WEBVTT

00:00:01.050 --> 00:00:04.040
Welcome to the presentation on
systems of linear equations.

00:00:04.040 --> 00:00:06.970
So let's get started and
see what it's all about.

00:00:06.970 --> 00:00:10.110
So let's say I had
two equations now.

00:00:10.110 --> 00:00:15.740
The first equation let
me write it as 9x minus

00:00:15.740 --> 00:00:21.760
4y equals minus 78.

00:00:21.760 --> 00:00:28.950
And the second equation
I will write as 4x plus

00:00:28.950 --> 00:00:33.390
y is equal to mine 18.

00:00:33.390 --> 00:00:35.411
Now what we're going to do now
is we're actually going to

00:00:35.411 --> 00:00:39.700
use both equations to
solve for x and y.

00:00:39.700 --> 00:00:41.900
We already know that if you
have one equation, it has one

00:00:41.900 --> 00:00:44.280
variable, it is very easy to
solve for that one variable.

00:00:44.280 --> 00:00:45.790
But now we have to equations.

00:00:45.790 --> 00:00:47.340
You can almost view them
as two constraints.

00:00:47.340 --> 00:00:50.340
And we're going to solve
for both variables.

00:00:50.340 --> 00:00:51.790
And you might be a
little confused.

00:00:51.790 --> 00:00:52.520
How does that work?

00:00:52.520 --> 00:00:54.910
Is it just magic that two
equations can solves

00:00:54.910 --> 00:00:55.900
for two variables?

00:00:55.900 --> 00:00:56.800
Well it's not.

00:00:56.800 --> 00:00:58.850
Because you can actually
rearranged each of these

00:00:58.850 --> 00:01:01.840
equations so that they
look kind of in normal y

00:01:01.840 --> 00:01:03.700
equals mx plus b format.

00:01:03.700 --> 00:01:06.200
And I'm not going to draw these
actual two equations because I

00:01:06.200 --> 00:01:08.860
don't know what they look like,
but if this was a coordinate

00:01:08.860 --> 00:01:11.620
axis-- and I don't know what
that first line actually does

00:01:11.620 --> 00:01:14.010
look like, we could do another
model where we figured it out

00:01:14.010 --> 00:01:16.500
--but lets just say for sake of
argument, that first line all

00:01:16.500 --> 00:01:20.540
the x's and y's that satisfy 9x
minus 4y equals negative

00:01:20.540 --> 00:01:22.690
78, let's say it looks
something like that.

00:01:22.690 --> 00:01:26.400
And let's say all of the x's
and y's that satisfy that

00:01:26.400 --> 00:01:31.340
second equation, 4x plus y
equals negative 18, let's say

00:01:31.340 --> 00:01:34.680
that looks something like this.

00:01:34.680 --> 00:01:35.620
Right?

00:01:35.620 --> 00:01:40.050
So, on the line is all of the
x's and y's that satisfy this

00:01:40.050 --> 00:01:42.555
equation, and on the green
line are all the x's and y's

00:01:42.555 --> 00:01:44.275
that satisfy this equation.

00:01:44.275 --> 00:01:48.170
But there's only one pair of
x and y's that satisfy both

00:01:48.170 --> 00:01:51.430
equations, and you can guess
where that is, that's

00:01:51.430 --> 00:01:52.560
right here right.

00:01:52.560 --> 00:01:57.660
Whatever that point is-- I'll
do it in pink for emphasis.

00:01:57.660 --> 00:02:00.800
Whatever this point is,
notice it's on both lines.

00:02:00.800 --> 00:02:05.260
So whatever x and y that is
would be the solution to

00:02:05.260 --> 00:02:06.670
this system of equations.

00:02:06.670 --> 00:02:09.860
So let's actually figure
out how to do that.

00:02:09.860 --> 00:02:12.080
So what we want to do is
eliminate a variable, because

00:02:12.080 --> 00:02:15.200
if you can eliminate a variable
then we can just solve for

00:02:15.200 --> 00:02:16.430
the one that's left over.

00:02:16.430 --> 00:02:19.930
And the way to do that-- let's
see, I want to eliminate, I

00:02:19.930 --> 00:02:22.210
feel like eliminating this y,
and I think you'll get

00:02:22.210 --> 00:02:24.630
an intuition for how we
can do that later on.

00:02:24.630 --> 00:02:26.620
And the way I'm going to do
that is I'm going to make

00:02:26.620 --> 00:02:29.250
it so that when I had this
to this, they cancel out.

00:02:29.250 --> 00:02:31.340
Well, they don't cancel out
right now, so I have to

00:02:31.340 --> 00:02:34.380
multiply this bottom equation
by 4, and I think it'll be

00:02:34.380 --> 00:02:35.520
obvious why I'm doing it.

00:02:35.520 --> 00:02:37.810
So let's multiply this
bottom equation by 4.

00:02:37.810 --> 00:02:50.820
And I get 16x plus 4y is equal
to 40 plus 32 minus 72.

00:02:50.820 --> 00:02:51.130
Right?

00:02:51.130 --> 00:02:53.950
All I did is I multiplied
both sides of the

00:02:53.950 --> 00:02:55.620
equation by 4, right?

00:02:55.620 --> 00:02:57.210
And you have to multiply
every term because

00:02:57.210 --> 00:02:59.500
it's the distributive
property on both sides.

00:02:59.500 --> 00:03:01.050
Whatever you do to one side
you have to do to the other.

00:03:01.050 --> 00:03:03.300
Let me rewrite top
equation again.

00:03:03.300 --> 00:03:05.230
And I'll write in the same
color so we can keep

00:03:05.230 --> 00:03:06.340
track of things.

00:03:06.340 --> 00:03:13.360
9x minus 4y is
equal to minus 78.

00:03:13.360 --> 00:03:18.580
OK, well now, if we were to add
these two equations, when you

00:03:18.580 --> 00:03:20.430
add equations, you just add
the left side and you

00:03:20.430 --> 00:03:22.270
add the right side.

00:03:22.270 --> 00:03:25.440
Well when you add, you
have 16x plus 9x.

00:03:25.440 --> 00:03:28.590
Well that equals 25x.

00:03:28.590 --> 00:03:28.950
Right?

00:03:28.950 --> 00:03:31.450
16 plus 9.

00:03:31.450 --> 00:03:34.910
4y minus 4, that just equals 0.

00:03:34.910 --> 00:03:43.680
So that's plus 0 equals, and
then we have minus 72 minus 78.

00:03:43.680 --> 00:03:51.490
So, let's see that's minus
150, minus 150, right?

00:03:51.490 --> 00:03:53.060
Just adding them all together.

00:03:53.060 --> 00:03:58.820
So we have 25x equals 150.

00:03:58.820 --> 00:04:03.420
Well, we could just divide both
sides by 25 or multiply both

00:04:03.420 --> 00:04:05.380
sides by 1/25, it's
the same thing.

00:04:05.380 --> 00:04:08.470
And you get x equals--
that's a negative 150

00:04:08.470 --> 00:04:11.500
--x equals minus 6.

00:04:11.500 --> 00:04:14.870
There we solved
the x-coordinate.

00:04:14.870 --> 00:04:16.950
Now to solve the y-coordinate
we can just use either one of

00:04:16.950 --> 00:04:18.500
these equations up at top.

00:04:18.500 --> 00:04:20.810
So let's use this one,
it seems a little bit,

00:04:20.810 --> 00:04:23.020
marginally simpler.

00:04:23.020 --> 00:04:26.090
So we just substitute the x
back in there and we get

00:04:26.090 --> 00:04:34.716
4 time minus 6 plus y
is equal to minus 18.

00:04:34.716 --> 00:04:35.730
Go up here.

00:04:35.730 --> 00:04:42.565
4 times minus 6 we get minus 24
plus y is equal to minus 18.

00:04:42.565 --> 00:04:47.406
And then get y is
equal to 24 minus 18.

00:04:47.406 --> 00:04:50.510
So y is equal to 6.

00:04:50.510 --> 00:04:54.100
So these two lines or these two
equations, you could even say,

00:04:54.100 --> 00:05:00.300
intersect at the point x is
m inus six and y is plus 6.

00:05:00.300 --> 00:05:02.520
So they actually intersect
someplace around here instead.

00:05:02.520 --> 00:05:05.640
I drew these, the line probably
look something more like that.

00:05:05.640 --> 00:05:06.950
But that's pretty cool, no?

00:05:06.950 --> 00:05:11.830
We actually solved for two
variables using two equations.

00:05:11.830 --> 00:05:12.640
Let's see how much time I have.

00:05:12.640 --> 00:05:14.470
I think we have enough time
to do another problem.

00:05:14.470 --> 00:05:20.200

105
00:05:20,2 --> 00:05:23,02
So let's say I had the points--
and I'm going to write them in

00:05:23.020 --> 00:05:32.940
two different colors again
--minus 7x minus 4y equals 9,

00:05:32.940 --> 00:05:39.150
and then the second equation is
going to be x plus

00:05:39.150 --> 00:05:42.460
2y is equal to 3.

00:05:42.460 --> 00:05:45.140
Now if I were doing this as
fast as possible, I'd probably

00:05:45.140 --> 00:05:47.990
multiply this equation times 7
and it would automatically

00:05:47.990 --> 00:05:49.020
cancel out.

00:05:49.020 --> 00:05:49.850
But that's easy way.

00:05:49.850 --> 00:05:51.290
I'm going to show you that
sometimes you might have to

00:05:51.290 --> 00:05:54.780
multiply both equations--
actually, not in this case.

00:05:54.780 --> 00:05:56.800
Actually let's just do it
the fast way real fast.

00:05:56.800 --> 00:05:59.380
So let's multiply this
bottom equation by 7.

00:05:59.380 --> 00:06:00.830
And the whole reason why I want
to the, multiply it with 7,

00:06:00.830 --> 00:06:03.440
because I want this to
cancel out with this.

00:06:03.440 --> 00:06:10.150
If you multiply it by 7 you get
7x plus 14y is equal to 21.

00:06:10.150 --> 00:06:12.930
Let's write that first
equation down again.

00:06:12.930 --> 00:06:19.065
Minus 7x minus 4y
is equal to 9.

00:06:19.065 --> 00:06:20.330
Now we just add.

00:06:20.330 --> 00:06:24.260
This is a positive 7x, it just
always looks like a negative.

00:06:24.260 --> 00:06:25.900
OK, so that's 0.

00:06:25.900 --> 00:06:32.460
14 minus 4y plus 10y
is equal to 30.

00:06:32.460 --> 00:06:34.750
y is equal to 3.

00:06:34.750 --> 00:06:36.350
Now we just substitute back
into either equation,

00:06:36.350 --> 00:06:37.980
lets do that one.

00:06:37.980 --> 00:06:42.110
x plus 2 times y, 2 times 3.

00:06:42.110 --> 00:06:43.880
x plus 6 equals 3.

00:06:43.880 --> 00:06:45.900
We get x equals negative 3.

00:06:45.900 --> 00:06:48.470
That one was super easy.

00:06:48.470 --> 00:06:49.550
The intercept.

00:06:49.550 --> 00:06:51.210
Hope I didn't do it to fast.

00:06:51.210 --> 00:06:54.430
Well, you can pause it and
watch it again if you have.

00:06:54.430 --> 00:07:00.270
OK, so these two lines
intersect at the point

00:07:00.270 --> 00:07:03.182
negative 3 comma 3.

00:07:03.182 --> 00:07:04.250
Let's do one more.

00:07:04.250 --> 00:07:07.456

140
00:07:07,456 --> 00:07:10,71
Hope this one's harder.

00:07:10.710 --> 00:07:11.510
I think it will.

00:07:11.510 --> 00:07:20.300
OK, negative 3x minus
9y is equal to 66.

00:07:20.300 --> 00:07:27.200
We have minus 7x plus 4y
is equal to minus 71.

00:07:27.200 --> 00:07:28.370
So here it's not obvious.

00:07:28.370 --> 00:07:31.540
What we have to do is, let's
say we want to cancel

00:07:31.540 --> 00:07:33.980
out the y's first.

00:07:33.980 --> 00:07:36.500
What we do is we try to make
both of them equal to the least

00:07:36.500 --> 00:07:38.660
common multiple of 9 and 4.

00:07:38.660 --> 00:07:43.340
So, if we multiply the top
equation by 4 we get--

00:07:43.340 --> 00:07:44.520
I'll do it right here.

00:07:44.520 --> 00:07:45.870
Let's multiply it by 4.

00:07:45.870 --> 00:07:47.960
Times 4.

00:07:47.960 --> 00:07:59.200
We'll get minus 12x minus
36y is equal to 4 times

00:07:59.200 --> 00:08:05.400
240 plus 24 is 264.

00:08:05.400 --> 00:08:06.930
Right, I hope that's right.

00:08:06.930 --> 00:08:09.220
We multiply the second
equation by 9.

00:08:09.220 --> 00:08:25.420
So it's minus 63x plus 36y is
equal to, let's see, 639.

00:08:25.420 --> 00:08:26.030
Big numbers.

00:08:26.030 --> 00:08:29.350
639.

00:08:29.350 --> 00:08:31.540
OK, now we add the
two equations.

00:08:31.540 --> 00:08:43.570
Minus 12 minus 63 thats minus
75x-- these cancel out --equals

00:08:43.570 --> 00:08:50.130
264, let's see what's
639 minus 264.

00:08:50.130 --> 00:08:51.160
See I do this in real time.

00:08:51.160 --> 00:08:55.100
I don't use some kind of
solution manual or something.

00:08:55.100 --> 00:08:59.710
13 and 5, 70.

00:08:59.710 --> 00:09:02.260
I don't know if I'm
right, but we'll see.

00:09:02.260 --> 00:09:06.360
Since it's actually the
negative 639, this is minus

00:09:06.360 --> 00:09:12.440
375, and I know that seventy
five goes into 300 4

00:09:12.440 --> 00:09:16.450
times, so x is equal to 5.

00:09:16.450 --> 00:09:19.515
75 times 5 is 375.

00:09:19.515 --> 00:09:22.460
We just divided
both sides by 75.

00:09:22.460 --> 00:09:25.367
So if x is 5 we just substitute
it back into-- let's

00:09:25.367 --> 00:09:27.890
use this equation.

00:09:27.890 --> 00:09:36.380
So we get minus 3 times 5
minus 9y is equal to 66.

00:09:36.380 --> 00:09:41.920
We get minus 15
minus 9y equals 66.

00:09:41.920 --> 00:09:45.880
Minus 9y is equal to 81.

00:09:45.880 --> 00:09:49.840
And then we get y is
equal to minus 9.

00:09:49.840 --> 00:09:53.530
So the answer is
5 comma minus 9.

00:09:53.530 --> 00:09:55.530
I think you're ready to do some
systems of equations now.

00:09:55.530 --> 00:09:57.090
Have Fun.