Analyzing solutions to linear systems graphically 2

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.740
So that we don't get stumped
by the Arbeglas in our life

00:00:03.740 --> 00:00:06.600
and especially when we don't
have talking birds to help us,

00:00:06.600 --> 00:00:09.350
we should be able to
identify when things

00:00:09.350 --> 00:00:12.280
get a little bit weird with
our systems of equations.

00:00:12.280 --> 00:00:15.210
When we have scenarios that have
an infinite number of solutions

00:00:15.210 --> 00:00:17.810
or that have no
solutions at all.

00:00:17.810 --> 00:00:21.300
And just as a little bit of a
review of what could happen,

00:00:21.300 --> 00:00:23.630
these are the-- think
about the three scenarios.

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You have the first
scenario which

00:00:25.190 --> 00:00:29.750
is kind of where we started off,
where you have two systems that

00:00:29.750 --> 00:00:31.470
just intersect in one place.

00:00:31.470 --> 00:00:33.300
And then you have
essentially one solution.

00:00:33.300 --> 00:00:35.150
So if you were to
graphically represent it

00:00:35.150 --> 00:00:40.650
you have one solution right
over there, one solution.

00:00:40.650 --> 00:00:46.580
And this means that the two
constraints are consistent

00:00:46.580 --> 00:00:49.610
and the two constraints are
independent of each other.

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They're not the exact same line,
consistent and independent.

00:00:57.330 --> 00:01:03.712
Then you have the other scenario
where they're consistent,

00:01:03.712 --> 00:01:05.920
they intersect,but they're
essentially the same line.

00:01:05.920 --> 00:01:07.310
They intersect everywhere.

00:01:07.310 --> 00:01:09.312
So this is one of the
constraints for one

00:01:09.312 --> 00:01:10.770
of the equations,
and the other one

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if you look at it,
if you graph it,

00:01:12.440 --> 00:01:14.740
it is actually the
exact same one.

00:01:14.740 --> 00:01:19.650
So here you have an infinite
number of solutions.

00:01:19.650 --> 00:01:23.820
It's consistent, you
do have solutions here,

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but they're dependent equations.

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It's a dependent system.

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And then the last
scenario, and this

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is when you're dealing
in two dimensions,

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the last scenario is where
your two constraints just

00:01:37.910 --> 00:01:39.670
don't intersect with each other.

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One might look like
this, and then the other

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might look like this.

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They have the exact
same slope but they

00:01:47.620 --> 00:01:49.440
have different intercepts.

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So this there is no solution,
they never intersect.

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And we call this an
inconsistent system.

00:02:00.409 --> 00:02:02.700
And if you wanted to think
about what would happen just

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think about what's
going on here.

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Here you have different slopes.

00:02:10.949 --> 00:02:13.640
And if you think about
it, two different lines

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with different
slopes are definitely

00:02:15.520 --> 00:02:20.120
going to intersect
in exactly one place.

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Here they have the same
slope and same y-intercept,

00:02:28.270 --> 00:02:30.250
so you have an infinite
number of solutions.

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Here you have the same slope
but different y-intercepts,

00:02:35.670 --> 00:02:37.520
and you get no solutions.

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So the times when
you're solving systems

00:02:39.260 --> 00:02:41.260
where things are going
to get a little bit weird

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are when you have
the same slope.

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And if you think about it,
what defines the slope,

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and I encourage you to test this
out with different equations,

00:02:52.830 --> 00:02:55.281
is when you have-- if you
have your x's and y's, or you

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have your a's and b's or
you have your variables

00:02:57.280 --> 00:02:59.810
on the same side of
an equation, where

00:02:59.810 --> 00:03:03.600
they have the same ratio
with respect to each other.

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So with that, kind of
keeping that in mind,

00:03:05.800 --> 00:03:09.350
let's see if we can think
about what types of solutions

00:03:09.350 --> 00:03:10.740
we might find.

00:03:10.740 --> 00:03:12.720
So let's take this down.

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So they say determine
how many solutions

00:03:14.430 --> 00:03:18.120
exist for the
system of equations.

00:03:18.120 --> 00:03:20.610
So you have 10x minus
2y is equal to 4,

00:03:20.610 --> 00:03:23.770
and 10x minus 2y is equal to 16.

00:03:23.770 --> 00:03:27.190
So just based on what we
just talked about the x's

00:03:27.190 --> 00:03:29.550
and the y's are on the
same side of the equation

00:03:29.550 --> 00:03:32.480
and the ratio is
10 to negative 2.

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Same ratio.

00:03:33.270 --> 00:03:35.730
So something strange is
going to happen here.

00:03:35.730 --> 00:03:37.930
But when we have the
same kind of combination

00:03:37.930 --> 00:03:40.980
of x's and y's in the first one
we get 4, and on the second one

00:03:40.980 --> 00:03:41.950
we get 16.

00:03:41.950 --> 00:03:43.915
So that seems a
little bit bizarre.

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Another way to
think about it, we

00:03:45.290 --> 00:03:47.670
have the same number of
x's, the same number of y's

00:03:47.670 --> 00:03:50.997
but we got a different number
on the right hand side.

00:03:50.997 --> 00:03:53.080
So if you were to simplify
this, and we could even

00:03:53.080 --> 00:03:54.550
look at the hints
to see what it says,

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you'll see that
you're going to end up

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with the same slope but
different y-intercepts.

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So we convert both the slope
intercept form right over here

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and you see one, the blue one
is y is equal to 5x minus 2,

00:04:06.630 --> 00:04:09.630
and the green one is y
is equal to 5x minus 8.

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Same slope, same ratio
between the x's and the Y's,

00:04:13.050 --> 00:04:17.290
but you have different
values right over here.

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You have different y-intercepts.

00:04:20.170 --> 00:04:23.670
So here you have no solutions.

00:04:23.670 --> 00:04:28.710
That is this scenario right over
here if you were to graph it.

00:04:28.710 --> 00:04:31.610
So no solutions,
check our answer.

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Let's go to the next question.

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So let's look at this
one right over here.

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So we have negative 5 times
x and negative 1 times y.

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We have 4 times x and 1 times y.

00:04:43.400 --> 00:04:45.329
So it looks like the
ratio if then we're

00:04:45.329 --> 00:04:47.620
looking at the x's and y's
always on the left hand side

00:04:47.620 --> 00:04:50.660
right over here, it looks like
the ratios of x's and y's are

00:04:50.660 --> 00:04:51.380
different.

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You have essentially
5 x's for every one y,

00:04:54.330 --> 00:04:57.170
or you could say negative 5
x's for every negative 1 y,

00:04:57.170 --> 00:05:00.170
and here you have 4
x's for every 1 y.

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So this is fundamentally
a different ratio.

00:05:02.080 --> 00:05:03.140
So right off the bat
you could say well these

00:05:03.140 --> 00:05:05.430
are going to intersect
in exactly one place.

00:05:05.430 --> 00:05:07.920
If you were to put this
into slope intercept form,

00:05:07.920 --> 00:05:10.520
you will see that they
have different slopes.

00:05:10.520 --> 00:05:12.380
So you could say
this has one solution

00:05:12.380 --> 00:05:13.960
and you can check your answer.

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And you could look at the
solution just to verify.

00:05:17.040 --> 00:05:18.780
And I encourage you to do this.

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So you see the blue one if you
put in the slope intercept form

00:05:21.740 --> 00:05:25.580
negative 5x plus 10 and
you take the green one

00:05:25.580 --> 00:05:29.040
into slope intercept
form negative 4x minus 8.

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So different slopes,
they're definitely

00:05:30.970 --> 00:05:33.310
going to intersect
in exactly one place.

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You're going to
have one solution.

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Let's try another one.

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So here we have 2x plus
y is equal to negative 3.

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And this is pretty
clear, you have

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2x plus y is equal
to negative 3.

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These are the exact
same equations.

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So it's consistent information,
there's definitely solutions.

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But there's an infinite number
of solutions right over here.

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This is a dependent system.

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So there are infinite
number of solutions here

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and we can check our answer.

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Let's do one more because that
was a little bit too easy.

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OK so this is interesting
right over here,

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we have it in different forms.

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2x plus y is equal
to negative 4,

00:06:02.580 --> 00:06:05.650
y is equal to
negative 2x minus 4.

00:06:05.650 --> 00:06:09.260
So let's take this
first blue equation

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and put it into
slope intercept form.

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If we did that you would
get, if you just subtract 2x

00:06:14.210 --> 00:06:17.810
from both sides you get y is
equal to negative 2x minus 4,

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which is the exact same thing as
this equation right over here.

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So once again they're
the exact same equation.

00:06:23.510 --> 00:06:25.550
You have an infinite
number of solutions.

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Check our answer, and you can
look at the solution right

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here.

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You convert the blue
one into slope intercept

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and you get the exact
same thing as what

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you saw in the green one.


Infinite solutions to systems

WEBVTT
Kind: captions
Language: en

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Arbegla starts to feel
angry and embarrassed

00:00:03.880 --> 00:00:08.130
that he was shown up by you and
the bird in front of the King

00:00:08.130 --> 00:00:10.200
and so he storms
out of the room.

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And then a few seconds
later he storms back in.

00:00:12.390 --> 00:00:13.560
He says, my fault.

00:00:13.560 --> 00:00:14.590
My apologies.

00:00:14.590 --> 00:00:18.420
I realize now what
the mistake was.

00:00:18.420 --> 00:00:22.650
There was a slight, I guess,
typing error or writing error.

00:00:22.650 --> 00:00:25.696
In the first week, when
they went to the market

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and bought two pounds of apples
and one pound of bananas,

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it wasn't a $3 cost.

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It was a $5 cost.

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Now surely considering how smart
you and this bird seem to be,

00:00:40.760 --> 00:00:45.550
you surely could figure out what
is the per pound cost of apples

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and what is the per
pound cost of bananas.

00:00:48.140 --> 00:00:51.270
So you think for a
little bit, is there now

00:00:51.270 --> 00:00:54.550
going to be a solution?

00:00:54.550 --> 00:00:57.440
So let's break it down using
the exact same variables.

00:00:57.440 --> 00:01:00.570
You say, well if a is the
cost of apples per pound

00:01:00.570 --> 00:01:05.519
and b is the cost of bananas,
this first constraint tells us

00:01:05.519 --> 00:01:09.416
that two pounds of apples
are going to cost 2a,

00:01:09.416 --> 00:01:11.420
because it's b
dollars per pound.

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And one pound of
bananas is going

00:01:13.520 --> 00:01:17.730
to cost b dollars because
it's one pound times

00:01:17.730 --> 00:01:20.823
b dollars per pound is
now going to cost $5.

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This is the corrected number.

00:01:27.720 --> 00:01:30.050
And we saw from
the last scenario,

00:01:30.050 --> 00:01:31.760
this information hasn't changed.

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Six pounds of apples is
going to cost 6a, six

00:01:36.030 --> 00:01:38.340
pounds times a
dollars per pound.

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And three pounds of bananas is
going to cost 3b, three pounds

00:01:42.820 --> 00:01:44.800
times b dollars per pound.

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The total cost of the
apples and bananas

00:01:46.890 --> 00:01:50.200
in this trip we
are given is $15.

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So once again, you
say, well let me

00:01:54.280 --> 00:01:58.140
try to solve this maybe
through elimination.

00:01:58.140 --> 00:02:00.470
And once again, you say well
let me cancel out the a's.

00:02:00.470 --> 00:02:01.280
I have 2a here.

00:02:01.280 --> 00:02:02.560
I have 6a here.

00:02:02.560 --> 00:02:05.295
If I multiply the 2a
here by negative 3,

00:02:05.295 --> 00:02:06.990
then this will
become a negative 6a.

00:02:06.990 --> 00:02:09.770
And it might be able to cancel
out with all of this business.

00:02:09.770 --> 00:02:10.820
So you do that.

00:02:10.820 --> 00:02:12.340
You multiply this
entire equation.

00:02:12.340 --> 00:02:13.770
You can't just
multiply one term.

00:02:13.770 --> 00:02:17.210
You have to multiply the entire
equation times negative 3

00:02:17.210 --> 00:02:19.270
if you want the
equation to still hold.

00:02:19.270 --> 00:02:21.330
And so we're multiplying
by negative 3

00:02:21.330 --> 00:02:25.790
so 2a times negative
3 is negative 6a.

00:02:25.790 --> 00:02:29.790
b times negative
3 is negative 3b.

00:02:29.790 --> 00:02:34.930
And then 5 times negative
3 is negative 15.

00:02:34.930 --> 00:02:37.220
And now something
fishy starts to look

00:02:37.220 --> 00:02:38.990
like it's about to happen.

00:02:38.990 --> 00:02:40.470
Because when you
add the left hand

00:02:40.470 --> 00:02:43.960
side of this blue equation
or this purplish equation

00:02:43.960 --> 00:02:46.720
to the green one, you get 0.

00:02:46.720 --> 00:02:50.320
All of these things right
over here just cancel out.

00:02:50.320 --> 00:02:53.900
And on the right hand
side, 15 minus 15,

00:02:53.900 --> 00:02:56.450
that is also equal to 0.

00:02:56.450 --> 00:03:00.922
And you get 0 equals 0, which
seems a little bit better

00:03:00.922 --> 00:03:02.630
than the last time
you worked through it.

00:03:02.630 --> 00:03:04.620
Last time we got 0 equals 6.

00:03:04.620 --> 00:03:07.130
But 0 equals 0 doesn't really
tell you anything about

00:03:07.130 --> 00:03:07.890
the x's and y's.

00:03:07.890 --> 00:03:08.890
This is true.

00:03:08.890 --> 00:03:13.040
This is absolutely true that
0 does definitely equals 0,

00:03:13.040 --> 00:03:16.110
but it doesn't tell you any
information about x and y.

00:03:16.110 --> 00:03:18.309
And so then the bird
whispers in the King's ear,

00:03:18.309 --> 00:03:19.850
and then the King
says, well the bird

00:03:19.850 --> 00:03:21.460
says you should graph
it to figure out

00:03:21.460 --> 00:03:23.450
what's actually going on.

00:03:23.450 --> 00:03:26.440
And so you've learned
that listening to the bird

00:03:26.440 --> 00:03:28.620
actually makes a lot of sense.

00:03:28.620 --> 00:03:32.644
So you try to graph
these two constraints.

00:03:32.644 --> 00:03:33.810
So let's do it the same way.

00:03:33.810 --> 00:03:34.643
We'll have a b axis.

00:03:37.290 --> 00:03:39.170
That's our b axis.

00:03:39.170 --> 00:03:42.980
And we will have our a axis.

00:03:42.980 --> 00:03:46.430
Let we mark off some markers
here-- one, two, three, four,

00:03:46.430 --> 00:03:50.190
five and one, two,
three, four, five.

00:03:50.190 --> 00:03:52.390
So this first equation
right over here,

00:03:52.390 --> 00:03:54.610
if we subtract 2a
from both sides,

00:03:54.610 --> 00:03:57.220
I'm just going to put it
into slope intercept form,

00:03:57.220 --> 00:04:03.940
you get b is equal to
negative 2a plus 5.

00:04:03.940 --> 00:04:06.430
All I did is subtract
2a from both sides.

00:04:06.430 --> 00:04:08.700
And if we were to graph
that, our b-intercept when

00:04:08.700 --> 00:04:10.890
a is equal to 0,
b is equal to 5.

00:04:10.890 --> 00:04:12.390
So that's right over here.

00:04:12.390 --> 00:04:13.770
And our slope is negative 2.

00:04:13.770 --> 00:04:18.079
Every time you add 1 to a--
so if a goes from 0 to 1-- b

00:04:18.079 --> 00:04:19.529
is going to go down by 2.

00:04:19.529 --> 00:04:23.510
So go down by two, go down by 2.

00:04:23.510 --> 00:04:26.970
So this first white
equation looks like this

00:04:26.970 --> 00:04:29.270
if we graph the solution set.

00:04:29.270 --> 00:04:34.900
These are all of the prices
for bananas and apples

00:04:34.900 --> 00:04:37.060
that meet this constraint.

00:04:37.060 --> 00:04:39.520
Now let's graph this
second equation.

00:04:39.520 --> 00:04:44.360
If we subtract 6a
from both sides,

00:04:44.360 --> 00:04:51.220
we get 3b is equal to
negative 6a plus 15.

00:04:51.220 --> 00:04:54.950
And now we could divide
both sides by 3, divide

00:04:54.950 --> 00:04:56.760
everything by 3.

00:04:56.760 --> 00:05:03.280
We are left with b is equal
to negative 2a plus 5.

00:05:03.280 --> 00:05:04.750
Well this is interesting.

00:05:04.750 --> 00:05:07.680
This looks very similar, or
it looks exactly the same.

00:05:07.680 --> 00:05:11.360
Our b-intercept is 5 and
our slope is negative 2a.

00:05:11.360 --> 00:05:16.390
So this is essentially
the same line.

00:05:16.390 --> 00:05:19.112
So these are essentially
the same constraints.

00:05:19.112 --> 00:05:21.320
And so you start to look at
it a little bit confused,

00:05:21.320 --> 00:05:25.510
and you say, OK, I see
why we got 0 equals 0.

00:05:25.510 --> 00:05:28.000
There's actually an infinite
number of solutions.

00:05:28.000 --> 00:05:30.750
You pick any x and then
the corresponding y

00:05:30.750 --> 00:05:32.920
for each of these
could be a solution

00:05:32.920 --> 00:05:34.880
for either of these things.

00:05:34.880 --> 00:05:37.040
So there's an infinite
number of solutions.

00:05:37.040 --> 00:05:39.270
But you start to wonder,
why is this happening?

00:05:39.270 --> 00:05:41.400
And so the bird whispers
again into the King's ear

00:05:41.400 --> 00:05:42.941
and the King says,
well the bird says

00:05:42.941 --> 00:05:46.270
this is because in both
trips to the market

00:05:46.270 --> 00:05:49.110
the same ratio of apples
and bananas was bought.

00:05:49.110 --> 00:05:52.520
In the green trip
versus the white trip,

00:05:52.520 --> 00:05:56.610
you bought three times as many
apples, bought three times

00:05:56.610 --> 00:06:00.460
as many bananas, and you
had three times the cost.

00:06:00.460 --> 00:06:04.560
So in any situation for any
per pound prices of apples

00:06:04.560 --> 00:06:07.750
and bananas, if you
buy exactly three times

00:06:07.750 --> 00:06:10.150
the number of apples, three
times the number bananas,

00:06:10.150 --> 00:06:12.750
and have three
times the cost, that

00:06:12.750 --> 00:06:14.810
could be true for any prices.

00:06:14.810 --> 00:06:18.220
And so this is actually
it's consistent.

00:06:18.220 --> 00:06:22.750
We can't say that
Arbegla is lying to us,

00:06:22.750 --> 00:06:25.440
but it's not giving
us enough information.

00:06:25.440 --> 00:06:27.680
This is what we call, this
is a consistent system.

00:06:27.680 --> 00:06:29.430
It's consistent
information here.

00:06:29.430 --> 00:06:31.270
So let me write this down.

00:06:31.270 --> 00:06:34.370
This is consistent.

00:06:34.370 --> 00:06:36.420
And it is consistent,
0 equals 0.

00:06:36.420 --> 00:06:39.300
There's no shadiness
going on here.

00:06:39.300 --> 00:06:41.060
But it's not enough information.

00:06:41.060 --> 00:06:44.260
This system of
equations is dependent.

00:06:44.260 --> 00:06:45.850
It is dependent.

00:06:45.850 --> 00:06:52.190
And you have an infinite
number of solutions.

00:06:52.190 --> 00:06:55.970
Any point this line
represents a solution.

00:06:55.970 --> 00:06:57.970
So you tell Arbegla,
well, if you really

00:06:57.970 --> 00:07:00.050
want us to figure
this out, you need

00:07:00.050 --> 00:07:01.290
to give us more information.

00:07:01.290 --> 00:07:06.328
And preferably buy a different
ratio of apples to bananas.


Inconsistent systems of equations

WEBVTT
Kind: captions
Language: en

00:00:02.504 --> 00:00:10.138
The king's advisor, Arbegla, is watching all of this discourse between you, the king, the bird. And he's starting to feel a little bit jealous

00:00:10.138 --> 00:00:14.354
'cause he's supposed to be the wise man in the kingdom, the king's closest advisor.

00:00:14.354 --> 00:00:17.221
So he steps in and says, "okay, so if you and this bird"

00:00:17.221 --> 00:00:22.904
"are so smart, how about you tackle the Riddle of the Fruit Prices?"

00:00:22.904 --> 00:00:27.838
And the king says, "Yes, that is something that we haven't been able to figure out."

00:00:27.838 --> 00:00:32.521
"The fruit prices. Arbegla, tell them the riddle of the fruit prices."

00:00:32.521 --> 00:00:34.739
And so Arbegla says, "Well,"

00:00:34.739 --> 00:00:40.221
"we wanna keep track of how much our fruit costs, but we forgot"

00:00:40.221 --> 00:00:45.622
"to actually log how much it costs when we went to the market but we know how much in total we spent"

00:00:45.622 --> 00:00:50.238
"we know much we got. We know that one week ago, when we went to"

00:00:50.238 --> 00:00:55.987
"the fruit market, we bought two, two, pounds of"

00:00:55.987 --> 00:01:01.387
"apples, we bought two pounds of apples, and one pound of bananas."

00:01:01.387 --> 00:01:09.087
"one pound, I guess, of bananans, bananas. And the total cost"

00:01:09.087 --> 00:01:15.254
"that, time, was three dollars, so there was three dollars, three dollars"

00:01:15.254 --> 00:01:21.636
in total cost. And then when we went the time before that we went the time before that

00:01:21.636 --> 00:01:26.321
we bought six pounds of bananas or six pounds of apples I should say

00:01:26.321 --> 00:01:32.487
Six pounds of apples. And three pounds three pounds of

00:01:32.487 --> 00:01:36.954
bananas. Ba-nanas. And

00:01:36.954 --> 00:01:42.705
the total cost at that point was fifteen dollars. So

00:01:42.705 --> 00:01:46.271
what is the cost of apples and bananas?

00:01:46.271 --> 00:01:48.353
So you look at the bird:

00:01:48.353 --> 00:01:51.654
The bird looks at you, the bird whispers into the king's ear, and the king says

00:01:51.654 --> 00:01:56.671
Well the bird says we'll just start defining some variables here, so we'll start expressing this thing algebraically

00:01:56.671 --> 00:02:01.955
So you go about doing that. What we want to figure out is the cost of apples and the cost of bananas.

00:02:01.955 --> 00:02:06.688
Per pound. So we set some variables. So let's...

00:02:06.688 --> 00:02:17.405
let a= the cost cost of apples, apple per pound. Per pound

00:02:17.405 --> 00:02:23.054
And let's let b = the cost of bananas.

00:02:23.054 --> 00:02:29.072
Ba-nanas. Bananas per pound. So how could we interpret

00:02:29.072 --> 00:02:34.138
this first information right here? Two pounds of apples and a pound of bananas cost

00:02:34.138 --> 00:02:43.054
$3. So how much are the apples going to cost? Well it's going to cost 2, two pounds times

00:02:43.054 --> 00:02:48.338
the cost per pound, times a, that's going to be the total cost of apples

00:02:48.338 --> 00:02:53.820
in this scenarios, and what's the total cost of the banana? Well it's one pound times the cost

00:02:53.820 --> 00:02:58.639
per pound. So, you're just going to have b, that's the total cost

00:02:58.639 --> 00:03:03.404
of the bananas, cause we know we bought one the total cost of the apples and bananas

00:03:03.404 --> 00:03:07.304
are going to be 2a+b and we know what that total cost is

00:03:07.304 --> 00:03:14.938
it is, it is $3. Now let's do the same thing for the other time we went to the market. Simply

00:03:14.938 --> 00:03:20.137
Six pounds of apple, the total cost is going to be six pounds times A dollars

00:03:20.137 --> 00:03:23.605
per pound and the total cost of banas is going to be

00:03:23.605 --> 00:03:25.570
well we bought three poiund of bananas.

00:03:25.570 --> 00:03:27.938
and the cost per pound is b

00:03:27.938 --> 00:03:31.488
and so the total cost of apples and bananas

00:03:31.488 --> 00:03:35.936
this scenario is going to be = to 15

00:03:35.936 --> 00:03:38.721
is going to be = to $15

00:03:38.721 --> 00:03:41.204
so let's think about how we might want to solve it

00:03:41.204 --> 00:03:44.053
we could use elimination we could use substitution

00:03:44.053 --> 00:03:45.753
whatever we want, we might do it graphically

00:03:45.753 --> 00:03:47.754
let's try it first with elimination.

00:03:47.754 --> 00:03:50.804
so the first thing I might want to do is

00:03:50.804 --> 00:03:54.338
is maybe I want to eliminate let's say I want to eliminate

00:03:54.338 --> 00:03:57.656
the a variable right over here so I have two

00:03:57.656 --> 00:04:00.039
a over here, I have six a over here

00:04:00.039 --> 00:04:03.838
so if I multiply this entire right equation by

00:04:03.838 --> 00:04:09.088
-3 then this 2a would become a -6a and then it might

00:04:09.088 --> 00:04:10.504
be able to cancel out with that

00:04:10.504 --> 00:04:11.421
so let me do that

00:04:11.421 --> 00:04:13.589
let me multiply this entire equation

00:04:13.589 --> 00:04:16.638
times -3

00:04:16.638 --> 00:04:19.337
times negative three

00:04:19.337 --> 00:04:23.071
so -3 * 2a is -6a

00:04:23.071 --> 00:04:27.256
-3 * b is -3b

00:04:27.256 --> 00:04:30.555
and then -3 * 3 is -9

00:04:30.555 --> 00:04:32.821
is -9

00:04:32.821 --> 00:04:35.888
and now we can essentially add the two equations

00:04:35.888 --> 00:04:38.071
or essentially add the left side of this to the left side of that

00:04:38.071 --> 00:04:40.372
or the right side of this equation to the right side of that

00:04:40.372 --> 00:04:44.188
we're essentially adding the same thing to both sides of this equation

00:04:44.188 --> 00:04:46.054
because we know this is equal to that

00:04:46.054 --> 00:04:47.288
So let's do that

00:04:47.288 --> 00:04:48.888
let's do it

00:04:48.888 --> 00:04:51.671
So on the left hand side, 6a and 6a cancel out.

00:04:51.671 --> 00:04:57.320
But something else interesting happens, the 3b and the 3b cancels out as well.

00:04:57.320 --> 00:05:01.771
So we're just left with 0 on the left hand side.

00:05:01.771 --> 00:05:04.654
And on the right hand side, what do we have?

00:05:04.654 --> 00:05:07.987
15 - 9 = 6.

00:05:07.987 --> 00:05:12.271
So we get this bizarre statement! All of our variables have gone away

00:05:12.271 --> 00:05:15.738
And we're left with this bizarre nonsensical statement that

00:05:15.738 --> 00:05:22.289
0 = 6, which we know is definitely not the case.

00:05:22.289 --> 00:05:25.472
So what's going on over here? What's going on?

00:05:25.472 --> 00:05:29.455
And then, you you you say, what's going on and you look at the bird

00:05:29.455 --> 00:05:32.188
'cause the bird seems to be the most knowledgeable person in the room

00:05:32.188 --> 00:05:34.455
or at least the most knowledgeable vertebrate

00:05:34.455 --> 00:05:37.236
in the room. And so the bird whispers into the king's ear

00:05:37.236 --> 00:05:42.471
and the king says, "Well, he says that there's no solution

00:05:42.471 --> 00:05:45.088
and you should at least try to graph it to see why."

00:05:45.088 --> 00:05:47.822
And you say, well, the bird seems to know what he's talking about

00:05:47.822 --> 00:05:51.604
So let me attempt to graph these two equations and see

00:05:51.604 --> 00:05:52.621
what's going on.

00:05:52.621 --> 00:05:57.238
And so what you do is, you take each of the equation

00:05:57.238 --> 00:05:59.321
and you like, when you graph it, you like to put

00:05:59.321 --> 00:06:02.671
it in kind of the y-intercept form or slope intercept form

00:06:02.671 --> 00:06:05.488
and so you do that, so you say, well let me

00:06:05.488 --> 00:06:07.290
solve both of these for b

00:06:07.290 --> 00:06:09.904
so if you want to solve this first equation for b

00:06:09.904 --> 00:06:12.304
you just subtract 2a from both sides

00:06:12.304 --> 00:06:14.887
if you subtract 2a from both sides of this first equation

00:06:14.887 --> 00:06:18.354
you get b is = to -2a

00:06:18.354 --> 00:06:23.204
+ 3. Now solve this second equation for b.

00:06:23.204 --> 00:06:27.770
So the first thing you might wanna do is subtract 6a from both sides.

00:06:27.770 --> 00:06:33.271
So you would get, you would get, I'll do it right over, let me do it right over here.

00:06:33.271 --> 00:06:38.955
You would get 3b, 3b is = to -6a plus 15

00:06:38.955 --> 00:06:41.638
and then you can divide both sides by 3

00:06:41.638 --> 00:06:44.122
you get b is = to -2a

00:06:44.122 --> 00:06:49.688
plus, plus 5. So the second equation, let me revert back

00:06:49.688 --> 00:06:53.471
to that other shade of green, is b is = to -2a

00:06:53.471 --> 00:06:57.504
plus 5. And we haven't even graphed it yet, but it looks like

00:06:57.504 --> 00:06:59.253
something interesting is going on.

00:06:59.253 --> 00:07:01.938
They both have the exact same slope

00:07:01.938 --> 00:07:04.921
when you solve in terms, when you solve for b

00:07:04.921 --> 00:07:08.404
but they seem to have different, let's call them, b-intercepts

00:07:08.404 --> 00:07:10.539
let's graph it to actually see what's going on

00:07:10.539 --> 00:07:16.804
so let me get, draw some axes over here, let's call that my b-axis

00:07:16.804 --> 00:07:20.372
and then this could be my, a axis

00:07:20.372 --> 00:07:24.504
And this first equation has a b-intercept of positive 3

00:07:24.504 --> 00:07:28.104
so let's see, one, two, three four five

00:07:28.104 --> 00:07:31.354
the first one has a b-intercept of positive three

00:07:31.354 --> 00:07:33.755
and it has a slope of negative 2

00:07:33.755 --> 00:07:36.455
So you go down or you go to the right one

00:07:36.455 --> 00:07:39.504
you go down two. Go to the right one, you go down two.

00:07:39.504 --> 00:07:41.054
So the line looks something like this.

00:07:41.054 --> 00:07:44.071
I'm trying my best to draw it straight. So it looks

00:07:44.071 --> 00:07:47.804
it looks something something like that

00:07:47.804 --> 00:07:49.603
And I'll just draw this green one.

00:07:49.603 --> 00:07:52.904
This green one, our b-intercept is 5

00:07:52.904 --> 00:07:55.671
so it's right over here. but we have the exact same slope

00:07:55.671 --> 00:07:57.154
the slope of -2, so it looks

00:07:57.154 --> 00:08:04.123
it looks something something like that right over there

00:08:04.123 --> 00:08:07.421
and you immediately see now that the bird was right

00:08:07.421 --> 00:08:10.805
There is no solution because these two constraints

00:08:10.805 --> 00:08:14.622
represent or can be represented by lines that

00:08:14.622 --> 00:08:20.004
don't intersect. So the lines don't, don't intersect.

00:08:20.004 --> 00:08:22.254
In-ter-sect.

00:08:22.254 --> 00:08:25.588
They don't intersect, and so the bird is right

00:08:25.588 --> 00:08:30.138
there's no solution, there's no x and y that can make this statement equal true!

00:08:30.138 --> 00:08:35.171
Or that can make 0 = 6, there is no possible, there is no overlap between these two

00:08:35.171 --> 00:08:37.855
things. And so something gets into your brain.

00:08:37.855 --> 00:08:40.938
You realize that Arbegla is trying to stump you.

00:08:40.938 --> 00:08:43.421
And you say, Arbegla, you have given me

00:08:43.421 --> 00:08:45.871
in-con-sistent information!

00:08:45.871 --> 00:08:48.904
This is an in-con-sistent system of equations!

00:08:48.904 --> 00:08:53.138
In. In...con...sistent. Which happens to be the word

00:08:53.138 --> 00:08:55.220
that is sometimes used to refer to a system

00:08:55.220 --> 00:08:58.304
that has no solutions, where the lines do not

00:08:58.304 --> 00:09:01.720
intersect. And there fore this information is incorrect

00:09:01.720 --> 00:09:04.939
We cannot assume that the apple or banana

00:09:04.939 --> 00:09:08.687
Either you are lying, which is possible, or you accounted for it wrong

00:09:08.687 --> 00:09:12.304
Or maybe the prices of apples and bananas actually changed

00:09:12.304 --> 00:09:14.789
between the two visits of the market.

00:09:14.789 --> 00:09:18.539
At which point the bird whispered into the King's ear,

00:09:18.539 --> 99:59:59.999
and says, oh, this character isn't so bad at this algebra stuff.


Solving systems of linear equations with substitution example

WEBVTT
Kind: captions
Language: en

00:00:00.053 --> 00:00:11.204
So that it's less likely that we get shown up by talking birds in the future, we've set a little bit of exercise for solving systems of equations with substitution.

00:00:11.265 --> 00:00:14.864
And so this is the first exercise or the first problem that they give us.

00:00:14.887 --> 00:00:19.294
-3x-4y=-2 and y=2x-5

00:00:19.325 --> 00:00:23.924
So let me get out my little scratch pad and let me rewrite the problem.

00:00:24.285 --> 00:00:32.177
So this is -3x-4y=-2

00:00:32.239 --> 00:00:38.429
and then they tell us y=2x-5.

00:00:38.429 --> 00:00:42.942
So, what's neat about this is that they've already solved the second equation.

00:00:42.972 --> 00:00:46.264
They've already made it explicitly solved for y

00:00:46.264 --> 00:00:50.969
which makes it very easy to substitute for. We can take this constraint, the constraint on y

00:00:50.976 --> 00:00:53.854
in terms of x and substitute it for y

00:00:53.916 --> 00:00:58.342
in this first blue equation and then solve for x.

00:00:58.342 --> 00:00:59.824
So let's try it out.

00:00:59.824 --> 00:01:01.985
So this first blue equation would then become

00:01:01.985 --> 00:01:07.353
-3x-4 but instead of putting a y there

00:01:07.353 --> 00:01:11.742
the second constraint tells us that y needs to be equal to 2x-5.

00:01:11.742 --> 00:01:22.963
So it's 4(2x-5) and all of that is going to be equal to -2.

00:01:22.963 --> 00:01:26.198
So now we get just one equation with one unknown.

00:01:26.198 --> 00:01:28.313
and now we just have to solve for x.

00:01:28.313 --> 00:01:30.259
So, let's see if we can do that.

00:01:30.259 --> 00:01:34.891
So, it's -3x and then this part right over here

00:01:34.891 --> 00:01:39.266
we have a -4, be careful, we have a -4 we want to distribute.

00:01:39.266 --> 00:01:45.530
We are going to multiply -4*2x which is -8x

00:01:45.530 --> 00:01:52.767
and -4*-5 is positive 20 and thats going to equal -2.

00:01:52.767 --> 00:01:55.802
And now we can combine all the x terms

00:01:55.849 --> 00:02:02.992
so -3x-8x, that's going to be -11x

00:02:02.992 --> 00:02:09.965
and then we have -11x+20=-2.

00:02:09.965 --> 00:02:13.704
Now to solve for x, we'll subtract 20 from both sides

00:02:13.704 --> 00:02:18.945
to get rid of the 20 on the left hand side.

00:02:18.963 --> 00:02:21.244
On the left hand side, we're just left with

00:02:21.252 --> 00:02:30.938
the -11x and then on the right hand side we are left with -22.

00:02:31.046 --> 00:02:37.410
Now we can divide both sides by -11.

00:02:37.456 --> 00:02:47.646
And we are left with x is equal to 22 divided by 11 is 2, and the negatives cancel out. x = 2.

00:02:47.646 --> 00:02:51.024
So we are not quite done yet. We've done, I guess you can say

00:02:51.039 --> 00:02:54.856
the hard part, we have solved for x but now we have to solve for y.

00:02:54.858 --> 00:02:59.279
We could take this x value to either one of these equations and solve for y.

00:02:59.279 --> 00:03:03.319
But this second one has already explicitly solved for y

00:03:03.319 --> 00:03:07.337
so let's use that one, so it says y = 2 times

00:03:07.337 --> 00:03:10.895
and instead of x, we now know that the x value

00:03:10.895 --> 00:03:14.564
where these two intersect, you could view it that way

00:03:14.564 --> 00:03:19.825
is going to be equal to 2, so 2 * 2 - 5

00:03:19.856 --> 00:03:25.062
let's figure out the corresponding y value.

00:03:25.984 --> 00:03:37.898
So you get y=2(2)-5 and y = 4 - 5 so y = -1.

00:03:37.898 --> 00:03:40.767
And you can verify that it'll work in this top equation

00:03:40.767 --> 00:03:50.435
If y = -1 and x=2, this top equation becomes -3(2) which is -6-4(-1)

00:03:50.435 --> 00:03:56.292
which would be plus 4. And -6+4 is indeed -2.

00:03:56.292 --> 00:03:59.100
So it satisfies both of these equations and now

00:03:59.100 --> 00:04:02.029
we can type it in to verify that we got it right,

00:04:02.029 --> 00:04:06.167
although, we know that we did, so x=2 and y=-1.

00:04:06.167 --> 00:04:14.071
So, let's type it in... x=2 and y=-1.

00:04:14.071 --> 00:04:18.071
Excellent, now we're much less likely to be embarassed by talking birds.


Talking bird solves systems with substitution

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:02.960
Just as you were
solving the potato chip

00:00:02.960 --> 00:00:06.700
conundrum in the last video,
the king's favorite magical bird

00:00:06.700 --> 00:00:11.072
comes flying along and starts
whispering into the king's ear.

00:00:11.072 --> 00:00:13.030
And this makes you a
little bit self-conscious,

00:00:13.030 --> 00:00:14.900
a little bit
insecure, so you tell

00:00:14.900 --> 00:00:17.580
the king, what is the
bird talking about.

00:00:17.580 --> 00:00:19.080
And the king says,
well, the bird

00:00:19.080 --> 00:00:22.550
says that he thinks that there's
another way to do the problem.

00:00:22.550 --> 00:00:24.970
And you're not used to
taking advice from birds.

00:00:24.970 --> 00:00:26.480
And so being a
little bit defensive,

00:00:26.480 --> 00:00:29.160
you say, well, if the bird
thinks he knows so much,

00:00:29.160 --> 00:00:30.500
let him do this problem.

00:00:30.500 --> 00:00:32.250
And so the bird whispers
a little bit more

00:00:32.250 --> 00:00:33.850
in the king's ear and
says, OK, well I'll

00:00:33.850 --> 00:00:35.849
have to do the writing
because the bird does not

00:00:35.849 --> 00:00:39.060
have any hands, or at least
can't manipulate chalk.

00:00:39.060 --> 00:00:41.970
And so the bird continues to
whisper in the king's ear.

00:00:41.970 --> 00:00:43.800
And the king
translates and says,

00:00:43.800 --> 00:00:46.430
well, the bird says, let's
use one of these equations

00:00:46.430 --> 00:00:47.990
to solve for a variable.

00:00:47.990 --> 00:00:50.590
So let's say, let's us this
blue equation right over here

00:00:50.590 --> 00:00:52.089
to solve for a variable.

00:00:52.089 --> 00:00:53.505
And that's essentially
going to be

00:00:53.505 --> 00:00:55.730
a constraint of one variable
in terms of another.

00:00:55.730 --> 00:00:57.250
So let's see if we can do that.

00:00:57.250 --> 00:00:59.410
So here, if we want
to solve for m,

00:00:59.410 --> 00:01:02.890
we could subtract 400
w from both sides.

00:01:02.890 --> 00:01:06.910
And we would have 100 m.

00:01:06.910 --> 00:01:09.970
If we subtract 400w from the
left, this 400w goes away.

00:01:09.970 --> 00:01:12.120
If we subtract 400w
from the right,

00:01:12.120 --> 00:01:19.600
we have is equal to
negative 400w plus 1,100.

00:01:19.600 --> 00:01:21.810
So what got us from
here to here is just

00:01:21.810 --> 00:01:25.080
subtracting 400w
from both sides.

00:01:25.080 --> 00:01:26.750
And then if we want
to solve for m,

00:01:26.750 --> 00:01:30.000
we just divide
both sides by 100.

00:01:30.000 --> 00:01:33.740
So we just divide all
of the terms by 100.

00:01:33.740 --> 00:01:42.650
And then we get m is equal to
negative 400 divided by 100,

00:01:42.650 --> 00:01:45.070
is negative 4w.

00:01:45.070 --> 00:01:47.280
1,100 divided by 100 is 11.

00:01:47.280 --> 00:01:49.030
Plus 11.

00:01:49.030 --> 00:01:51.740
So now we've constrained
m in terms of w.

00:01:51.740 --> 00:01:53.680
This is what the
bird is saying, using

00:01:53.680 --> 00:01:55.450
the king as his translator.

00:01:55.450 --> 00:01:56.850
Why don't we take
this constraint

00:01:56.850 --> 00:02:00.490
and substitute it back for
m in the first equation?

00:02:00.490 --> 00:02:04.540
And then we will have one
equation with one unknown.

00:02:04.540 --> 00:02:07.410
And so the king starts to
write at the bird's direction.

00:02:07.410 --> 00:02:12.580
200, so he's looking at
that first equation now,

00:02:12.580 --> 00:02:15.480
he says 200.

00:02:15.480 --> 00:02:18.010
Instead of putting
an m there, the bird

00:02:18.010 --> 00:02:20.100
says well, by the
second constraint,

00:02:20.100 --> 00:02:23.680
m is equal to
negative 4w plus 11.

00:02:23.680 --> 00:02:29.320
So instead of writing an
m, we substitute for m

00:02:29.320 --> 00:02:31.480
the expression
negative 4w plus 11.

00:02:35.490 --> 00:02:43.700
And then we have the rest of it,
plus 300w, is equal to 1,200.

00:02:43.700 --> 00:02:45.970
So just to be clear,
everywhere we saw an m,

00:02:45.970 --> 00:02:48.770
we replaced it with
this right over here,

00:02:48.770 --> 00:02:50.065
in that first equation.

00:02:50.065 --> 00:02:52.190
So the first thing, you
start to scratch your head.

00:02:52.190 --> 00:02:54.240
And you say, is this a
legitimate thing to do.

00:02:54.240 --> 00:02:56.360
Will I get the same
answer as I got

00:02:56.360 --> 00:02:59.300
when I solved the same
problem with elimination?

00:02:59.300 --> 00:03:02.080
And I want you to sit and
think about that for a second.

00:03:02.080 --> 00:03:04.330
But then the bird starts
whispering in the king's ear.

00:03:04.330 --> 00:03:06.970
And the king just progresses to
just work through the algebra.

00:03:06.970 --> 00:03:10.130
This is one equation
with one unknown now.

00:03:10.130 --> 00:03:14.280
So the first step would
be to distribute the 200.

00:03:14.280 --> 00:03:20.360
So 200 times negative
4w is negative 800w.

00:03:20.360 --> 00:03:22.510
200 times 11 is 2,200.

00:03:22.510 --> 00:03:24.770
Plus 2,200.

00:03:24.770 --> 00:03:27.250
And then we have the plus 300w.

00:03:27.250 --> 00:03:32.050
Plus 300w is equal
to positive 1,200.

00:03:32.050 --> 00:03:33.830
Now we just need to solve for w.

00:03:33.830 --> 00:03:36.950
We first might want to
group this negative 800w

00:03:36.950 --> 00:03:39.600
with this 300w.

00:03:39.600 --> 00:03:42.620
Negative 800 of something
plus 300 of something

00:03:42.620 --> 00:03:46.520
is going to be negative 500w.

00:03:46.520 --> 00:03:52.930
And then we still have this
plus 2,200 is equal to 1,200.

00:03:52.930 --> 00:03:57.470
Now to solve for w, we'd want to
subtract 2,200 from both sides.

00:03:57.470 --> 00:04:03.120
So subtract 2,200,
subtract 2,200.

00:04:03.120 --> 00:04:05.780
On the left-hand
side, you're left just

00:04:05.780 --> 00:04:07.405
with the negative 500w.

00:04:11.370 --> 00:04:16.290
And on the right-hand side, you
are left with negative 1,000.

00:04:16.290 --> 00:04:18.269
This is starting to
look interesting,

00:04:18.269 --> 00:04:23.650
because if we divide both
sides by negative 500,

00:04:23.650 --> 00:04:29.650
we get w is equal to 2, which
is the exact same answer that we

00:04:29.650 --> 00:04:33.267
got when we tried to figure
out how many bags of chips

00:04:33.267 --> 00:04:34.600
each woman would eat on average.

00:04:34.600 --> 00:04:36.980
When we tried to solve
it with elimination,

00:04:36.980 --> 00:04:39.710
we got the exact right answer.

00:04:39.710 --> 00:04:44.390
So at least for this example,
it seems like the substitution

00:04:44.390 --> 00:04:48.220
method that this bird came
up with worked just as well

00:04:48.220 --> 00:04:51.041
as the elimination method
that you had originally

00:04:51.041 --> 00:04:53.540
done the first time you wanted
to figure out the potato chip

00:04:53.540 --> 00:04:54.470
conundrum.

00:04:54.470 --> 00:04:57.032
And if now, you actually
wanted to figure out

00:04:57.032 --> 00:04:58.730
how many chips
the men would eat,

00:04:58.730 --> 00:05:00.080
well, you could do
exactly the same thing

00:05:00.080 --> 00:05:00.730
you did the last time.

00:05:00.730 --> 00:05:01.980
You know one of the variables.

00:05:01.980 --> 00:05:04.410
You can substitute it back
into one of the equations

00:05:04.410 --> 00:05:06.040
and then solve for m.

00:05:06.040 --> 00:05:08.510
And you could try that yourself
to verify that you actually

00:05:08.510 --> 00:05:11.750
will get the same
value for m as well.

00:05:11.750 --> 00:05:14.690
And in fact, this would
probably be the easiest equation

00:05:14.690 --> 00:05:16.890
to substitute into,
because it explicitly

00:05:16.890 --> 00:05:20.329
solves for m already.


Systems with elimination practice

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:03.840
We never know when we might have
to do a little bit more party

00:00:03.840 --> 00:00:04.630
planning.

00:00:04.630 --> 00:00:07.370
So it doesn't hurt to
have some practice.

00:00:07.370 --> 00:00:09.580
And that's what this
exercise is doing for us,

00:00:09.580 --> 00:00:11.480
is generating
problems so that we

00:00:11.480 --> 00:00:15.279
can try solving systems of
equations with elimination.

00:00:15.279 --> 00:00:16.820
And so in this first
problem, it says

00:00:16.820 --> 00:00:19.016
solve for x and y
using elimination.

00:00:19.016 --> 00:00:20.390
And then this is
what they have--

00:00:20.390 --> 00:00:23.170
6x minus 6y is equal
to negative 24.

00:00:23.170 --> 00:00:25.900
Negative 5x minus 5y is
equal to negative 60.

00:00:25.900 --> 00:00:29.300
So let me get my scratch
pad out to solve this.

00:00:29.300 --> 00:00:30.710
Let me rewrite it.

00:00:30.710 --> 00:00:39.580
So they gave us 6x minus
6y is equal to negative 24.

00:00:39.580 --> 00:00:47.097
And negative 5x minus 5y
is equal to negative 60.

00:00:47.097 --> 00:00:48.680
So what we have to
think about, and we

00:00:48.680 --> 00:00:50.620
saw this in several
of the other videos,

00:00:50.620 --> 00:00:52.720
is when we want to
eliminate a variable,

00:00:52.720 --> 00:00:54.730
we want to manipulate
these two equations.

00:00:54.730 --> 00:00:56.840
And if we were to add
the corresponding sides,

00:00:56.840 --> 00:00:58.960
that variable might disappear.

00:00:58.960 --> 00:01:01.740
So if we just added a
6x to a negative 5x,

00:01:01.740 --> 00:01:03.180
that's not going
to cancel it out.

00:01:03.180 --> 00:01:05.390
If this was a negative
6x, that would work out.

00:01:05.390 --> 00:01:07.710
Or if this was a positive
5x, that would work out.

00:01:07.710 --> 00:01:09.490
But this isn't exactly right.

00:01:09.490 --> 00:01:11.550
So if I want to
eliminate the x, I

00:01:11.550 --> 00:01:14.160
have to manipulate these
equations so that these two

00:01:14.160 --> 00:01:15.949
characters might cancel out.

00:01:15.949 --> 00:01:17.490
And one thing that
pops into my brain

00:01:17.490 --> 00:01:18.906
is it looks like
all of this stuff

00:01:18.906 --> 00:01:21.100
up here is divisible by
6, and all of this stuff

00:01:21.100 --> 00:01:22.812
down here is divisible by 5.

00:01:22.812 --> 00:01:25.020
And if we were to divide
all this stuff up here by 6,

00:01:25.020 --> 00:01:26.930
we'd be left with
an x over here.

00:01:26.930 --> 00:01:29.530
And if we were to divide
all this bottom stuff by 5,

00:01:29.530 --> 00:01:31.700
we'd be left with a
negative x right over here.

00:01:31.700 --> 00:01:33.790
And then they just
might cancel out.

00:01:33.790 --> 00:01:35.390
So let's try that out.

00:01:35.390 --> 00:01:36.970
Let's take this first equation.

00:01:36.970 --> 00:01:41.040
And we're going to
multiply both sides by 1/6.

00:01:41.040 --> 00:01:42.790
Or another way you
could think about it is

00:01:42.790 --> 00:01:45.950
we're dividing both sides by 6.

00:01:45.950 --> 00:01:48.800
And as long as we do the
same thing to both sides,

00:01:48.800 --> 00:01:50.105
the equation holds.

00:01:50.105 --> 00:01:51.970
The equality holds.

00:01:51.970 --> 00:01:55.370
So if you multiply everything
by 1/6, 6x times 1/6

00:01:55.370 --> 00:01:57.270
is just going to be x.

00:01:57.270 --> 00:01:59.440
6y times 1/6 is just y.

00:01:59.440 --> 00:02:03.160
So it's negative y.

00:02:03.160 --> 00:02:07.150
Negative 24 times
1/6 is negative 4.

00:02:07.150 --> 00:02:09.720
Or you could just view it
as negative 24 divided by 6

00:02:09.720 --> 00:02:10.840
is negative 4.

00:02:10.840 --> 00:02:14.110
So this equation, the blue one,
we've simplified as x minus y

00:02:14.110 --> 00:02:15.670
is equal to negative 4.

00:02:15.670 --> 00:02:18.150
Let's do something similar
with the second one.

00:02:18.150 --> 00:02:21.750
Here we could say we're going to
multiply everything times 1/5.

00:02:21.750 --> 00:02:24.750
Or you could say that we're
dividing everything by 5.

00:02:24.750 --> 00:02:27.310
If we do that, negative
5x divided by 5

00:02:27.310 --> 00:02:29.910
is just negative x.

00:02:29.910 --> 00:02:33.980
Negative 5y divided
by 5 is negative y.

00:02:33.980 --> 00:02:40.120
And then negative 60
divided by 5 is negative 12.

00:02:40.120 --> 00:02:43.830
And now, this looks
pretty interesting.

00:02:43.830 --> 00:02:46.525
If we add the two left-hand
sides-- and remember,

00:02:46.525 --> 00:02:48.650
we can keep the equality,
because we're essentially

00:02:48.650 --> 00:02:50.160
adding the same
thing to both sides.

00:02:50.160 --> 00:02:52.540
You can imagine we're starting
with the blue equation.

00:02:52.540 --> 00:02:55.840
And on the left-hand side,
we're adding negative x minus y.

00:02:55.840 --> 00:02:58.300
And on the right-hand side,
we're adding negative 12.

00:02:58.300 --> 00:03:01.190
But the second equation tells
us that those two things

00:03:01.190 --> 00:03:01.916
are equal.

00:03:01.916 --> 00:03:03.290
So we're doing
the same principle

00:03:03.290 --> 00:03:06.090
that we saw when we first
started looking to algebra,

00:03:06.090 --> 00:03:08.280
that you can maintain your
equality as long as you

00:03:08.280 --> 00:03:09.272
add the same thing.

00:03:09.272 --> 00:03:11.230
On the left-hand side,
we're going to add this.

00:03:11.230 --> 00:03:13.396
And on the right-hand side,
we're going to add this.

00:03:13.396 --> 00:03:14.980
But this second
equation tells us

00:03:14.980 --> 00:03:16.380
that those two things are equal.

00:03:16.380 --> 00:03:18.540
So we can maintain our equality.

00:03:18.540 --> 00:03:19.310
So let's do that.

00:03:19.310 --> 00:03:21.045
What do we get on
the left-hand side?

00:03:21.045 --> 00:03:22.920
Well, you have a positive
x and a negative x.

00:03:22.920 --> 00:03:24.050
They cancel out.

00:03:24.050 --> 00:03:27.030
That was the whole point behind
manipulating them in this way.

00:03:27.030 --> 00:03:31.910
And then you have negative y
minus y, which is negative 2y.

00:03:31.910 --> 00:03:34.130
And then on the right-hand
side, you have negative 4

00:03:34.130 --> 00:03:36.894
minus 12, which is negative 16.

00:03:36.894 --> 00:03:38.810
And these are going to
be equal to each other.

00:03:38.810 --> 00:03:42.130
Once again, we're adding the
same thing to both sides.

00:03:42.130 --> 00:03:47.330
To solve for y, we can divide
both sides by negative 2.

00:03:47.330 --> 00:03:51.990
And we are left with y
is equal to positive 8.

00:03:51.990 --> 00:03:53.480
But we are not done yet.

00:03:53.480 --> 00:03:56.654
We want to go and substitute
back into one of the equations.

00:03:56.654 --> 00:03:58.320
And we can substitute
back into this one

00:03:58.320 --> 00:04:00.650
and to this one, or
this one and this one.

00:04:00.650 --> 00:04:03.060
The solutions need to satisfy
all of these essentially.

00:04:03.060 --> 00:04:06.474
This blue one is another way of
expressing this blue equation.

00:04:06.474 --> 00:04:07.890
This green equation
is another way

00:04:07.890 --> 00:04:09.700
of expressing this
green equation.

00:04:09.700 --> 00:04:12.340
So I'll go for whichever one
seems to be the simplest.

00:04:12.340 --> 00:04:15.250
And this one seems to be
pretty simple right over here.

00:04:15.250 --> 00:04:21.180
So let's take x
minus y-- we just

00:04:21.180 --> 00:04:26.075
solved that y would be positive
8-- is equal to negative 4.

00:04:30.110 --> 00:04:35.810
And now to solve for x, we just
have to add 8 to both sides.

00:04:35.810 --> 00:04:37.910
And we are left with,
on the left-hand side,

00:04:37.910 --> 00:04:39.540
negative 8 plus 8 cancels out.

00:04:39.540 --> 00:04:41.000
You're just left with an x.

00:04:41.000 --> 00:04:44.950
And negative 4 plus 8
is equal to positive 4.

00:04:44.950 --> 00:04:49.939
So you get x is equal
to 4, y is equal to 8.

00:04:49.939 --> 00:04:51.480
And you can verify
that it would work

00:04:51.480 --> 00:04:53.430
with either one of
these equations.

00:04:53.430 --> 00:04:58.670
6 times 4 is 24 minus 6 times
8-- so it's 24 minus 48--

00:04:58.670 --> 00:05:00.810
is, indeed, negative 24.

00:05:00.810 --> 00:05:06.160
Negative 5 times 4 is negative
20, minus negative 40,

00:05:06.160 --> 00:05:09.100
if y is equal to 8, does,
indeed, get you negative 60.

00:05:09.100 --> 00:05:10.900
So it works out
for both of these.

00:05:10.900 --> 00:05:13.370
And we can try it out by
inputting our answers.

00:05:13.370 --> 00:05:15.130
So x is 4, y is 8.

00:05:15.130 --> 00:05:17.000
So let's do that.

00:05:17.000 --> 00:05:19.300
So let me type this in.

00:05:19.300 --> 00:05:22.180
x is going to be equal to 4.

00:05:22.180 --> 00:05:24.460
y is going to be equal to 8.

00:05:24.460 --> 00:05:26.730
And let's check our answer.

00:05:26.730 --> 00:05:28.680
It is correct.

00:05:28.680 --> 00:05:30.390
Very good.