 00:10:13.641
It's like 1 plus
1 is equal to 2.

00:10:13.641 --> 00:10:15.890
And the paradigm is, once
you get 10 in a row on that,

00:10:15.890 --> 00:10:20.690
then it keeps forwarding you to
more and more advanced modules.

00:10:20.690 --> 00:10:22.750
So keep-- this is further
down the knowledge map.

00:10:22.750 --> 00:10:24.720
We're getting into more
advanced arithmetic.

00:10:24.720 --> 00:10:26.220
Further down, you
start getting into

00:10:26.220 --> 00:10:27.720
pre-algebra and early algebra.

00:10:27.720 --> 00:10:31.500
Further down, you start getting
into algebra one, algebra two,

00:10:31.500 --> 00:10:33.520
a little bit of precalculus.

00:10:33.520 --> 00:10:36.930
And the idea is, from this, we
can actually teach everything.

00:10:36.930 --> 00:10:39.330
Well, everything
that can be taught

00:10:39.330 --> 00:10:40.830
in this type of a framework.

00:10:40.830 --> 00:10:41.810
So you can imagine.

00:10:41.810 --> 00:10:43.470
And this is what
we are working on--

00:10:43.470 --> 00:10:45.610
is from this knowledge
map, you have logic.

00:10:45.610 --> 00:10:47.120
You have computer programming.

00:10:47.120 --> 00:10:48.430
You have grammar.

00:10:48.430 --> 00:10:49.540
You have genetics.

00:10:49.540 --> 00:10:52.740
All based off of that core of,
OK, If you know this and that,

00:10:52.740 --> 00:10:56.200
now you're ready for
this next concept.

00:10:56.200 --> 00:10:58.487
Now that can work well
for an individual learner.

00:10:58.487 --> 00:11:00.570
And I encourage one, for
you to do with your kids.

00:11:00.570 --> 00:11:03.236
But I also encourage everyone in
the audience to do it yourself.

00:11:03.236 --> 00:11:05.660
It'll change what happens
at the dinner table.

00:11:05.660 --> 00:11:08.650
But what we want to do is
use the natural conclusion

00:11:08.650 --> 00:11:11.200
of the flipping of the classroom
that those early teachers

00:11:11.200 --> 00:11:12.326
had emailed me about.

00:11:12.326 --> 00:11:13.700
And so what I'm
showing you here,

00:11:13.700 --> 00:11:16.610
this is actually data from a
pilot in the Los Altos school

00:11:16.610 --> 00:11:19.130
district, where they took two
fifth-grade classes, and two

00:11:19.130 --> 00:11:21.200
seventh-grade classes,
and completely gutted

00:11:21.200 --> 00:11:22.390
their old math curriculum.

00:11:22.390 --> 00:11:23.690
These kids aren't
using textbooks.

00:11:23.690 --> 00:11:25.490
They're not getting "one
size fits all" lectures.

00:11:25.490 --> 00:11:26.710
They're doing Khan Academy.

00:11:26.710 --> 00:11:28.990
They're doing that
software for roughly half

00:11:28.990 --> 00:11:30.140
of their math class.

00:11:30.140 --> 00:11:31.350
And I want to make it clear.

00:11:31.350 --> 00:11:33.160
We don't view this as a
complete math education.

00:11:33.160 --> 00:11:34.660
What it does is--
and this is what's

00:11:34.660 --> 00:11:36.774
happening Los Altos--
it frees up time.

00:11:36.774 --> 00:11:38.190
This is the blocking
and tackling.

00:11:38.190 --> 00:11:40.481
Making sure you know how to
do the system of equations.

00:11:40.481 --> 00:11:43.140
And it frees up time for the
simulations, for the games,

00:11:43.140 --> 00:11:45.390
for the mechanics, for
the robot building,

00:11:45.390 --> 00:11:49.370
for the estimating how high that
hill is based on its shadow.

00:11:49.370 --> 00:11:51.950
And so the paradigm is the
teacher walks in every day.

00:11:51.950 --> 00:11:53.630
Every kid works
at their own pace.

00:11:53.630 --> 00:11:56.410
And a teacher-- this is actually
a live dashboard from Los Altos

00:11:56.410 --> 00:11:58.890
school district-- and they
look at this dashboard.

00:11:58.890 --> 00:12:00.610
Every row is a student.

00:12:00.610 --> 00:12:02.300
Every column is one
of those concepts.

00:12:02.300 --> 00:12:04.360
Green means the student's
already proficient.

00:12:04.360 --> 00:12:06.960
Blue means that they're working
on it, no need to worry.

00:12:06.960 --> 00:12:08.960
Red means they're stuck.

00:12:08.960 --> 00:12:12.320
And what the teacher does
is literally just says,

00:12:12.320 --> 00:12:13.760
let me intervene
on the red kids.

00:12:13.760 --> 00:12:16.040
Or even better, let me get
one of the green kids who

00:12:16.040 --> 00:12:17.680
are already proficient
in that concept

00:12:17.680 --> 00:12:22.340
to be the first line of attack
and actually tutor their peer.

00:12:22.340 --> 00:12:29.000
[APPLAUSE]

00:12:29.000 --> 00:12:32.976
Now I come from a very
data-centric reality.

00:12:32.976 --> 00:12:35.600
So we don't want that teacher to
even go and intervene and have

00:12:35.600 --> 00:12:37.060
to ask the kid
awkward questions.

00:12:37.060 --> 00:12:39.476
Oh, what do you not understand,
or what do you understand,

00:12:39.476 --> 00:12:40.440
and all of the rest.

00:12:40.440 --> 00:12:42.330
So our paradigm is to
really arm the teachers

00:12:42.330 --> 00:12:43.580
with as much data as possible.

00:12:43.580 --> 00:12:46.000
Really data that, in almost
any other field, is expected.

00:12:46.000 --> 00:12:48.240
If you're in finance, or
marketing, or manufacturing.

00:12:48.240 --> 00:12:49.800
And so the teachers
can actually diagnose

00:12:49.800 --> 00:12:50.930
what's wrong with the
students, so that they

00:12:50.930 --> 00:12:53.630
can make their interaction
as productive as possible.

00:12:53.630 --> 00:12:55.410
So now the teachers
know exactly what

00:12:55.410 --> 00:12:57.200
the student's been up
to, how long they've

00:12:57.200 --> 00:12:58.200
been spending every day.

00:12:58.200 --> 00:12:59.310
What videos have
they been watching?

00:12:59.310 --> 00:13:00.720
When did they pause the videos?

00:13:00.720 --> 00:13:01.970
What did they stop watching?

00:13:01.970 --> 00:13:03.590
What exercises are they using?

00:13:03.590 --> 00:13:04.920
What have they been focused on?

00:13:04.920 --> 00:13:07.336
The outer circle shows the
exercises they were focused on.

00:13:07.336 --> 00:13:11.314
The inner circle shows the
videos they're focused on.

00:13:11.314 --> 00:13:12.730
And the data gets
pretty granular.

00:13:12.730 --> 00:13:14.320
So you can actually
see the exact problems

00:13:14.320 --> 00:13:15.819
that the student
got right or wrong.

00:13:15.819 --> 00:13:16.580
Red is wrong.

00:13:16.580 --> 00:13:17.660
Blue is right.

00:13:17.660 --> 00:13:19.460
The leftmost question
is the first question

00:13:19.460 --> 00:13:20.584
that the student attempted.

00:13:20.584 --> 00:13:22.251
They watched the video
right over there.

00:13:22.251 --> 00:13:23.833
And then you could
see eventually they

00:13:23.833 --> 00:13:25.100
were able to get 10 in a row.

00:13:25.100 --> 00:13:26.070
It's almost like
you can almost see

00:13:26.070 --> 00:13:28.150
them learning over
those last 10 problems.

00:13:28.150 --> 00:13:29.110
They also got faster.

00:13:29.110 --> 00:13:30.770
The height is how
long it took them.

00:13:33.650 --> 00:13:37.119
So when you talk about
self-paced learning,

00:13:37.119 --> 00:13:39.160
it makes sense for everyone--
in education speak,

00:13:39.160 --> 00:13:40.700
differentiated learning.

00:13:40.700 --> 00:13:42.940
But it's kind of crazy what
happens when you actually

00:13:42.940 --> 00:13:44.370
see it in a classroom.

00:13:44.370 --> 00:13:46.360
Because every time
we've done this,

00:13:46.360 --> 00:13:49.960
in every classroom we've done,
over and over again, if you

00:13:49.960 --> 00:13:51.602
go five days into
it, there's a group

00:13:51.602 --> 00:13:52.810
of kids who have raced ahead.

00:13:52.810 --> 00:13:55.260
And there's a group of kids
who are a little bit slower.

00:13:55.260 --> 00:13:57.430
And in a traditional model, if
you did a snapshot assessment,

00:13:57.430 --> 00:13:59.200
you say, oh, these
are the gifted kids.

00:13:59.200 --> 00:14:00.350
These are the slow kids.

00:14:00.350 --> 00:14:02.120
Maybe they should be
tracked differently.

00:14:02.120 --> 00:14:04.590
Maybe we should put them
in different classes.

00:14:04.590 --> 00:14:06.840
But when you let every student
work at their own pace,

00:14:06.840 --> 00:14:08.850
and we see it over and
over and over again.

00:14:08.850 --> 00:14:11.460
You see students who took
a little bit extra time

00:14:11.460 --> 00:14:13.440
on one concept or the other.

00:14:13.440 --> 00:14:16.469
But once they get through that
concept, they just race ahead.

00:14:16.469 --> 00:14:18.010
And so the same kids
that you thought

00:14:18.010 --> 00:14:21.840
were slow six weeks ago, you
now would think are gifted.

00:14:21.840 --> 00:14:23.890
And we're seeing it over
and over and over again.

00:14:23.890 --> 00:14:26.290
And it makes you
really wonder how much

00:14:26.290 --> 00:14:29.080
all of the labels a lot
of us have benefited from

00:14:29.080 --> 00:14:31.480
were really just due to
a coincidence of time.

00:14:34.950 --> 00:14:36.690
Now, as valuable as
something like this

00:14:36.690 --> 00:14:40.400
is in a district like
Los Altos, our goal

00:14:40.400 --> 00:14:42.520
is to use technology
to humanize,

00:14:42.520 --> 00:14:45.630
not just in Los Altos, but kind
of on a global scale, what's

00:14:45.630 --> 00:14:47.030
happening in education.

00:14:47.030 --> 00:14:49.630
And actually that brings
an interesting point.

00:14:49.630 --> 00:14:52.310
A lot of the effort in
humanizing the classroom

00:14:52.310 --> 00:14:55.300
is focused on student
to teacher ratios.

00:14:55.300 --> 00:14:57.360
In our mind the
relevant metric is

00:14:57.360 --> 00:15:01.272
"student to valuable human
time with the teacher" ratio.

00:15:01.272 --> 00:15:03.480
So in a traditional model,
most of the teacher's time

00:15:03.480 --> 00:15:06.470
is spent doing lectures, and
grading tests, and whatnot.

00:15:06.470 --> 00:15:08.010
Maybe 5% of their
time is actually

00:15:08.010 --> 00:15:10.600
sitting next to students and
actually working with them.

00:15:10.600 --> 00:15:12.530
Now 100% of their time is.

00:15:12.530 --> 00:15:14.440
So once again, using
technology, not just

00:15:14.440 --> 00:15:17.550
flipping the classroom, you're
humanizing the classroom,

00:15:17.550 --> 00:15:20.797
I'd argue, by a
factor of five or 10.

00:15:20.797 --> 00:15:22.380
And as valuable as
it is in Los Altos,

00:15:22.380 --> 00:15:24.421
imagine what that does to
the adult learner who's

00:15:24.421 --> 00:15:27.180
embarrassed to go back and learn
stuff that they should have

00:15:27.180 --> 00:15:28.990
known before, before
going back to college.

00:15:28.990 --> 00:15:35.440
Imagine what it does to a
street kid in Calcutta who

00:15:35.440 --> 00:15:37.790
has to help his
family during the day.

00:15:37.790 --> 00:15:40.250
And that's the reason why he
or she can't go to school.

00:15:40.250 --> 00:15:41.960
Now they can spend
two hours a day

00:15:41.960 --> 00:15:44.590
and remediate or get
up to speed and not

00:15:44.590 --> 00:15:47.910
feel embarrassed about
what they do or don't know.

00:15:47.910 --> 00:15:50.870
Now imagine what happens where--
we talked about the peers

00:15:50.870 --> 00:15:54.350
teaching each other
inside of a classroom.

00:15:54.350 --> 00:15:56.190
But this is all one system.

00:15:56.190 --> 00:15:58.250
There's no reason why
you can't have that peer

00:15:58.250 --> 00:16:01.510
to peer tutoring beyond
that one classroom.

00:16:01.510 --> 00:16:04.540
Imagine what happens if
that student in Calcutta

00:16:04.540 --> 00:16:06.670
all of a sudden
can tutor your son.

00:16:06.670 --> 00:16:09.600
Or your son can tutor
that kid in Calcutta.

00:16:09.600 --> 00:16:11.290
And I think what
you'll see emerging

00:16:11.290 --> 00:16:19.030
is this notion of a global
one world classroom.

00:16:19.030 --> 00:16:22.060
And that's essentially
what we're trying to build.

00:16:22.060 --> 00:16:22.975
Thank you.

00:16:22.975 --> 00:16:31.885
[APPLAUSE]

00:16:31.885 --> 00:16:34.360
[SIDE CONVERSATION]

00:16:34.360 --> 00:17:04.579
[APPLAUSE]

00:17:04.579 --> 00:17:06.119
BILL GATES: I've
seen some things

00:17:06.119 --> 00:17:08.600
you're doing in the system
that have to do with motivation

00:17:08.600 --> 00:17:12.440
and feedback-- energy
points, merit badges.

00:17:12.440 --> 00:17:14.449
Tell me what you're
thinking there.

00:17:14.449 --> 00:17:16.490
SALMAN KHAN: Oh yeah, no,
we have an awesome team

00:17:16.490 --> 00:17:17.280
working on that.

00:17:17.280 --> 00:17:18.069
And I have to be clear.

00:17:18.069 --> 00:17:19.150
It's not just me anymore.

00:17:19.150 --> 00:17:20.540
I'm still doing all the videos.

00:17:20.540 --> 00:17:23.012
We have a rock star
team doing the software.

00:17:23.012 --> 00:17:24.720
Yeah, we've put a
bunch of game mechanics

00:17:24.720 --> 00:17:25.930
in there, where you
get these badges.

00:17:25.930 --> 00:17:28.054
We're going to start having
leader boards by areas,

00:17:28.054 --> 00:17:29.360
and you get points.

00:17:29.360 --> 00:17:31.040
It's actually been
pretty interesting.

00:17:31.040 --> 00:17:33.580
Just the wording of the
badging, or how many points you

00:17:33.580 --> 00:17:36.320
get for doing something, we
see on the system-wide basis

00:17:36.320 --> 00:17:37.944
tens of thousands
of fifth graders

00:17:37.944 --> 00:17:40.360
or sixth graders going one
direction or another, depending

00:17:40.360 --> 00:17:43.500
on what badge you give them.

00:17:43.500 --> 00:17:44.384
[LAUGHTER]

00:17:44.384 --> 00:17:46.050
BILL GATES: And the
collaboration you're

00:17:46.050 --> 00:17:49.730
doing with Los Altos,
how did that come about?

00:17:49.730 --> 00:17:52.560
SALMAN KHAN: Yeah, Los
Altos was kind of crazy.

00:17:52.560 --> 00:17:55.540
Once again, I didn't expect
it to be used in classrooms.

00:17:55.540 --> 00:17:57.920
Someone from their board
came and said, what would you

00:17:57.920 --> 00:18:00.240
do if you had carte
blanche in a classroom?

00:18:00.240 --> 00:18:02.274
And I said, well, I would
just-- every student

00:18:02.274 --> 00:18:04.190
work at their own pace
on something like this.

00:18:04.190 --> 00:18:05.107
We'd give a dashboard.

00:18:05.107 --> 00:18:06.898
And they said, oh, this
is kind of radical,

00:18:06.898 --> 00:18:08.060
we have to think about it.

00:18:08.060 --> 00:18:10.600
And me and the rest of team
were like, they're never

00:18:10.600 --> 00:18:11.850
going to want to do this.

00:18:11.850 --> 00:18:13.600
But literally the next
day they were like,

00:18:13.600 --> 00:18:15.311
can you start in two weeks?

00:18:15.311 --> 00:18:17.650
[LAUGHTER]

00:18:17.650 --> 00:18:19.150
BILL GATES: So it's
fifth-grade math

00:18:19.150 --> 00:18:20.460
is where that's
going on right now?

00:18:20.460 --> 00:18:22.168
SALMAN KHAN: It's two
fifth-grade classes

00:18:22.168 --> 00:18:23.530
and two seventh-grade classes.

00:18:23.530 --> 00:18:24.590
They're doing it at
the district level.

00:18:24.590 --> 00:18:26.173
And I think what
they're excited about

00:18:26.173 --> 00:18:27.830
is they can now
follow these kids.

00:18:27.830 --> 00:18:29.601
It's not an only
in-school thing.

00:18:29.601 --> 00:18:31.850
Even on Christmas, we saw
some of the kids were doing.

00:18:31.850 --> 00:18:32.850
And we track everything.

00:18:32.850 --> 00:18:34.610
So they can actually
track them as they

00:18:34.610 --> 00:18:36.530
go through the entire district,
through the summers, as they

00:18:36.530 --> 00:18:37.940
go from one teacher to a next.

00:18:37.940 --> 00:18:41.060
You have this continuity of
data that, even at the district

00:18:41.060 --> 00:18:42.100
level, they can see.

00:18:42.100 --> 00:18:43.516
BILL GATES: So
some of those views

00:18:43.516 --> 00:18:46.940
we saw were for the teacher
to go in and track actually

00:18:46.940 --> 00:18:49.289
what's going on with those kids.

00:18:49.289 --> 00:18:51.330
So you're getting feedback
on those teacher views

00:18:51.330 --> 00:18:53.050
to see what they
think they need?

00:18:53.050 --> 00:18:53.925
SALMAN KHAN: Oh yeah.

00:18:53.925 --> 00:18:57.494
Actually, most of those
were specs by the teachers.

00:18:57.494 --> 00:18:58.910
We made some of
those for students

00:18:58.910 --> 00:18:59.980
so they could their data.

00:18:59.980 --> 00:19:01.602
But we have a very
tight design loop

00:19:01.602 --> 00:19:02.810
with the teachers themselves.

00:19:02.810 --> 00:19:05.080
And they're literally
saying, hey, this is nice.

00:19:05.080 --> 00:19:07.424
But that focus graph,
a lot of the teachers

00:19:07.424 --> 00:19:10.090
said I have a feeling that a lot
of the kids are jumping around,

00:19:10.090 --> 00:19:11.380
and not focusing on one topic.

00:19:11.380 --> 00:19:13.660
So we made that focus
diagram for them.

00:19:13.660 --> 00:19:15.290
So it's all been teacher driven.

00:19:15.290 --> 00:19:16.422
It's been pretty crazy.

00:19:16.422 --> 00:19:18.130
BILL GATES: Is this
ready for prime time?

00:19:18.130 --> 00:19:21.330
Do you think a lot of
classes next school

00:19:21.330 --> 00:19:22.780
year should try this thing out?

00:19:22.780 --> 00:19:23.530
SALMAN KHAN: Yeah.

00:19:23.530 --> 00:19:25.400
It's ready.

00:19:25.400 --> 00:19:27.530
We've got a million people
on the site already.

00:19:27.530 --> 00:19:28.730
So we can handle a few more.

00:19:28.730 --> 00:19:33.250
And no, no reason
why it really can't

00:19:33.250 --> 00:19:36.180
happen in every classroom
in America tomorrow.

00:19:36.180 --> 00:19:38.640
BILL GATES: And the vision
of the tutoring thing.

00:19:38.640 --> 00:19:41.550
The idea there is, if I'm
confused about a topic,

00:19:41.550 --> 00:19:43.810
somehow right in
the user interface,

00:19:43.810 --> 00:19:45.760
I'd find people who
are volunteering,

00:19:45.760 --> 00:19:47.610
maybe see their reputation.

00:19:47.610 --> 00:19:50.500
And I could schedule and
connect up with those people.

00:19:50.500 --> 00:19:51.500
SALMAN KHAN: Absolutely.

00:19:51.500 --> 00:19:53.250
And this is something
I recommend everyone

00:19:53.250 --> 00:19:54.542
in this audience to do.

00:19:54.542 --> 00:19:57.000
Those dashboards the teachers
have, you can go log in right

00:19:57.000 --> 00:19:57.500
now.

00:19:57.500 --> 00:19:59.390
And you could essentially
become a coach

00:19:59.390 --> 00:20:02.850
for your kids, your nephews,
your cousins, or maybe

00:20:02.850 --> 00:20:05.380
some kids at the
Boys & Girls Club.

00:20:05.380 --> 00:20:08.790
And you can start becoming
a mentor or tutor really

00:20:08.790 --> 00:20:09.340
immediately.

00:20:09.340 --> 00:20:10.969
But yeah, it's all there.

00:20:10.969 --> 00:20:12.260
BILL GATES: Well, it's amazing.

00:20:12.260 --> 00:20:13.870
I think you've
just got a glimpse

00:20:13.870 --> 00:20:15.680
of the future of education.

00:20:15.680 --> 00:20:16.192
Thank you.

00:20:16.192 --> 00:20:17.150
SALMAN KHAN: Thank you.

00:20:17.150 --> 00:20:27.034
[APPLAUSE]


Graphing systems of equations

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.225
Solve the system of linear
equations by graphing,

00:00:03.225 --> 00:00:05.130
and they give us
two equations here.

00:00:05.130 --> 00:00:10.210
5x plus 3y is equal to 7, and
3x minus 2y is equal to 8.

00:00:10.210 --> 00:00:12.460
When they say, "Solve the
system of linear equations,"

00:00:12.460 --> 00:00:14.380
they're really just
saying find an x

00:00:14.380 --> 00:00:17.615
and a y that satisfies
both of these equations.

00:00:17.615 --> 00:00:19.600
And when they say to
do it by graphing,

00:00:19.600 --> 00:00:21.830
we're essentially going to
graph this first equation.

00:00:21.830 --> 00:00:24.010
Remember, the graph is
really just depicting

00:00:24.010 --> 00:00:26.929
all of the x's and y's that
satisfy this first equation,

00:00:26.929 --> 00:00:28.470
and then we graph
the second equation

00:00:28.470 --> 00:00:31.700
that's depicting all of the x's
and y's that satisfy that one.

00:00:31.700 --> 00:00:35.200
So if we were looking for an
x and a y that satisfies both,

00:00:35.200 --> 00:00:38.740
that point needs to
be on both equations

00:00:38.740 --> 00:00:40.760
or it has to be on both graphs.

00:00:40.760 --> 00:00:43.170
So it'll be the intersection
of the two graphs.

00:00:43.170 --> 00:00:45.170
So let's try to see
if we can do that.

00:00:45.170 --> 00:00:48.360
So let's focus on
this first equation,

00:00:48.360 --> 00:00:49.400
and I want to graph it.

00:00:49.400 --> 00:00:57.086
So I have 5x plus
3y is equal to 7.

00:00:57.086 --> 00:00:58.960
There's a couple of ways
we could graph this.

00:00:58.960 --> 00:01:01.820
We could put this in
slope-intercept form,

00:01:01.820 --> 00:01:03.480
or we could just pick
some points here.

00:01:03.480 --> 00:01:05.920
You just really need two
points to graph a line.

00:01:05.920 --> 00:01:09.390
So let me just set
some points over here.

00:01:09.390 --> 00:01:11.710
Let's say x and y.

00:01:11.710 --> 00:01:16.210
When x is equal to 0, what
does y need to be equal to?

00:01:16.210 --> 00:01:18.950
So when x is equal
to 0, we have--

00:01:18.950 --> 00:01:23.990
let me do it over here-- we
have 5 times 0, plus 3 times y,

00:01:23.990 --> 00:01:26.130
is equal to 7.

00:01:26.130 --> 00:01:27.710
That's just 0 over there.

00:01:27.710 --> 00:01:29.800
So you have 3y is equal to 7.

00:01:29.800 --> 00:01:32.950
Divide both sides
by 3, you get y

00:01:32.950 --> 00:01:40.310
is equal to 7/3, which is
the same thing as 2 and 1/3

00:01:40.310 --> 00:01:42.500
if we want to write
it as a mixed number.

00:01:42.500 --> 00:01:44.100
Now let's set y equal to 0.

00:01:44.100 --> 00:01:52.270
So if we set y as equal to
0, we get 5x, plus 3 times 0,

00:01:52.270 --> 00:01:54.560
is equal to 7.

00:01:54.560 --> 00:01:56.950
Or in this part right over
here, it just becomes 0.

00:01:56.950 --> 00:01:58.410
So we have 5x is equal to 7.

00:01:58.410 --> 00:02:03.360
Divide both sides
by 5, and we get

00:02:03.360 --> 00:02:13.280
x equals 7/5, which is the
same thing as 1 and 2/5.

00:02:13.280 --> 00:02:14.780
So let's graph both
of these points,

00:02:14.780 --> 00:02:16.600
and then we should be able to
graph this line, or at least

00:02:16.600 --> 00:02:18.690
a pretty good
approximation of that line.

00:02:18.690 --> 00:02:20.520
So we have the
point, 0, 2 and 1/3.

00:02:24.250 --> 00:02:26.460
So that's that point
right over there.

00:02:26.460 --> 00:02:31.380
So I'll call it 0,
7/3 right over there,

00:02:31.380 --> 00:02:35.520
and then we have the point,
7/5, 0, or 1 and 2/5, 0.

00:02:35.520 --> 00:02:38.290
So 1 and 2/5.

00:02:38.290 --> 00:02:40.260
2/5 is a little
less than a half.

00:02:40.260 --> 00:02:44.420
So 1 and 2/5, 0.

00:02:44.420 --> 00:02:47.270
So our line is going to
look something like this.

00:02:47.270 --> 00:02:51.050
I just have to connect the dots.

00:02:51.050 --> 00:02:53.860
It's always hard to
draw the straight line.

00:02:53.860 --> 00:02:56.130
I'll draw it as a dotted line.

00:02:56.130 --> 00:02:59.850
So it would look
something like this.

00:02:59.850 --> 00:03:02.380
Normally, when you have to
solve a system of equations

00:03:02.380 --> 00:03:04.720
by graphing, they normally
give you a little bit cleaner

00:03:04.720 --> 00:03:06.690
numbers, but we'll
try our best and see

00:03:06.690 --> 00:03:11.250
if we can see where these
two lines intersect.

00:03:11.250 --> 00:03:13.890
So now, let's worry about the
second one right over there.

00:03:13.890 --> 00:03:19.000
So we have 3x minus
2y is equal to 8.

00:03:19.000 --> 00:03:20.250
So I'll do the same thing.

00:03:20.250 --> 00:03:25.860
So 3x minus 2y is equal to 8.

00:03:25.860 --> 00:03:29.320
We'll just look at the
x- and the y-intercepts.

00:03:29.320 --> 00:03:30.440
So first, the y-intercept.

00:03:30.440 --> 00:03:37.250
When x is equal to 0, this whole
thing boils down to 3 times

00:03:37.250 --> 00:03:40.520
0 minus 2y is equal to 8.

00:03:40.520 --> 00:03:41.690
That's just 0.

00:03:41.690 --> 00:03:44.580
So you have negative
2y is equal to 8.

00:03:44.580 --> 00:03:47.190
Divide both sides
by negative 2, we

00:03:47.190 --> 00:03:49.900
get y is equal to negative 4.

00:03:52.430 --> 00:03:56.930
So the y-intercept
is 0, negative 4.

00:03:56.930 --> 00:04:01.520
Right over here, and we
mark it 0, negative 4.

00:04:01.520 --> 00:04:03.920
And then, let's
set y equal to 0.

00:04:03.920 --> 00:04:06.960
So when y is equal to 0,
this term right over here

00:04:06.960 --> 00:04:08.770
just becomes 0.

00:04:08.770 --> 00:04:12.410
So we get 3x minus 2
times 0, so that's just 0.

00:04:12.410 --> 00:04:14.470
So 3x is equal to 8.

00:04:14.470 --> 00:04:18.575
Divide both sides by 3,
you get x is equal to 8/3.

00:04:22.270 --> 00:04:25.240
And 8/3 is the same
thing as 2 and 2/3.

00:04:27.770 --> 00:04:31.080
So it puts it right about there.

00:04:31.080 --> 00:04:34.410
That's the point, 0, 8/3.

00:04:34.410 --> 00:04:36.520
Now let me try my
best to graph it.

00:04:36.520 --> 00:04:38.580
Connect these two dots.

00:04:38.580 --> 00:04:41.656
So let me do my very best.

00:04:41.656 --> 00:04:44.790
I drew a dotted line there.

00:04:44.790 --> 00:04:48.580
It goes something like that.

00:04:48.580 --> 00:04:54.650
And just eyeballing it, it looks
like these two lines intersect

00:04:54.650 --> 00:04:55.910
right over there.

00:04:55.910 --> 00:04:58.300
I'm hoping that this will
give us a clean answer.

00:04:58.300 --> 00:05:01.790
And this is the
point 2, negative 1.

00:05:01.790 --> 00:05:05.460
So that is the
point, 2, negative 1.

00:05:05.460 --> 00:05:05.960
Right?

00:05:05.960 --> 00:05:09.780
The x value here is 2.
y value is negative 1.

00:05:09.780 --> 00:05:12.070
Now, that's what we got
just by eyeballing it,

00:05:12.070 --> 00:05:14.340
and clearly these are
hand-drawn graphs.

00:05:14.340 --> 00:05:15.670
Not very precise.

00:05:15.670 --> 00:05:19.300
Let's verify or let's
see if 2, negative 1

00:05:19.300 --> 00:05:21.380
does satisfy both
of these equations,

00:05:21.380 --> 00:05:24.980
if it's an x and y value
that satisfies both and lies

00:05:24.980 --> 00:05:26.570
on both the graphs.

00:05:26.570 --> 00:05:29.090
So if you put 2, negative
1 in this first equation,

00:05:29.090 --> 00:05:35.090
you get 5 times 2, plus
3 times negative 1,

00:05:35.090 --> 00:05:37.810
and we're going to see
if that is equal to 7.

00:05:37.810 --> 00:05:41.460
So this is 10 plus negative 3.

00:05:41.460 --> 00:05:43.940
So that's the same
thing as 10 minus 3.

00:05:43.940 --> 00:05:45.290
Does that equal 7?

00:05:45.290 --> 00:05:46.040
And yeah, it does.

00:05:46.040 --> 00:05:47.650
10 minus 3 is equal to 7.

00:05:47.650 --> 00:05:52.770
So 2, negative 1 is
definitely on that graph

00:05:52.770 --> 00:05:55.220
or definitely satisfies
that equation.

00:05:55.220 --> 00:05:57.120
And then, let's do it
with the other one.

00:05:57.120 --> 00:06:00.080
So if we do 2, negative
1, you have 3 times

00:06:00.080 --> 00:06:03.750
2, minus 2 times negative 1.

00:06:03.750 --> 00:06:06.610
And we're testing to see
if that is equal to 8.

00:06:06.610 --> 00:06:08.450
So 3 times 2.

00:06:08.450 --> 00:06:10.300
3 times 2 is 6.

00:06:10.300 --> 00:06:12.840
And then 2 times
negative 1 is negative 2.

00:06:12.840 --> 00:06:16.270
But then, we're subtracting
that, so it's 6 plus 2.

00:06:16.270 --> 00:06:18.830
6 plus 2 is equal to 8.

00:06:18.830 --> 00:06:20.600
And it definitely does equal 8.

00:06:20.600 --> 00:06:25.540
So we have the coordinate
or we have the point,

00:06:25.540 --> 00:06:29.320
2, negative 1 satisfying
both equations.

00:06:29.320 --> 00:06:33.727
So we've solved the system of
linear equations by graphing.


Consistent and inconsistent systems

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.700
Is the system of
linear equations below

00:00:02.700 --> 00:00:05.370
consistent or inconsistent?

00:00:05.370 --> 00:00:08.330
And they give us x
plus 2y is equal to 13

00:00:08.330 --> 00:00:11.590
and 3x minus y is
equal to negative 11.

00:00:11.590 --> 00:00:13.205
So to answer this
question, we need

00:00:13.205 --> 00:00:16.430
to know what it means to be
consistent or inconsistent.

00:00:16.430 --> 00:00:18.460
So a consistent
system of equations.

00:00:20.980 --> 00:00:24.815
has at least one solution.

00:00:28.810 --> 00:00:31.510
And an inconsistent system of
equations, as you can imagine,

00:00:31.510 --> 00:00:32.685
has no solutions.

00:00:36.370 --> 00:00:38.300
So if we think about
it graphically,

00:00:38.300 --> 00:00:43.600
what would the graph of a
consistent system look like?

00:00:43.600 --> 00:00:45.620
Let me just draw a
really rough graph.

00:00:45.620 --> 00:00:50.750
So that's my x-axis,
and that is my y-axis.

00:00:50.750 --> 00:00:53.480
So if I have just two
different lines that intersect,

00:00:53.480 --> 00:00:55.250
that would be consistent.

00:00:55.250 --> 00:00:58.860
So that's one line, and
then that's another line.

00:00:58.860 --> 00:01:01.300
They clearly have
that one solution

00:01:01.300 --> 00:01:03.300
where they both
intersect, so that

00:01:03.300 --> 00:01:04.980
would be a consistent system.

00:01:04.980 --> 00:01:07.179
Another consistent
system would be

00:01:07.179 --> 00:01:08.970
if they're the same
line, because then they

00:01:08.970 --> 00:01:12.430
would intersect at
a ton of points,

00:01:12.430 --> 00:01:14.250
actually at an infinite
number of points.

00:01:14.250 --> 00:01:16.520
So let's say one of the
lines looks like that.

00:01:16.520 --> 00:01:19.190
And then the other line is
actually the exact same line.

00:01:19.190 --> 00:01:21.200
So it's exactly
right on top of it.

00:01:21.200 --> 00:01:24.220
So those two intersect at
every point along those lines,

00:01:24.220 --> 00:01:26.500
so that also would
be consistent.

00:01:26.500 --> 00:01:30.010
An inconsistent system
would have no solutions.

00:01:30.010 --> 00:01:34.230
So let me again draw my axes.

00:01:34.230 --> 00:01:36.940
Let me once again draw my axes.

00:01:36.940 --> 00:01:38.470
It will have no solutions.

00:01:38.470 --> 00:01:40.300
And so the only way
that you're going

00:01:40.300 --> 00:01:43.030
to have two lines
in two dimensions

00:01:43.030 --> 00:01:46.280
have no solutions is if
they don't intersect,

00:01:46.280 --> 00:01:47.990
or if they are parallel.

00:01:47.990 --> 00:01:50.580
So one line could
look like this.

00:01:50.580 --> 00:01:52.802
And then the other line
would have the same slope,

00:01:52.802 --> 00:01:54.010
but it would be shifted over.

00:01:54.010 --> 00:01:56.056
It would have a
different y-intercept,

00:01:56.056 --> 00:01:57.180
so it would look like this.

00:01:57.180 --> 00:02:00.900
So that's what an inconsistent
system would look like.

00:02:00.900 --> 00:02:02.720
You have parallel lines.

00:02:02.720 --> 00:02:05.650
This right here is inconsistent.

00:02:05.650 --> 00:02:07.610
So what we could do is
just do a rough graph

00:02:07.610 --> 00:02:11.250
of both of these lines
and see if they intersect.

00:02:11.250 --> 00:02:13.620
Another way to do it is,
you could look at the slope.

00:02:13.620 --> 00:02:16.700
And if they have the same slope
and different y-intercepts,

00:02:16.700 --> 00:02:18.610
then you'd also have
an inconsistent system.

00:02:18.610 --> 00:02:20.410
But let's just graph them.

00:02:20.410 --> 00:02:27.506
So let me draw my x-axis
and let me draw my y-axis.

00:02:30.350 --> 00:02:34.389
So this is x and then this is y.

00:02:34.389 --> 00:02:36.430
And then there's a couple
of ways we could do it.

00:02:36.430 --> 00:02:38.346
The easiest way is really
just find two points

00:02:38.346 --> 00:02:41.730
on each of these that satisfy
each of these equations,

00:02:41.730 --> 00:02:43.590
and that's enough
to define a line.

00:02:43.590 --> 00:02:47.230
So for this first one, let's
just make a little table

00:02:47.230 --> 00:02:48.890
of x's and y's.

00:02:48.890 --> 00:02:56.940
When x is 0, you have
2y is equal to 13,

00:02:56.940 --> 00:03:05.170
or y is equal to 13/2, which
is the same thing as 6 and 1/2.

00:03:05.170 --> 00:03:08.710
So when x is 0, y is 6 and 1/2.

00:03:08.710 --> 00:03:10.390
I'll just put it
right over here.

00:03:10.390 --> 00:03:14.440
So this is 0 comma 13/2.

00:03:14.440 --> 00:03:16.890
And then let's just see
what happens when y is 0.

00:03:16.890 --> 00:03:19.950
When y is 0, then
2 times y is 0.

00:03:19.950 --> 00:03:22.230
You have x equaling 13.

00:03:22.230 --> 00:03:24.150
x equals 13.

00:03:24.150 --> 00:03:26.810
So we have the point 13 comma 0.

00:03:26.810 --> 00:03:29.920
So this is 0, 6 and
1/2, so 13 comma 0

00:03:29.920 --> 00:03:31.420
would be right about there.

00:03:31.420 --> 00:03:34.780
We're just trying to
approximate-- 13 comma 0.

00:03:34.780 --> 00:03:37.750
And so this line right
up here, this equation

00:03:37.750 --> 00:03:39.480
can be represented by this line.

00:03:39.480 --> 00:03:41.880
Let me try my best to draw it.

00:03:41.880 --> 00:03:45.050
It would look
something like that.

00:03:45.050 --> 00:03:47.950
Now let's worry about this one.

00:03:47.950 --> 00:03:49.400
Let's worry about that one.

00:03:49.400 --> 00:03:51.980
So once again, let's make a
little table, x's and y's.

00:03:51.980 --> 00:03:54.840
I'm really just looking for
two points on this graph.

00:03:54.840 --> 00:03:59.970
So when x is equal to
0, 3 times 0 is just 0.

00:03:59.970 --> 00:04:02.460
So you get negative y
is equal to negative 11,

00:04:02.460 --> 00:04:04.890
or you get y is equal to 11.

00:04:04.890 --> 00:04:08.380
So you have the point 0, 11, so
that's maybe right over there.

00:04:08.380 --> 00:04:11.110
0 comma 11 is on that line.

00:04:11.110 --> 00:04:16.680
And then when y is 0,
you have 3x minus 0

00:04:16.680 --> 00:04:20.640
is equal to negative 11, or
3x is equal to negative 11.

00:04:20.640 --> 00:04:22.250
Or if you divide
both sides by 3,

00:04:22.250 --> 00:04:24.060
you get x is equal
to negative 11/3.

00:04:28.220 --> 00:04:33.320
And this is the exact same
thing as negative 3 and 2/3.

00:04:33.320 --> 00:04:40.490
So when y is 0, you have x
being negative 3 and 2/3.

00:04:40.490 --> 00:04:43.600
So maybe this is about
6, so negative 3 and 2/3

00:04:43.600 --> 00:04:46.090
would be right about here.

00:04:46.090 --> 00:04:51.440
So this is the point
negative 11/3 comma 0.

00:04:51.440 --> 00:04:55.380
And so the second equation will
look like something like this.

00:04:55.380 --> 00:04:57.790
Will look something like that.

00:04:57.790 --> 00:05:01.270
Now clearly-- and I might have
not been completely precise

00:05:01.270 --> 00:05:04.230
when I did this hand-drawn
graph-- clearly these two

00:05:04.230 --> 00:05:05.790
guys intersect.

00:05:05.790 --> 00:05:07.360
They intersect right over here.

00:05:07.360 --> 00:05:08.734
And to answer
their question, you

00:05:08.734 --> 00:05:11.380
don't even have to find the
point that they intersect at.

00:05:11.380 --> 00:05:13.060
We just have to
see, very clearly,

00:05:13.060 --> 00:05:14.950
that these two lines intersect.

00:05:14.950 --> 00:05:18.480
So this is a consistent
system of equations.

00:05:18.480 --> 00:05:20.510
It has one solution.

00:05:20.510 --> 00:05:23.840
You just have to have at least
one in order to be consistent.

00:05:23.840 --> 00:05:27.567
So once again, consistent
system of equations.


Independent and dependent systems

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:02.880
Is the system of
linear equations below

00:00:02.880 --> 00:00:05.600
dependent or independent?

00:00:05.600 --> 00:00:07.430
And they give us two
equations right here.

00:00:07.430 --> 00:00:09.650
And before I tackle
this specific problem,

00:00:09.650 --> 00:00:11.590
let's just do a little
bit a review of what

00:00:11.590 --> 00:00:13.050
dependent or independent means.

00:00:13.050 --> 00:00:16.770
And actually, I'll compare that
to consistent and inconsistent.

00:00:16.770 --> 00:00:18.380
So just to start
off with, if we're

00:00:18.380 --> 00:00:22.250
dealing with systems of linear
equations in two dimensions,

00:00:22.250 --> 00:00:26.620
there's only three possibilities
that the lines or the equations

00:00:26.620 --> 00:00:28.015
can have relative to each other.

00:00:28.015 --> 00:00:29.640
So let me draw the
three possibilities.

00:00:29.640 --> 00:00:32.189
So let me draw three
coordinate axes.

00:00:32.189 --> 00:00:37.140
So that's my first
x-axis and y-axis.

00:00:37.140 --> 00:00:39.620
Let me draw another one.

00:00:39.620 --> 00:00:44.220
That is x and that is y.

00:00:44.220 --> 00:00:45.950
Let me draw one
more, because there's

00:00:45.950 --> 00:00:49.130
only three possibilities
in two dimensions.

00:00:49.130 --> 00:00:51.100
x and y if we're dealing
with linear equations.

00:00:54.420 --> 00:00:57.290
So you can have the situation
where the lines just

00:00:57.290 --> 00:00:59.310
intersect in one point.

00:00:59.310 --> 00:01:00.720
Let me do this.

00:01:00.720 --> 00:01:03.040
So you could have
one line like that

00:01:03.040 --> 00:01:05.630
and maybe the other line
does something like that

00:01:05.630 --> 00:01:08.360
and they intersect at one point.

00:01:08.360 --> 00:01:09.870
You could have the
situation where

00:01:09.870 --> 00:01:11.880
the two lines are parallel.

00:01:11.880 --> 00:01:14.380
So you could have a situation--
actually let me draw it over

00:01:14.380 --> 00:01:17.580
here-- where you have one
line that goes like that

00:01:17.580 --> 00:01:20.120
and the other line has the
same slope but it's shifted.

00:01:20.120 --> 00:01:22.840
It has a different y-intercept,
so maybe it looks like this.

00:01:22.840 --> 00:01:24.800
And you have no points
of intersection.

00:01:24.800 --> 00:01:27.050
And then you could
have the situation

00:01:27.050 --> 00:01:30.660
where they're actually the same
line, so that both lines have

00:01:30.660 --> 00:01:33.540
the same slope and
the same y-intercept.

00:01:33.540 --> 00:01:35.520
So really they
are the same line.

00:01:35.520 --> 00:01:38.290
They intersect on an
infinite number of points.

00:01:38.290 --> 00:01:39.790
Every point on
either of those lines

00:01:39.790 --> 00:01:42.060
is also a point
on the other line.

00:01:42.060 --> 00:01:45.210
So just to give you a little
bit of the terminology here,

00:01:45.210 --> 00:01:49.520
and we learned this in the
last video, this type of system

00:01:49.520 --> 00:01:53.030
where they don't intersect,
where you have no solutions,

00:01:53.030 --> 00:01:56.575
this is an inconsistent system.

00:01:59.530 --> 00:02:01.270
And by definition, or
I guess just taking

00:02:01.270 --> 00:02:03.100
the opposite of
inconsistent, both of these

00:02:03.100 --> 00:02:04.391
would be considered consistent.

00:02:08.440 --> 00:02:11.140
But then within consistent,
there's obviously a difference.

00:02:11.140 --> 00:02:12.830
Here we only have one solution.

00:02:12.830 --> 00:02:15.440
These are two different lines
that intersect in one place.

00:02:15.440 --> 00:02:18.800
And here they're essentially
the same exact line.

00:02:18.800 --> 00:02:21.450
And so we differentiate
between these two scenarios

00:02:21.450 --> 00:02:27.440
by calling this one
over here independent

00:02:27.440 --> 00:02:28.830
and this one over
here dependent.

00:02:32.310 --> 00:02:34.980
So independent-- both lines
are doing their own thing.

00:02:34.980 --> 00:02:37.260
They're not dependent
on each other.

00:02:37.260 --> 00:02:38.360
They're not the same line.

00:02:38.360 --> 00:02:40.410
They will intersect
at one place.

00:02:40.410 --> 00:02:42.440
Dependent-- they're
the exact same line.

00:02:42.440 --> 00:02:44.950
Any point that satisfies one
line will satisfy the other.

00:02:44.950 --> 00:02:46.533
Any points that
satisfies one equation

00:02:46.533 --> 00:02:48.500
will satisfy the other.

00:02:48.500 --> 00:02:53.540
So with that said, let's see if
this system of linear equations

00:02:53.540 --> 00:02:55.532
right here is dependent
or independent.

00:02:55.532 --> 00:02:56.990
So they're kind of
having us assume

00:02:56.990 --> 00:02:58.364
that it's going
to be consistent,

00:02:58.364 --> 00:03:00.430
that we're going to
intersect in one place

00:03:00.430 --> 00:03:03.360
or going to intersect in an
infinite number of places.

00:03:03.360 --> 00:03:05.640
And the easiest way to
do this-- we already

00:03:05.640 --> 00:03:07.710
have this second equation here.

00:03:07.710 --> 00:03:09.500
It's already in
slope-intercept form.

00:03:09.500 --> 00:03:13.450
We know the slope is negative
2, the y-intercept is 8.

00:03:13.450 --> 00:03:17.709
Let's put this first equation
up here in slope-intercept form

00:03:17.709 --> 00:03:20.250
and see if it has a different
slope or a different intercept.

00:03:20.250 --> 00:03:21.930
Or maybe it's the same line.

00:03:21.930 --> 00:03:27.370
So we have 4x plus
2y is equal to 16.

00:03:27.370 --> 00:03:29.640
We can subtract 4x
from both sides.

00:03:29.640 --> 00:03:32.750
What we want to do is isolate
the y on the left hand side.

00:03:32.750 --> 00:03:37.020
So let's subtract
4x from both sides.

00:03:37.020 --> 00:03:40.470
The left hand side-- we
are just left with a 2y.

00:03:40.470 --> 00:03:45.810
And then the right hand side,
we have a negative 4x plus 16.

00:03:45.810 --> 00:03:48.270
I just wrote the negative
4 in front of the 16,

00:03:48.270 --> 00:03:50.880
just so that we have it in the
traditional slope-intercept

00:03:50.880 --> 00:03:51.439
form.

00:03:51.439 --> 00:03:53.480
And now we can divide both
sides of this equation

00:03:53.480 --> 00:03:56.910
by 2, so that we can isolate
the y on the left hand side.

00:03:56.910 --> 00:03:58.180
Divide both sides by 2.

00:03:58.180 --> 00:04:02.160
We are left with y is equal
to negative 4 divided by 2

00:04:02.160 --> 00:04:06.830
is negative 2x plus
16 over 2 plus 8.

00:04:06.830 --> 00:04:10.780
So all I did is algebraically
manipulate this top equation up

00:04:10.780 --> 00:04:11.480
here.

00:04:11.480 --> 00:04:14.610
And when I did that, when
I solved essentially for y,

00:04:14.610 --> 00:04:16.450
I got this right
over here, which

00:04:16.450 --> 00:04:19.149
is the exact same thing
as the second equation.

00:04:19.149 --> 00:04:23.590
We have the exact same slope,
negative 2, negative 2,

00:04:23.590 --> 00:04:26.870
and we have the exact
same y-intercept, 8 and 8.

00:04:26.870 --> 00:04:33.070
If I were to graph these
equations-- that's my x-axis,

00:04:33.070 --> 00:04:39.520
and that is my y-axis-- both
of them have a y-intercept at 8

00:04:39.520 --> 00:04:41.090
and then have a
slope of negative 2.

00:04:41.090 --> 00:04:42.750
So they look
something-- I'm just

00:04:42.750 --> 00:04:46.270
drawing an approximation of it--
but they would look something

00:04:46.270 --> 00:04:47.100
like that.

00:04:47.100 --> 00:04:49.530
So maybe this is the graph
of this equation right

00:04:49.530 --> 00:04:51.174
here, this first equation.

00:04:51.174 --> 00:04:53.590
And then the second equation
will be the exact same graph.

00:04:53.590 --> 00:04:55.110
It has the exact
same y-intercept

00:04:55.110 --> 00:04:56.850
and the exact same slope.

00:04:56.850 --> 00:04:59.426
So clearly these two
lines are dependent.

00:05:02.360 --> 00:05:05.010
They have an infinite
number of points

00:05:05.010 --> 00:05:09.393
that are common to both of them,
because they're the same line.


Checking solutions to systems of equations example

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:04.260
Is negative 1 comma 7 a
solution for the system

00:00:04.260 --> 00:00:06.132
of linear equations below?

00:00:06.132 --> 00:00:07.590
And they give us
the first equation

00:00:07.590 --> 00:00:10.100
is x plus 2y is equal to 13.

00:00:10.100 --> 00:00:14.430
Second equation is 3x minus
y is equal to negative 11.

00:00:14.430 --> 00:00:17.190
In order for negative
1 comma 7 to be

00:00:17.190 --> 00:00:19.670
a solution for the
system, it needs

00:00:19.670 --> 00:00:21.970
to satisfy both equations.

00:00:21.970 --> 00:00:27.880
Or another way of thinking about
it, x equals 7, and y-- sorry,

00:00:27.880 --> 00:00:28.939
x is equal to negative 1.

00:00:28.939 --> 00:00:29.980
This is the x coordinate.

00:00:29.980 --> 00:00:34.520
X equals negative 1, and
y is equal to 7, need

00:00:34.520 --> 00:00:37.340
to satisfy both of
these equations in order

00:00:37.340 --> 00:00:39.367
for it to be a Solution.

00:00:39.367 --> 00:00:40.200
So let's try it out.

00:00:40.200 --> 00:00:42.960
Let's try it out with
the first equation.

00:00:42.960 --> 00:00:45.477
So we have x plus
2y is equal to 13.

00:00:45.477 --> 00:00:47.060
So if we're thinking
about that, we're

00:00:47.060 --> 00:00:49.530
testing to see if when x
is equal to negative 1,

00:00:49.530 --> 00:00:53.210
and y is equal to 7,
will x plus 2y equals 13?

00:00:53.210 --> 00:00:57.110
So we have negative
1 plus 2 times

00:00:57.110 --> 00:01:01.774
7-- y should be 7-- this
needs to be equal to 13.

00:01:01.774 --> 00:01:03.190
And I'll put a
question mark there

00:01:03.190 --> 00:01:04.830
because we don't
know whether it does.

00:01:04.830 --> 00:01:06.880
So this is the same
thing as negative 1

00:01:06.880 --> 00:01:09.760
plus 2 times 7 plus 14.

00:01:09.760 --> 00:01:12.310
That does, indeed, equal 13.

00:01:12.310 --> 00:01:15.200
Negative 1 plus 14, this is 13.

00:01:15.200 --> 00:01:20.370
So 13 does definitely equal 13.

00:01:20.370 --> 00:01:24.960
So this point it does, at least,
satisfy this first equation.

00:01:24.960 --> 00:01:28.820
This point does sit on the
graph of this first equation,

00:01:28.820 --> 00:01:30.637
or on the line of
this first equation.

00:01:30.637 --> 00:01:32.220
Now let's look at
the second equation.

00:01:32.220 --> 00:01:34.810
I'll do that one in blue.

00:01:34.810 --> 00:01:44.300
We have 3 times negative
1 minus y, so minus 7,

00:01:44.300 --> 00:01:46.244
needs to be equal
to negative 11.

00:01:46.244 --> 00:01:47.910
I'll put a question
mark here because we

00:01:47.910 --> 00:01:49.770
don't know whether
it's true or not.

00:01:49.770 --> 00:01:52.840
So let's see, we have 3 times
negative 1 is negative 3.

00:01:52.840 --> 00:01:57.850
And then we have minus 7 needs
to be equal to negative 11--

00:01:57.850 --> 00:01:59.390
I put the question mark there.

00:01:59.390 --> 00:02:02.700
Negative 3 minus 7,
that's negative 10.

00:02:02.700 --> 00:02:06.340
So we get negative 10
equaling negative 11.

00:02:06.340 --> 00:02:09.729
No, negative 10 does
not equal a negative 11.

00:02:09.729 --> 00:02:14.450
So x equaling negative
1, and y equaling 7

00:02:14.450 --> 00:02:17.200
does not satisfy
the second equation.

00:02:17.200 --> 00:02:19.190
So it does not sit on its graph.

00:02:19.190 --> 00:02:25.120
So this over here is not
a solution for the system.

00:02:25.120 --> 00:02:27.240
So the answer is no.

00:02:27.240 --> 00:02:29.280
It satisfies the
first equation, but it

00:02:29.280 --> 00:02:30.760
doesn't satisfy the second.

00:02:30.760 --> 00:02:32.910
In order to be a
solution for the system,

00:02:32.910 --> 00:02:36.590
it has to satisfy
both equations.


System of Inequalities Application

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:05.070
Luis receives a gift card worth
$25 to an online retailer that

00:00:05.070 --> 00:00:07.270
sells digital music and games.

00:00:07.270 --> 00:00:11.880
Each song costs $1,
and each game costs $2.

00:00:11.880 --> 00:00:15.450
He wants to buy at least
15 items with his card.

00:00:15.450 --> 00:00:18.920
Set up a system of inequalities
that represents this scenario

00:00:18.920 --> 00:00:21.310
and identify the range
of possible purchases

00:00:21.310 --> 00:00:22.650
using a graph.

00:00:22.650 --> 00:00:24.330
So let's define
some variables here.

00:00:27.125 --> 00:00:29.030
So he's going to
buy music and games.

00:00:29.030 --> 00:00:38.910
So let's let m equal the number
of songs he buys, and let g

00:00:38.910 --> 00:00:41.939
be equal to the number of games.

00:00:41.939 --> 00:00:43.480
Actually, just so
it's more intuitive

00:00:43.480 --> 00:00:45.070
let's let s be the
number of songs.

00:00:53.190 --> 00:01:00.707
And let's let g be equal to the
number of games that he buys.

00:01:00.707 --> 00:01:02.290
So they give us some
information here.

00:01:02.290 --> 00:01:07.770
They say that he wants to buy at
least 15 items with this card.

00:01:07.770 --> 00:01:09.290
So that means that
the total number

00:01:09.290 --> 00:01:12.120
of items, the number of songs
plus the number of games,

00:01:12.120 --> 00:01:14.210
have to be at least 15.

00:01:14.210 --> 00:01:16.380
So this sentence
right here tells us

00:01:16.380 --> 00:01:19.960
that s plus g-- that's
the total number

00:01:19.960 --> 00:01:22.720
of items-- it has
to be at least 15.

00:01:22.720 --> 00:01:25.720
So it has to be greater
than or equal to 15.

00:01:25.720 --> 00:01:27.920
It can be 15 or
larger, it could be

00:01:27.920 --> 00:01:30.560
16, 17 18, so on and so forth.

00:01:30.560 --> 00:01:36.720
It also tells us that he
has a gift card worth $25,

00:01:36.720 --> 00:01:41.490
and each song costs $1
and each game costs $2.

00:01:41.490 --> 00:01:43.620
So he can't spend
any more than $25.

00:01:43.620 --> 00:01:45.840
So how much is he going
to spend as a function

00:01:45.840 --> 00:01:47.400
of the songs and the games?

00:01:47.400 --> 00:01:49.130
So each song costs $1.

00:01:49.130 --> 00:01:51.580
So he's going to spend
the number of dollars

00:01:51.580 --> 00:01:53.230
times the number of songs.

00:01:53.230 --> 00:01:55.750
So if he buys 10 songs
is going to spend $10.

00:01:55.750 --> 00:01:58.430
Plus, for each game,
he is going to spend

00:01:58.430 --> 00:02:01.580
$2, plus 2 times
the number of games.

00:02:01.580 --> 00:02:04.250
So if he buys five
songs and five games,

00:02:04.250 --> 00:02:09.550
the five songs are going to cost
$5, 5 times 1, plus 5 times 2

00:02:09.550 --> 00:02:11.039
for the games, plus 10.

00:02:11.039 --> 00:02:12.540
So it's going to be $15.

00:02:12.540 --> 00:02:15.140
So this is how much he
spends on songs and games.

00:02:15.140 --> 00:02:17.040
I could have put a 1
out here for dollars,

00:02:17.040 --> 00:02:21.020
but this is how much he spent--
or 1 for it's $1 per song.

00:02:21.020 --> 00:02:23.870
But this is how much he spends
in dollars on songs and games.

00:02:23.870 --> 00:02:26.820
And he only has a
gift card worth $25.

00:02:26.820 --> 00:02:28.870
The total amount he
spends has to be less than

00:02:28.870 --> 00:02:30.690
or equal to $25.

00:02:30.690 --> 00:02:32.980
He can spend exactly
$25 if he has to.

00:02:32.980 --> 00:02:36.000
He just can't spend $25.01.

00:02:36.000 --> 00:02:38.320
So we have a system
of inequalities here,

00:02:38.320 --> 00:02:40.800
and now we can
essentially graph them.

00:02:40.800 --> 00:02:43.360
And so to graph
them we just have

00:02:43.360 --> 00:02:45.380
to pick one something to
go on the vertical axis

00:02:45.380 --> 00:02:48.250
and the horizontal axis.

00:02:48.250 --> 00:02:51.020
And actually I'm only going to
draw the first quadrant here

00:02:51.020 --> 00:02:53.580
since he's only going to buy
a positive number of songs

00:02:53.580 --> 00:02:54.630
and games.

00:02:54.630 --> 00:02:57.510
So let's let the songs
on the vertical axis,

00:02:57.510 --> 00:03:02.050
and let's let the games
on the horizontal axis.

00:03:02.050 --> 00:03:05.240
And then what I like to do
is, I like to express it

00:03:05.240 --> 00:03:08.840
in terms of-- or if I'm
putting s as the vertical axis,

00:03:08.840 --> 00:03:11.860
I like to solve for
s, so it kind of

00:03:11.860 --> 00:03:14.810
goes in the traditional
slope intercept form.

00:03:14.810 --> 00:03:17.430
So this first thing up
here, if we subtract g

00:03:17.430 --> 00:03:20.220
from both sides of this,
we get that the songs

00:03:20.220 --> 00:03:25.570
have to be greater than or
equal to negative g plus 15.

00:03:25.570 --> 00:03:28.446
And then this one over
here, if we subtract 2g

00:03:28.446 --> 00:03:30.820
from both sides-- I'm going
to do it in that same magenta

00:03:30.820 --> 00:03:34.520
color-- if we subtract
2g from both sides,

00:03:34.520 --> 00:03:36.490
so that we isolate
the s on the left,

00:03:36.490 --> 00:03:45.300
we get s is less than or
equal to negative 2g plus 25.

00:03:45.300 --> 00:03:46.770
And now we can try
to graph these,

00:03:46.770 --> 00:03:48.590
and we could just
look at points.

00:03:48.590 --> 00:03:53.430
Or we could look at the slope
intercept of the boundary line.

00:03:53.430 --> 00:03:58.014
So over here, if
he buys no games,

00:03:58.014 --> 00:04:00.430
then s is going to be-- well
just think about the boundary

00:04:00.430 --> 00:04:00.930
line here.

00:04:00.930 --> 00:04:03.550
So the boundary line for
this constraint over here

00:04:03.550 --> 00:04:07.380
is going to be s is equal
to negative g plus 15.

00:04:07.380 --> 00:04:10.390
So if he buys no games
he will buy 15 songs.

00:04:10.390 --> 00:04:15.780
So let's make this
5, 10, 15 songs.

00:04:15.780 --> 00:04:18.690
So the s intercept,
we could say, is 15.

00:04:18.690 --> 00:04:22.070
If he buys no games
he'll buy 15 songs,

00:04:22.070 --> 00:04:24.350
based on that constraint.

00:04:24.350 --> 00:04:26.940
Now what happens if
he buys no songs.

00:04:26.940 --> 00:04:28.720
If he buys no
songs, then you have

00:04:28.720 --> 00:04:31.250
0 is equal to
negative g plus 15,

00:04:31.250 --> 00:04:34.680
or you could say that
g is equal to 15.

00:04:34.680 --> 00:04:38.630
So if he buys no songs then he's
going to have to buy 15 games.

00:04:38.630 --> 00:04:43.560
So it's 5, 10, and 15.

00:04:43.560 --> 00:04:45.650
So that would be a point
on the boundary line.

00:04:45.650 --> 00:04:47.900
And we're actually going to
include the boundary line,

00:04:47.900 --> 00:04:50.280
because it's greater
than or equal to.

00:04:50.280 --> 00:04:52.000
So I'll draw a solid line here.

00:04:54.930 --> 00:04:56.980
So this is our boundary
line, but it's not just

00:04:56.980 --> 00:04:57.729
the boundary line.

00:04:57.729 --> 00:04:59.930
It's not s plus
g is equal to 15.

00:04:59.930 --> 00:05:02.620
It's s plus g has to be
greater than or equal to 15.

00:05:02.620 --> 00:05:06.520
Or s is greater than or
equal to negative g plus 15.

00:05:06.520 --> 00:05:11.320
So when you look at it this
way, you see s for any g,

00:05:11.320 --> 00:05:14.540
negative g plus 15, is
going to hit the line,

00:05:14.540 --> 00:05:17.110
or it's going to hit right at
the boundary line for the s.

00:05:17.110 --> 00:05:19.160
But it's that s
and any of the s's

00:05:19.160 --> 00:05:22.255
that are greater, so we're
going to be above the line.

00:05:22.255 --> 00:05:23.630
And if you just
want to make sure

00:05:23.630 --> 00:05:25.350
you can verify that
we're above the line.

00:05:25.350 --> 00:05:27.891
You could try something that's
on the other side of the line,

00:05:27.891 --> 00:05:29.290
maybe 0, 0.

00:05:29.290 --> 00:05:30.600
0, 0 definitely doesn't work.

00:05:30.600 --> 00:05:33.290
0 plus 0 is not greater
than or equal to 15.

00:05:33.290 --> 00:05:35.320
And if you try something
out here like 15 comma

00:05:35.320 --> 00:05:39.700
15-- 15 plus 15 is 30, which
is greater than or equal to 15.

00:05:39.700 --> 00:05:44.280
So we feel pretty good that it's
this whole area above the line.

00:05:44.280 --> 00:05:46.890
And now let's look at
this second constraint.

00:05:46.890 --> 00:05:52.340
The boundary line is s is
equal to negative 2g plus 25.

00:05:52.340 --> 00:05:55.330
So once again, the s intercept,
or if g is equal to 0,

00:05:55.330 --> 00:05:57.330
s is equal to 25.

00:05:57.330 --> 00:06:01.520
So we go to 20, and then 25.

00:06:01.520 --> 00:06:04.070
If g is 0, s is 25.

00:06:04.070 --> 00:06:06.785
And then if s is 0, what is g?

00:06:06.785 --> 00:06:10.720
If s is 0, we get 0 is equal
to negative 2g plus 25.

00:06:10.720 --> 00:06:12.830
We can add 2g to both sides.

00:06:12.830 --> 00:06:14.780
You get 2g is equal to 25.

00:06:14.780 --> 00:06:17.280
You can divide both sides by 2.

00:06:17.280 --> 00:06:20.610
You get g is equal
to 25/2, which

00:06:20.610 --> 00:06:23.690
is the same thing as 12 and 1/2.

00:06:23.690 --> 00:06:26.130
So if s is 0, g is 12 and 1/2.

00:06:29.240 --> 00:06:33.250
Something like this-- 12
and 1/2 would put us-- well

00:06:33.250 --> 00:06:34.969
it's actually going
to be-- that's 5,

00:06:34.969 --> 00:06:37.260
so 12 and 1/2 is going to be
right in between these two

00:06:37.260 --> 00:06:37.630
guys.

00:06:37.630 --> 00:06:39.120
It's going to be
right over there.

00:06:39.120 --> 00:06:41.430
And once again we're
including the boundary line

00:06:41.430 --> 00:06:43.240
because it's less
than or equal to,

00:06:43.240 --> 00:06:44.700
so we'll include
the boundary line.

00:06:44.700 --> 00:06:48.590
So I'll draw a solid straight
line, just like that.

00:06:48.590 --> 00:06:50.650
That's supposed to
intersect right over there.

00:06:50.650 --> 00:06:53.260
And then you can think
about it two ways.

00:06:53.260 --> 00:06:58.720
You could just say, well, for
any g, negative 2g plus 25

00:06:58.720 --> 00:07:00.350
will put us on
the boundary line,

00:07:00.350 --> 00:07:03.656
and we want all the s values
that are less than that.

00:07:03.656 --> 00:07:07.380
We are going to be
below this line.

00:07:07.380 --> 00:07:08.360
And you can verify it.

00:07:08.360 --> 00:07:11.000
Take some points that's on
either side of this magenta

00:07:11.000 --> 00:07:13.280
line, 0, 0 should work.

00:07:13.280 --> 00:07:16.610
If you have 0 plus 0, that
is less than or equal to 25,

00:07:16.610 --> 00:07:17.590
so that works.

00:07:17.590 --> 00:07:20.260
15 comma 15 shouldn't work.

00:07:20.260 --> 00:07:24.799
15 plus 2 times 15 is 3
times 15, so that's 45,

00:07:24.799 --> 00:07:26.340
and that's not less
than or equal 25.

00:07:26.340 --> 00:07:28.790
So stuff out here above the
magenta line won't work.

00:07:28.790 --> 00:07:31.040
Stuff below the
magenta line will work.

00:07:31.040 --> 00:07:34.370
Now the range of combinations
of songs and games

00:07:34.370 --> 00:07:37.990
that will actually
work are the overlap

00:07:37.990 --> 00:07:39.790
of these two inequalities.

00:07:39.790 --> 00:07:42.340
The way we've visually
drawn it right here,

00:07:42.340 --> 00:07:49.490
they only overlap right
over here, this region

00:07:49.490 --> 00:07:51.550
that I'm shading
in here in blue.

00:07:51.550 --> 00:07:53.780
But it will include
the boundaries

00:07:53.780 --> 00:07:55.170
because both of
those constraints

00:07:55.170 --> 00:07:56.745
were-- included the equal.

00:07:56.745 --> 00:07:58.120
This is greater
than or equal to,

00:07:58.120 --> 00:07:59.369
this is less than or equal to.

00:07:59.369 --> 00:08:01.150
So it will include
the boundaries.

00:08:01.150 --> 00:08:03.420
So any combination
of songs and games

00:08:03.420 --> 00:08:04.980
that fit in this
boundary right here

00:08:04.980 --> 00:08:07.898
will satisfy his constraints.


Graphical systems application problem

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.160
Use graphing to solve
the following problem.

00:00:05.770 --> 00:00:08.610
Abby and Ben did household
chores last weekend.

00:00:08.610 --> 00:00:13.690
Together they earned $50, and
Abby earned $10 more than Ben.

00:00:13.690 --> 00:00:15.480
How much did they each earn?

00:00:15.480 --> 00:00:16.990
So let's define
some variables here.

00:00:16.990 --> 00:00:21.780
Let's let A equal
Abby's earnings.

00:00:26.280 --> 00:00:30.495
And let's let B
equal Ben's earnings.

00:00:34.840 --> 00:00:37.500
Then they tell us how
these earnings relate.

00:00:37.500 --> 00:00:42.080
They first tell us that
together they earned $50.

00:00:42.080 --> 00:00:45.460
So that statement can be
converted mathematically

00:00:45.460 --> 00:00:46.890
into-- well,
together, that means

00:00:46.890 --> 00:00:48.280
the sum of the two earnings.

00:00:48.280 --> 00:00:53.800
So A plus B needs
to be equal to $50.

00:00:53.800 --> 00:00:56.590
Abby's plus Ben's
earnings is $50.

00:00:56.590 --> 00:01:01.640
And then they tell us Abby
earned $10 more than Ben.

00:01:01.640 --> 00:01:06.460
So we could translate
that into Abby's earnings

00:01:06.460 --> 00:01:09.290
is equal to Ben's
earnings plus 10.

00:01:09.290 --> 00:01:12.590
Abby earned $10 more than Ben.

00:01:12.590 --> 00:01:14.879
So we have a system
of two equations

00:01:14.879 --> 00:01:16.170
and actually with two unknowns.

00:01:16.170 --> 00:01:20.354
And then they say, how
much did each earn?

00:01:20.354 --> 00:01:22.770
So to do that, and they want
us to solve this graphically.

00:01:22.770 --> 00:01:24.186
There's multiple
ways to solve it,

00:01:24.186 --> 00:01:25.950
but we'll do what
they ask us to do.

00:01:25.950 --> 00:01:28.690
Let me draw some axes over here.

00:01:28.690 --> 00:01:30.160
And I'll be in
the first quadrant

00:01:30.160 --> 00:01:31.680
since we're dealing
with earnings,

00:01:31.680 --> 00:01:34.990
so neither of their
earnings can be negative.

00:01:34.990 --> 00:01:37.400
And let me just define
the vertical axis

00:01:37.400 --> 00:01:40.980
as Abby's axis or the
Abby's earnings axis.

00:01:40.980 --> 00:01:43.610
And let me define
the horizontal axis

00:01:43.610 --> 00:01:46.460
as Ben's axis or
Ben's earnings axis.

00:01:46.460 --> 00:01:49.420
And let me just graph
each of these equations.

00:01:49.420 --> 00:01:52.282
And to do that, I'm going
to take this first equation,

00:01:52.282 --> 00:01:53.990
and I'm going to put
it in the equivalent

00:01:53.990 --> 00:01:55.322
of slope-intercept form.

00:01:55.322 --> 00:01:57.030
It might look a little
unfamiliar to you,

00:01:57.030 --> 00:01:59.020
but it really is
slope-intercept form.

00:01:59.020 --> 00:02:00.020
Let me rewrite it first.

00:02:00.020 --> 00:02:03.180
So we have A plus
B is equal to 50.

00:02:03.180 --> 00:02:06.300
We can subtract B
from both sides.

00:02:06.300 --> 00:02:09.740
So let's subtract
B from both sides.

00:02:09.740 --> 00:02:17.920
And then we get A is equal
to negative B plus 50.

00:02:17.920 --> 00:02:20.830
So if you think about it this
way, when B is equal to 0,

00:02:20.830 --> 00:02:22.000
A is going to be 50.

00:02:22.000 --> 00:02:24.529
So we know our A intercept,
we could call it.

00:02:24.529 --> 00:02:26.820
We normally would call that
a y-intercept, but now this

00:02:26.820 --> 00:02:28.170
is the A axis.

00:02:28.170 --> 00:02:35.540
So this right here, let me call
this 10, 20, 30, 40, and 50.

00:02:35.540 --> 00:02:39.630
So if Ben made $0,
then Abby would

00:02:39.630 --> 00:02:42.800
have to make $50 based
on that first constraint.

00:02:42.800 --> 00:02:46.230
So we know that that's a point
on the line right over there.

00:02:46.230 --> 00:02:49.020
And we also know that
the slope is negative 1,

00:02:49.020 --> 00:02:51.560
that B is the independent
variable, the way I've written

00:02:51.560 --> 00:02:54.380
it over here, and this
coefficient is negative 1.

00:02:54.380 --> 00:02:58.540
Or another way to think
about it is if A is 0, then

00:02:58.540 --> 00:02:59.830
B is going to be 50.

00:02:59.830 --> 00:03:03.060
If Abby made no money, then
Ben would have to make $50.

00:03:03.060 --> 00:03:05.470
And that falls purely out
of this equation right here.

00:03:05.470 --> 00:03:06.970
If Abby made nothing,
then Ben would

00:03:06.970 --> 00:03:13.489
have to make $50, so
10, 20, 30, 40, 50.

00:03:13.489 --> 00:03:15.530
So those are those two
situations and every point

00:03:15.530 --> 00:03:18.640
in between will satisfy
this first constraint.

00:03:18.640 --> 00:03:21.440
So let me connect the dots.

00:03:21.440 --> 00:03:23.970
So it would look
something like that.

00:03:23.970 --> 00:03:26.190
That's due to this first
constraint, due to the fact

00:03:26.190 --> 00:03:29.077
that together they earned $50.

00:03:29.077 --> 00:03:30.660
Now, let's think
about the second one.

00:03:30.660 --> 00:03:33.140
Abby earned $10 more than Ben.

00:03:33.140 --> 00:03:35.090
So that's this
equation right here.

00:03:35.090 --> 00:03:37.530
And it's really already in
our slope-intercept form.

00:03:37.530 --> 00:03:41.490
If Ben made $0, then
Abby would make $10.

00:03:41.490 --> 00:03:42.720
So that's our A intercept.

00:03:42.720 --> 00:03:44.390
So it's right over there.

00:03:44.390 --> 00:03:45.680
We could keep doing that.

00:03:45.680 --> 00:03:47.020
Our slope is going to be 1 here.

00:03:47.020 --> 00:03:51.700
If Ben makes $10, then
Abby's going to make $20.

00:03:51.700 --> 00:03:56.574
If Ben makes $20, Abby's
going to make $30.

00:03:56.574 --> 00:03:58.490
We could keep going, but
I think this gives us

00:03:58.490 --> 00:03:59.406
the general direction.

00:03:59.406 --> 00:04:02.360
It already hints at a
point of intersection.

00:04:02.360 --> 00:04:05.480
So just eyeballing it, so we've
graphed to the two constraints,

00:04:05.480 --> 00:04:07.090
together they earned $50.

00:04:07.090 --> 00:04:09.480
That's the magenta
constraint right over here.

00:04:09.480 --> 00:04:11.810
Abby earned $10 more than Ben.

00:04:11.810 --> 00:04:14.417
That's this green
constraint right over here.

00:04:14.417 --> 00:04:16.500
And it looks like we have
a point of intersection.

00:04:16.500 --> 00:04:19.300
And it looks like we have
a point of intersection

00:04:19.300 --> 00:04:21.410
at Ben earning $20.

00:04:21.410 --> 00:04:25.540
Let me label this as
10, 20, 30, 40, and 50.

00:04:25.540 --> 00:04:29.460
So this is Ben earning
$20 and Abby earning--

00:04:29.460 --> 00:04:32.870
this is 10, 20, and 30.

00:04:32.870 --> 00:04:35.190
And Abby earning $30.

00:04:35.190 --> 00:04:40.940
So just eyeballing it off of
this, it looks like A is 30

00:04:40.940 --> 00:04:42.430
and B is 20.

00:04:42.430 --> 00:04:46.610
And let's go verify, make sure
that these levels of earnings

00:04:46.610 --> 00:04:50.370
for Abby and Ben actually
satisfy both constraints.

00:04:50.370 --> 00:04:54.030
So the first constraint is that
Abby plus Ben have to make $50.

00:04:54.030 --> 00:04:56.600
Well, $30 plus $20 is $50.

00:04:56.600 --> 00:04:58.550
So it meets our
first constraint.

00:04:58.550 --> 00:05:01.910
The second constraint is that
Abby earned $10 more than Ben,

00:05:01.910 --> 00:05:04.370
that Abby is equal
to Ben plus 10.

00:05:04.370 --> 00:05:05.810
Well, once again,
over here, Abby

00:05:05.810 --> 00:05:07.720
is making $10 more than Ben.

00:05:07.720 --> 00:05:09.970
So it meets our
second constraint.

00:05:09.970 --> 00:05:11.669
And we only have two of them.

00:05:11.669 --> 00:05:12.710
So it meets both of them.

00:05:12.710 --> 00:05:14.190
So that's our solution.

00:05:14.190 --> 00:05:15.740
Abby earned $30.

00:05:15.740 --> 00:05:17.833
Ben earned $20.