Mixture problems 2

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.110
We're told to make a
table and solve.

00:00:03.110 --> 00:00:07.090
So they tell us that we have
50 ounces of a 25% saline

00:00:07.090 --> 00:00:09.180
solution, a mixture
of water and salt.

00:00:09.180 --> 00:00:13.230
How many ounces of a 10% saline
solution must you add

00:00:13.230 --> 00:00:17.700
to make a new solution
that is 15% saline?

00:00:17.700 --> 00:00:21.190
So let's make this table that
they're talking about.

00:00:21.190 --> 00:00:23.180
Let's write amount
of solution.

00:00:23.180 --> 00:00:31.130
Let me write total amount of
solution or maybe I should say

00:00:31.130 --> 00:00:33.510
total volume of solution.

00:00:33.510 --> 00:00:36.195
And then the next column I'll
say percent saline.

00:00:40.490 --> 00:00:42.610
And then we can use this
information to figure out

00:00:42.610 --> 00:00:43.860
total amount of saline.

00:00:50.740 --> 00:00:53.380
And let's list it for each of
the two solutions that they

00:00:53.380 --> 00:00:54.210
talk about.

00:00:54.210 --> 00:00:58.900
We're starting with 50 ounces
of a 25% saline solution.

00:00:58.900 --> 00:01:02.220
So this is what we start with.

00:01:02.220 --> 00:01:09.050
Starting solution, we have
50 ounces of it, so

00:01:09.050 --> 00:01:09.850
I'll just write 50.

00:01:09.850 --> 00:01:12.220
We'll assume everything
is in ounces.

00:01:12.220 --> 00:01:17.000
It is 25% saline.

00:01:17.000 --> 00:01:20.190
So if we wanted to figure out
the total ounces of saline, we

00:01:20.190 --> 00:01:21.900
say, well, we have 50 ounces.

00:01:21.900 --> 00:01:25.240
Multiply that by 25% and we
have the total amount of

00:01:25.240 --> 00:01:26.940
saline in this solution.

00:01:26.940 --> 00:01:33.510
So 50 times 25%, that's the same
thing as 50 divided by 4,

00:01:33.510 --> 00:01:38.870
so that's 12.5 ounces
of saline in

00:01:38.870 --> 00:01:40.470
this 50 total ounces.

00:01:40.470 --> 00:01:42.560
It's 25% saline.

00:01:42.560 --> 00:01:48.760
Now, let's talk about what we're
going to add to it, so

00:01:48.760 --> 00:01:50.010
solution added.

00:01:54.310 --> 00:01:57.450
Now, they say how many ounces
of a 10% solution?

00:01:57.450 --> 00:01:59.600
So we don't even know how many
ounces we're going to add.

00:01:59.600 --> 00:02:01.480
That's what we have to
actually solve for.

00:02:01.480 --> 00:02:05.140
So let's call that x for the
amount of solution that we

00:02:05.140 --> 00:02:05.980
have to add.

00:02:05.980 --> 00:02:08.120
So we don't know how much we're
going to add, but we do

00:02:08.120 --> 00:02:10.506
know that it is a 10%
saline solution.

00:02:14.620 --> 00:02:17.750
And if we know what x is, we
know the total amount of

00:02:17.750 --> 00:02:22.630
saline is going to
be 10% of x.

00:02:22.630 --> 00:02:24.790
If we had 50 here, it
would be 10% of 50.

00:02:24.790 --> 00:02:27.200
If we had 10 here, it
would be 10% of 10.

00:02:27.200 --> 00:02:30.370
So the amount of saline we have
in this solution, in x

00:02:30.370 --> 00:02:37.040
ounces of this solution, is
going to be 0.1x, or 10%

00:02:37.040 --> 00:02:38.600
percent of x.

00:02:38.600 --> 00:02:41.560
That's what 10% of the solution
being saline means.

00:02:41.560 --> 00:02:44.070
Now, when we add it, what
do we end up with?

00:02:44.070 --> 00:02:46.450
So let me do this in
a different color.

00:02:46.450 --> 00:02:49.945
Resulting solution.

00:02:56.170 --> 00:03:00.230
Well, if we started with 50
ounces and we add x ounces,

00:03:00.230 --> 00:03:05.170
we're going to end up with
50 plus x ounces.

00:03:05.170 --> 00:03:08.600
That's our total volume of
the resulting solution.

00:03:08.600 --> 00:03:10.060
What percent saline is it?

00:03:10.060 --> 00:03:14.080
Well, our goal is to make a 15%
saline solution, so it has

00:03:14.080 --> 00:03:17.210
to be 15% saline.

00:03:17.210 --> 00:03:20.090
Now, what's the total amount
of saline in it?

00:03:20.090 --> 00:03:26.340
And this is kind of the main
box or cell in this table.

00:03:26.340 --> 00:03:29.140
There's two ways to get
to the total amount of

00:03:29.140 --> 00:03:30.590
saline in the solution.

00:03:30.590 --> 00:03:33.600
One, we could just multiply the
percent saline times the

00:03:33.600 --> 00:03:35.450
total value, so we
could do that.

00:03:35.450 --> 00:03:36.150
So let write that.

00:03:36.150 --> 00:03:41.270
It's 0.15 times 50 plus x.

00:03:41.270 --> 00:03:43.560
All I did, I multiplied
the percent saline

00:03:43.560 --> 00:03:44.720
times my total volume.

00:03:44.720 --> 00:03:47.170
That's one way to get the
total amount of saline.

00:03:47.170 --> 00:03:50.020
The other way to get the total
amount of saline is to add

00:03:50.020 --> 00:03:51.190
these two numbers.

00:03:51.190 --> 00:03:55.250
The 50-ounce solution had
12.5 ounces of saline.

00:03:55.250 --> 00:03:58.910
We added 0.1x ounces of saline,
so if we add these two

00:03:58.910 --> 00:04:00.860
numbers, it should also
be equal to the

00:04:00.860 --> 00:04:02.180
total amount of saline.

00:04:02.180 --> 00:04:07.180
So this has to be equal to the
sum of these two things.

00:04:07.180 --> 00:04:16.290
It has to be equal to
12.5 plus 0.1x.

00:04:16.290 --> 00:04:19.990
And just like that, because 15%
of this has to be the same

00:04:19.990 --> 00:04:22.340
thing as the sum of this, we
have one equation and one

00:04:22.340 --> 00:04:25.030
unknown, and we can solve
for x, which is what we

00:04:25.030 --> 00:04:25.570
need to solve for.

00:04:25.570 --> 00:04:27.840
How much solution do
we need to add?

00:04:27.840 --> 00:04:29.430
So let's just do that.

00:04:29.430 --> 00:04:32.630
So 0.15 times 50.

00:04:32.630 --> 00:04:37.090
Let's see, that's 7.5, if
I'm doing that right.

00:04:37.090 --> 00:04:40.430
Yeah, because 0.15 times
100 would be 15.

00:04:40.430 --> 00:04:49.650
So this is 7.5 plus 0.15x--
that's the left-hand side-- is

00:04:49.650 --> 00:04:54.080
equal to 12.5 plus 0.1x.

00:04:54.080 --> 00:04:55.560
Let me scroll to the
right a little bit.

00:04:58.370 --> 00:05:02.082
Now, we can subtract 7.5 from
both sides of this equation.

00:05:02.082 --> 00:05:04.590
Let me do it in a new color.

00:05:04.590 --> 00:05:09.390
So if we subtract 7.5 from both
sides of this equation,

00:05:09.390 --> 00:05:11.660
the left-hand side,
that cancels out.

00:05:11.660 --> 00:05:16.700
We're just left with 0.15x
is equal to-- 12.5

00:05:16.700 --> 00:05:19.580
minus 7.5 is just 5.

00:05:19.580 --> 00:05:23.630
5 plus 0.1x.

00:05:23.630 --> 00:05:25.890
Now, we could subtract
0.1x from both

00:05:25.890 --> 00:05:26.980
sides of this equation.

00:05:26.980 --> 00:05:28.950
Let me scroll down
a little bit.

00:05:28.950 --> 00:05:34.170
Let me subtract 0.1, or I could
say 0.10x from both

00:05:34.170 --> 00:05:35.420
sides of this equation.

00:05:37.920 --> 00:05:39.340
These are the same number.

00:05:39.340 --> 00:05:40.790
So they cancel out.

00:05:40.790 --> 00:05:42.640
The left-hand side, we
just end up with

00:05:42.640 --> 00:05:47.900
0.05x is equal to 5.

00:05:47.900 --> 00:05:56.850
And now we just divide
both sides by 0.05.

00:05:56.850 --> 00:06:01.160
And we get x is equal to
5 divided by 0.05.

00:06:01.160 --> 00:06:04.150
That's the same thing as 5
divided by 1/20, or the same

00:06:04.150 --> 00:06:05.970
thing as 5 times 20.

00:06:05.970 --> 00:06:10.030
So we get x is equal to 100.

00:06:10.030 --> 00:06:11.060
So we're done!

00:06:11.060 --> 00:06:15.380
If you add 100 ounces--
so we figured out that

00:06:15.380 --> 00:06:15.900
x is equal to 100.

00:06:15.900 --> 00:06:19.750
If you add 100 ounces of 10%
saline solution, you will end

00:06:19.750 --> 00:06:25.870
up with 150 ounces of
15% saline solution.

00:06:25.870 --> 00:06:27.920
So if you add 100,
10% of that, we

00:06:27.920 --> 00:06:30.960
actually added 10 ounces.

00:06:30.960 --> 00:06:35.700
So you have 12.5 plus 10 ounces
is 22.5 ounces of

00:06:35.700 --> 00:06:41.450
saline in a solution of 150
ounces, which will be 15%.

00:06:41.450 --> 00:06:43.060
And we're done.


Mixture problems 1

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:06.160
At the Grind House, Honduran
coffee sells for $8 per pound,

00:00:06.160 --> 00:00:10.860
and Costa Rican coffee sells
for $11.50 per pound.

00:00:10.860 --> 00:00:15.390
If 1 pound of Central American
Blend coffee contains 0.4

00:00:15.390 --> 00:00:18.570
pounds of Costa Rican-- so I did
Costa Rican in this orange

00:00:18.570 --> 00:00:22.360
color before, so I'll do it
again-- so it contains 0.4

00:00:22.360 --> 00:00:27.650
pounds of Costa Rican coffee
and 0.6 pounds of Honduran

00:00:27.650 --> 00:00:30.670
coffee-- I'll do the Honduran
coffee in blue.

00:00:30.670 --> 00:00:32.750
That was our original--
that first statement.

00:00:32.750 --> 00:00:34.570
How much should 1
pound sell for?

00:00:34.570 --> 00:00:36.680
So they're really just saying,
well, how much does it cost

00:00:36.680 --> 00:00:39.480
for me to buy 0.4 pounds
of Costa Rican and

00:00:39.480 --> 00:00:40.990
0.6 pounds of Honduran?

00:00:40.990 --> 00:00:43.490
And if I take that total, that's
how much it's going to

00:00:43.490 --> 00:00:47.350
cost me to get 1 pound of my
Central American Blend.

00:00:47.350 --> 00:00:48.750
So let's just figure that out.

00:00:48.750 --> 00:00:53.010
The Costa Rican coffee-- I'll
write it here-- Costa Rican

00:00:53.010 --> 00:00:56.690
coffee-- we're going to--
it's $8 per pound.

00:00:56.690 --> 00:00:59.720
Oh no, sorry, it's
$11.50 per pound.

00:00:59.720 --> 00:01:00.700
We have to be very careful.

00:01:00.700 --> 00:01:06.630
The Costa Rican coffee
is $11.50 per pound.

00:01:06.630 --> 00:01:09.970
We can write it like
this, per pound.

00:01:09.970 --> 00:01:11.820
And then we're going to multiply
that times the number

00:01:11.820 --> 00:01:14.650
of pounds we want of the Costa
Rican coffee in our blend.

00:01:14.650 --> 00:01:17.020
And we have 0.4 pounds of it.

00:01:17.020 --> 00:01:20.950
So times 0.4 pounds.

00:01:20.950 --> 00:01:25.760
The pounds cancel out, and
you have 0.4 times 11.50.

00:01:25.760 --> 00:01:32.090
4 times 11.50 would be 44 plus
2, which would be 46, so 0.4

00:01:32.090 --> 00:01:34.500
is going to be 4.6.

00:01:34.500 --> 00:01:41.390
So this is equal to $4.60.

00:01:41.390 --> 00:01:43.630
Now, we'll do the same thing
for the Honduran coffee.

00:01:48.310 --> 00:01:49.870
We're going to get 0.6 pounds.

00:01:49.870 --> 00:01:51.200
Well, first, let's put
the price here.

00:01:51.200 --> 00:01:52.930
It's $8 per pound.

00:01:52.930 --> 00:01:57.750
So it is $8 per pound.

00:01:57.750 --> 00:01:59.530
And multiply it times the
number of pounds.

00:01:59.530 --> 00:02:01.130
We need 0.6 pounds of it.

00:02:03.870 --> 00:02:09.550
And 8 times 6 is 48, so
8 times 0.6 is $4.80.

00:02:09.550 --> 00:02:13.950
So this is equal to $4.80.

00:02:13.950 --> 00:02:15.410
So that's how much we're going
to have to spend on Costa

00:02:15.410 --> 00:02:16.920
Rican, that's how much we're
going to have to spend on

00:02:16.920 --> 00:02:17.700
Honduran coffee.

00:02:17.700 --> 00:02:20.510
If we add them up, that's how
much a pound of the Central

00:02:20.510 --> 00:02:22.980
American Blend is going to cost.
Because we're getting--

00:02:22.980 --> 00:02:25.710
if you add up the pounds, you
get a pound here, so the total

00:02:25.710 --> 00:02:28.740
cost is the total cost of
a pound of the blend.

00:02:28.740 --> 00:02:30.410
Let's put it right here.

00:02:30.410 --> 00:02:31.540
0 plus 0 is 0.

00:02:31.540 --> 00:02:34.700
6 plus 8 is 14.

00:02:34.700 --> 00:02:37.810
1 plus 4 plus 4 is 9.

00:02:37.810 --> 00:02:39.050
Put the decimal in
the right place.

00:02:39.050 --> 00:02:44.390
It's going to cost us a total
of $9.40 for the blend.


Systems and rate problems 3

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.220
A jogger and a walker set out
at 9:00 AM from the same

00:00:04.220 --> 00:00:07.080
point, headed in the
same direction.

00:00:07.080 --> 00:00:10.660
The average speed of the jogger
is 1 mile per hour

00:00:10.660 --> 00:00:13.080
slower than twice the
speed of the walker.

00:00:13.080 --> 00:00:16.140
So that's going to be an
equation right there.

00:00:16.140 --> 00:00:19.420
The average speed of the jogger
is 1 mile per hour

00:00:19.420 --> 00:00:23.170
slower than twice the
speed of the walker.

00:00:23.170 --> 00:00:26.740
And then they say, in 2 hours,
the jogger is 3 miles ahead of

00:00:26.740 --> 00:00:27.200
the walker.

00:00:27.200 --> 00:00:30.350
That's going to be
another equation.

00:00:30.350 --> 00:00:34.240
In 2 hours, the jogger is 3
miles ahead of the walker.

00:00:34.240 --> 00:00:36.250
Find the rate of the jogger.

00:00:36.250 --> 00:00:39.580
So let's write down that first
statement in equation form.

00:00:39.580 --> 00:00:43.610
The average speed of the jogger,
let's call that the

00:00:43.610 --> 00:00:46.450
speed of the jogger.

00:00:46.450 --> 00:00:49.380
That's the average speed of the
jogger is 1 mile per hour

00:00:49.380 --> 00:00:51.700
slower than twice the
speed of the walker.

00:00:51.700 --> 00:00:55.550
So that's equal to 2 times
the speed of the

00:00:55.550 --> 00:00:57.940
walker, minus 1, right?

00:00:57.940 --> 00:01:02.370
It's 1 mile per hour slower
than twice the

00:01:02.370 --> 00:01:04.700
speed of the walker.

00:01:04.700 --> 00:01:06.730
So that's that first
statement.

00:01:06.730 --> 00:01:10.920
And then they say, in 2 hours,
the jogger is 3 miles ahead of

00:01:10.920 --> 00:01:12.200
the walker.

00:01:12.200 --> 00:01:14.890
So if we go 2 hours
into the future.

00:01:14.890 --> 00:01:18.780
So in 2 hours, the second
statement is the jogger is 3

00:01:18.780 --> 00:01:20.370
miles ahead of the walker.

00:01:20.370 --> 00:01:24.100
So we could say that the
distance that the jogger has

00:01:24.100 --> 00:01:28.550
traveled, so the distance of
the jogger, is 3 miles more

00:01:28.550 --> 00:01:29.685
than the distance
of the walker.

00:01:29.685 --> 00:01:34.510
It is equal to the distance
of the walker plus 3.

00:01:34.510 --> 00:01:35.130
That's all I'm saying.

00:01:35.130 --> 00:01:36.670
This is how far the
jogger's traveled.

00:01:36.670 --> 00:01:40.710
He's traveled 3 miles more
than the walker.

00:01:40.710 --> 00:01:43.640
Now right now I have two
equations with four unknowns.

00:01:43.640 --> 00:01:44.750
That's not cool.

00:01:44.750 --> 00:01:46.320
But they've given us
some information.

00:01:46.320 --> 00:01:48.840
They said this has happened
after 2 hours.

00:01:48.840 --> 00:01:51.830
So we can express the distances
they've traveled in

00:01:51.830 --> 00:01:54.900
terms of the speed
they traveled at.

00:01:54.900 --> 00:01:57.310
Just as a bit of a review,
distance is equal to rate

00:01:57.310 --> 00:01:58.560
times time.

00:02:00.850 --> 00:02:03.740
In this situation,
the time is 2.

00:02:03.740 --> 00:02:06.470
The rate is the speed
of each person.

00:02:06.470 --> 00:02:08.270
So we can figure out
the distance.

00:02:08.270 --> 00:02:10.560
So if you want to know the
distance that the jogger

00:02:10.560 --> 00:02:17.010
traveled, that's just going to
be 2 hours times the speed of

00:02:17.010 --> 00:02:18.020
the jogger.

00:02:18.020 --> 00:02:21.270
So it's just going to
be 2 times the speed

00:02:21.270 --> 00:02:22.510
of the jogger, right?

00:02:22.510 --> 00:02:25.580
Distance is equal to
rate times time,

00:02:25.580 --> 00:02:27.280
or time times rate.

00:02:27.280 --> 00:02:28.260
Either way.

00:02:28.260 --> 00:02:30.450
What's the distance
of the walker?

00:02:30.450 --> 00:02:33.350
Well, it's going to be 2
hours times the speed

00:02:33.350 --> 00:02:34.890
of the walker, right?

00:02:34.890 --> 00:02:37.750
Distance is equal to
rate times time.

00:02:37.750 --> 00:02:40.880
So now we can rewrite this
equation in terms of the

00:02:40.880 --> 00:02:42.850
speeds of the jogger
and the walker.

00:02:42.850 --> 00:02:44.670
So this equation right here.

00:02:44.670 --> 00:02:49.100
We can rewrite it as, instead of
writing the distance of the

00:02:49.100 --> 00:02:53.460
jogger, we could write 2 times
the speed of the jogger--

00:02:53.460 --> 00:02:56.230
that's that right there-- is
equal to-- instead of writing

00:02:56.230 --> 00:02:58.800
the distance of the walker, we
could write 2 times the speed

00:02:58.800 --> 00:03:04.900
of the walker, make sure that's
a subscript-- speed of

00:03:04.900 --> 00:03:07.770
the walker plus 3.

00:03:07.770 --> 00:03:10.830
Now we have two equations and
two unknowns, and it looks

00:03:10.830 --> 00:03:13.410
pretty tempting to eliminate
one of the variables.

00:03:13.410 --> 00:03:15.660
In particular, we have a 2 times
the speed of the walker,

00:03:15.660 --> 00:03:17.605
we have a 2 times the speed
of the walker here.

00:03:17.605 --> 00:03:21.040
What we can do is we can
multiply this second equation

00:03:21.040 --> 00:03:23.230
by negative 1 and then
add the 2's.

00:03:23.230 --> 00:03:24.720
Then these guys will
cancel out.

00:03:24.720 --> 00:03:25.300
So let's do that.

00:03:25.300 --> 00:03:26.380
Actually let's do it
even a better way.

00:03:26.380 --> 00:03:28.780
Let's multiply the top equation
by negative 1 and

00:03:28.780 --> 00:03:29.840
then add them.

00:03:29.840 --> 00:03:32.020
So let's multiply the top
equation by negative 1.

00:03:32.020 --> 00:03:34.250
This becomes a negative times
the speed of the jogger.

00:03:34.250 --> 00:03:37.270
This becomes a negative 2 times
the speed of the walker.

00:03:37.270 --> 00:03:43.260
Negative 1 times negative 1
times negative 1 is plus 1.

00:03:43.260 --> 00:03:45.040
And now if we add these
two equations.

00:03:45.040 --> 00:03:47.690
These two guys are going
to cancel out.

00:03:47.690 --> 00:03:48.990
And what do we get?

00:03:48.990 --> 00:03:52.650
We get 2 times the speed of the
jogger minus the speed of

00:03:52.650 --> 00:03:53.120
the jogger.

00:03:53.120 --> 00:03:54.160
Well that's just going
to be the speed

00:03:54.160 --> 00:03:55.850
of the jogger, right?

00:03:55.850 --> 00:03:57.420
2 of something minus
1 of something is

00:03:57.420 --> 00:04:00.430
just 1 of that thing.

00:04:00.430 --> 00:04:01.760
These two guys cancel out.

00:04:01.760 --> 00:04:03.740
That was the whole point behind
multiplying the top

00:04:03.740 --> 00:04:05.280
equation by negative 1.

00:04:05.280 --> 00:04:09.840
Then you have 1 plus 3, so the
speed of the jogger is 4.

00:04:09.840 --> 00:04:10.580
And we're done.

00:04:10.580 --> 00:04:12.090
That's all they want
us to find.

00:04:12.090 --> 00:04:15.360
Find the rate of the jogger.

00:04:15.360 --> 00:04:16.860
And we just figured it out.

00:04:16.860 --> 00:04:20.260
If we also wanted to figure out
the rate of the walker, we

00:04:20.260 --> 00:04:23.360
could substitute back into this
top equation right here,

00:04:23.360 --> 00:04:26.580
or the original version of it,
which was the speed of the

00:04:26.580 --> 00:04:31.350
jogger is equal to 2 times the
speed of the walker minus 1.

00:04:31.350 --> 00:04:33.880
We know the speed of the jogger
is 4, so you get 4 is

00:04:33.880 --> 00:04:37.340
equal to 2 times the speed
of the walker minus 1.

00:04:37.340 --> 00:04:41.480
Add 1 to both sides, you get 5
is equal to 2 times the speed

00:04:41.480 --> 00:04:42.530
of the walker.

00:04:42.530 --> 00:04:45.590
Divide both sides by 2, and
you get the speed of the

00:04:45.590 --> 00:04:51.080
walker is 2-1/2 miles
per hour.

00:04:51.080 --> 00:04:52.120
But we didn't even
have to do that.

00:04:52.120 --> 00:04:54.420
This was the answer that the
problem was looking for.


Systems and rate problems 2

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:04.360
Joelle has two after-school
jobs, babysitting and working

00:00:04.360 --> 00:00:05.990
at her parents' grocery store.

00:00:05.990 --> 00:00:09.290
Working at the grocery pays
her $12 an hour, and

00:00:09.290 --> 00:00:11.740
babysitting pays her
$9.50 an hour.

00:00:11.740 --> 00:00:14.170
Last week she worked
19 hours in total.

00:00:14.170 --> 00:00:15.800
That looks like a
statement that

00:00:15.800 --> 00:00:17.000
could lead to an equation.

00:00:17.000 --> 00:00:20.320
Last week she worked
19 hours in total.

00:00:20.320 --> 00:00:23.320
And she earned $56 more at the
grocery store than she did

00:00:23.320 --> 00:00:23.850
babysitting.

00:00:23.850 --> 00:00:25.390
That looks like another
statement

00:00:25.390 --> 00:00:26.620
that could be an equation.

00:00:26.620 --> 00:00:31.090
She earned $56 more at the
grocery store than she did

00:00:31.090 --> 00:00:32.200
babysitting.

00:00:32.200 --> 00:00:35.220
And then they ask us, how
many hours did she

00:00:35.220 --> 00:00:37.440
work at each job?

00:00:37.440 --> 00:00:40.330
So they want to know--
so let's define

00:00:40.330 --> 00:00:42.410
some variables here.

00:00:42.410 --> 00:00:48.490
Let's define time babysitting,
so t sub b, as equal to hours

00:00:48.490 --> 00:00:49.740
babysitting.

00:00:56.110 --> 00:01:00.476
And let's define t sub g as
hours at the grocery.

00:01:07.650 --> 00:01:11.640
So they want us to figure out
what was t sub b and t sub g,

00:01:11.640 --> 00:01:13.400
what were the hours of
babysitting and what were the

00:01:13.400 --> 00:01:14.700
hours at the grocery?

00:01:14.700 --> 00:01:18.150
Now, they tell us that she
worked 19 hours total.

00:01:18.150 --> 00:01:20.660
So that blue statement says that
the sum of these two is

00:01:20.660 --> 00:01:22.310
going to be 19.

00:01:22.310 --> 00:01:25.940
So the time she worked
babysitting plus the time she

00:01:25.940 --> 00:01:31.900
worked at the grocery store is
going to be equal to 19 hours.

00:01:31.900 --> 00:01:32.890
Now the other statement.

00:01:32.890 --> 00:01:35.000
This by itself obviously we
can't solve for these.

00:01:35.000 --> 00:01:36.370
We need another constraint.

00:01:36.370 --> 00:01:39.770
They said that she earned $56
more at the grocery store then

00:01:39.770 --> 00:01:41.480
she did babysitting.

00:01:41.480 --> 00:01:43.880
So let me define two more
variables here.

00:01:43.880 --> 00:01:52.020
So this is the pay babysitting,
her total pay

00:01:52.020 --> 00:01:53.910
babysitting.

00:01:53.910 --> 00:01:58.940
And let's say P sub g
is equal to the pay

00:01:58.940 --> 00:02:01.370
at the grocery store.

00:02:01.370 --> 00:02:03.220
Now, what are these terms?

00:02:03.220 --> 00:02:05.570
I mean, if we just look at this
second statement right

00:02:05.570 --> 00:02:08.160
there, we'll just say, OK, the
pay at babysitting, she earned

00:02:08.160 --> 00:02:11.320
$56 more at the grocery than
she did babysitting.

00:02:11.320 --> 00:02:14.350
So this second statement tells
us that, well, I guess the

00:02:14.350 --> 00:02:18.960
pay-- let me write it over
here-- the pay babysitting,

00:02:18.960 --> 00:02:21.260
she earned $56 more at the
grocery than she did

00:02:21.260 --> 00:02:21.780
babysitting.

00:02:21.780 --> 00:02:26.500
So if we take babysitting and
add $56, that will get us to

00:02:26.500 --> 00:02:28.910
how much she earned at
the grocery store.

00:02:28.910 --> 00:02:31.690
That's what that second
statement is telling us. $56

00:02:31.690 --> 00:02:33.260
more at the grocery
than babysitting.

00:02:33.260 --> 00:02:35.160
So if you add these two,
you get the grocery.

00:02:35.160 --> 00:02:36.700
Now, this still isn't cool.

00:02:36.700 --> 00:02:40.370
We have one, two, three,
four unknowns

00:02:40.370 --> 00:02:41.410
and only two equations.

00:02:41.410 --> 00:02:42.580
That's not going to help us.

00:02:42.580 --> 00:02:45.970
But what will help us we can
express these two variables in

00:02:45.970 --> 00:02:49.360
terms of these two variables,
and then we have two equations

00:02:49.360 --> 00:02:51.300
with two unknowns, so we can
solve them the way we've

00:02:51.300 --> 00:02:53.670
solved all systems
of equations.

00:02:53.670 --> 00:02:56.340
Now the other information they
give us is the rate, how much

00:02:56.340 --> 00:02:57.750
she earned per hour.

00:02:57.750 --> 00:03:00.950
Working at the grocery, she
makes $12 per hour.

00:03:00.950 --> 00:03:04.290
Babysitting, she makes
$9.50 per hour.

00:03:04.290 --> 00:03:06.890
So what is going to be
her pay babysitting?

00:03:06.890 --> 00:03:08.150
Let me write this over here.

00:03:08.150 --> 00:03:13.320
Her pay babysitting-- I'll do
it in this color-- her pay

00:03:13.320 --> 00:03:17.570
babysitting is going to
be equal to the time

00:03:17.570 --> 00:03:19.370
babysitting, right?

00:03:19.370 --> 00:03:22.620
The time babysitting times how
much she gets paid per hour.

00:03:22.620 --> 00:03:25.090
The time babysitting is how many
hours, times how much she

00:03:25.090 --> 00:03:25.900
gets paid per hour.

00:03:25.900 --> 00:03:27.030
It tells us right there.

00:03:27.030 --> 00:03:30.040
She gets paid $9.50 per
hour babysitting.

00:03:30.040 --> 00:03:33.830
So times $9.50.

00:03:33.830 --> 00:03:36.150
That is what her pay
babysitting is.

00:03:36.150 --> 00:03:38.870
And then, same argument, what's
her pay working at the

00:03:38.870 --> 00:03:40.370
grocery store?

00:03:40.370 --> 00:03:42.970
Her total pay working at
the grocery store.

00:03:42.970 --> 00:03:46.320
Her pay at the grocery is going
to be equal to the time

00:03:46.320 --> 00:03:50.910
at the grocery times the rate
that she gets paid at the

00:03:50.910 --> 00:03:53.050
grocery. $12 per hour.

00:03:53.050 --> 00:03:55.630
So times $12 per hour.

00:03:55.630 --> 00:03:58.320
And this hopefully
is intuitive.

00:03:58.320 --> 00:04:01.430
If I'm making $12 per hour and
I work for, I don't know, 3

00:04:01.430 --> 00:04:03.900
hours, I'm going to make $36.

00:04:03.900 --> 00:04:05.390
I would just multiply
these two.

00:04:05.390 --> 00:04:09.460
If I worked 2 hours at $9.50,
I'm going to make $19.

00:04:09.460 --> 00:04:10.150
That's all this is.

00:04:10.150 --> 00:04:11.280
A total pay.

00:04:11.280 --> 00:04:14.060
So now we can take these two
expressions-- we could take

00:04:14.060 --> 00:04:17.040
this expression and this
expression-- and substitute it

00:04:17.040 --> 00:04:20.779
back into that equation right
there, and then we'll have an

00:04:20.779 --> 00:04:24.190
equation that's only in terms
of the time babysitting and

00:04:24.190 --> 00:04:25.400
the time at the grocery store.

00:04:25.400 --> 00:04:26.510
So let's do that.

00:04:26.510 --> 00:04:30.940
So this first term right here,
her total pay babysitting,

00:04:30.940 --> 00:04:32.710
that is this expression
right there.

00:04:32.710 --> 00:04:34.200
So let me write it right here.

00:04:34.200 --> 00:04:36.420
So $9.50 times the
time babysitting.

00:04:36.420 --> 00:04:41.130
So I'll just write 9.5 times
time babysitting-- that's this

00:04:41.130 --> 00:04:48.670
term right here-- plus 56 is
equal to the pay that she got

00:04:48.670 --> 00:04:51.460
at the grocery, which is 12
times the time at the grocery.

00:04:55.080 --> 00:04:57.980
And now we need to solve for
both of these variables.

00:04:57.980 --> 00:04:59.680
We have two equations
and two unknowns.

00:04:59.680 --> 00:05:02.670
This is just straightforward
algebra at this point.

00:05:02.670 --> 00:05:04.060
So let's do it.

00:05:04.060 --> 00:05:06.950
So the easiest way to do this
might just be to do straight

00:05:06.950 --> 00:05:09.750
up substitution cause we have
all these crazy numbers here.

00:05:09.750 --> 00:05:12.870
Let's just solve for the time at
the babysitting in terms of

00:05:12.870 --> 00:05:14.330
the time at the grocery.

00:05:14.330 --> 00:05:19.140
So if you take this top
equation-- let me write it

00:05:19.140 --> 00:05:23.210
over here-- so the time
babysitting plus the time at

00:05:23.210 --> 00:05:25.800
the grocery is equal to 19.

00:05:25.800 --> 00:05:32.250
If you subtract the time at the
grocery from both sides,

00:05:32.250 --> 00:05:35.680
you get the time babysitting
is equal to-- these cancel

00:05:35.680 --> 00:05:38.920
out-- 19 minus the time
at the grocery.

00:05:38.920 --> 00:05:41.700
So we solved for the time
babysitting in terms of the

00:05:41.700 --> 00:05:42.570
time at the grocery.

00:05:42.570 --> 00:05:46.063
So everywhere we see the time
babysitting, we can substitute

00:05:46.063 --> 00:05:49.540
it with this expression
right here.

00:05:49.540 --> 00:05:51.250
That's what t sub b is.

00:05:51.250 --> 00:05:53.370
That is what the time
babysitting is.

00:05:53.370 --> 00:05:54.290
So let's do that.

00:05:54.290 --> 00:05:59.900
So this equation becomes 9.5
times the time babysitting,

00:05:59.900 --> 00:06:03.780
which we just figured out from
this top equation is 19 minus

00:06:03.780 --> 00:06:11.290
the time at the grocery, plus
56, is equal to 12 times the

00:06:11.290 --> 00:06:13.010
time at the grocery.

00:06:13.010 --> 00:06:15.890
And now we just have to solve
for the time at the grocery.

00:06:15.890 --> 00:06:21.650
So 9.5 times 19.

00:06:21.650 --> 00:06:22.770
That's a little bit of hairy.

00:06:22.770 --> 00:06:26.470
Let me just do it over here, on
the right side right here.

00:06:26.470 --> 00:06:32.430
So if I have 19 times 9.5.

00:06:32.430 --> 00:06:36.320
5 times 9 is 45.

00:06:36.320 --> 00:06:41.440
1 times 5 is 5, plus 4, is 9.

00:06:41.440 --> 00:06:43.420
And then we have a 0 here.

00:06:43.420 --> 00:06:46.410
9 times 9 is 81.

00:06:46.410 --> 00:06:51.520
9 times 1 is 9, plus 8, is 17.

00:06:51.520 --> 00:06:53.160
So 5 plus 0 is 5.

00:06:53.160 --> 00:06:55.110
9 plus 1 is 10.

00:06:55.110 --> 00:06:56.980
1 plus 7 is 8.

00:06:56.980 --> 00:06:59.290
You just have a 1 there, and you
have one number behind the

00:06:59.290 --> 00:07:00.030
decimal point.

00:07:00.030 --> 00:07:02.130
So it's 180.5.

00:07:02.130 --> 00:07:11.350
So 9.5 times 19 is 180.5 minus
9.5 times the time at the

00:07:11.350 --> 00:07:18.830
grocery, plus 56, is
equal to 12 times

00:07:18.830 --> 00:07:20.380
the time at the grocery.

00:07:20.380 --> 00:07:24.780
Now what's 180.5 plus 56?

00:07:24.780 --> 00:07:29.060
So 180 plus 50-- let's add just
those two terms-- 180.5

00:07:29.060 --> 00:07:31.410
plus 56, just to simplify it.

00:07:31.410 --> 00:07:35.300
180 plus 50 would be 230.

00:07:35.300 --> 00:07:46.810
So this'll be 236.5 minus 9.5
times the time at the grocery,

00:07:46.810 --> 00:07:50.400
is equal to 12 times the
time at the grocery.

00:07:50.400 --> 00:07:52.740
Now we can add the time at the
grocery to both sides of this

00:07:52.740 --> 00:07:54.270
equation, or the 9.5.

00:07:54.270 --> 00:07:55.000
So let's do that.

00:07:55.000 --> 00:08:04.800
Plus 9.5 times the time at the
grocery is equal-- or plus 9.5

00:08:04.800 --> 00:08:05.820
times time at the grocery.

00:08:05.820 --> 00:08:08.470
Just adding it to both sides
of that equation.

00:08:08.470 --> 00:08:11.030
And we are left with-- the
left-hand side, that cancels

00:08:11.030 --> 00:08:17.830
out-- we just have 236.5 is
equal to, 12 plus 9 is 21, so

00:08:17.830 --> 00:08:19.880
this is going to be 21.5.

00:08:19.880 --> 00:08:25.810
It's equal to 21.5 times the
time she spent at the grocery.

00:08:25.810 --> 00:08:28.500
So now we can just divide both
sides of this equation by

00:08:28.500 --> 00:08:30.970
21.5, and we've figured
out how long she

00:08:30.970 --> 00:08:32.970
spent at the grocery.

00:08:32.970 --> 00:08:34.230
Let's do that.

00:08:34.230 --> 00:08:38.309
Divide both sides by 21.5.

00:08:38.309 --> 00:08:39.460
So that cancels out.

00:08:39.460 --> 00:08:42.309
The time she spent at the
grocery is going to be equal

00:08:42.309 --> 00:08:43.940
to that thing.

00:08:43.940 --> 00:08:45.440
And let's see what that is.

00:08:45.440 --> 00:08:56.420
We get 21.5 goes into 236.5.

00:08:56.420 --> 00:08:58.860
Let's multiply both of
these numbers by 10.

00:08:58.860 --> 00:09:04.165
So this is the same thing
as 215 going into 2,365.

00:09:04.165 --> 00:09:06.200
The decimal goes right there.

00:09:06.200 --> 00:09:09.270
215 goes into 236 one time.

00:09:09.270 --> 00:09:13.060
1 times 215 is 215.

00:09:13.060 --> 00:09:15.730
And then we can subtract-- let
me scroll down more-- this

00:09:15.730 --> 00:09:17.910
problem is hairier than
I thought it would be.

00:09:17.910 --> 00:09:21.000
So 36 minus 15 is 21.

00:09:21.000 --> 00:09:24.400
So 21-- bring down the 5-- and
lucky for us that's pretty

00:09:24.400 --> 00:09:24.890
straightforward.

00:09:24.890 --> 00:09:27.290
215 goes into 215 one time.

00:09:27.290 --> 00:09:29.890
1 times 215 is 215.

00:09:29.890 --> 00:09:30.660
Subtract.

00:09:30.660 --> 00:09:32.030
You get no remainder.

00:09:32.030 --> 00:09:36.590
So the time at the grocery
store is 11, or 11 hours.

00:09:36.590 --> 00:09:39.260
Now we can go and substitute
back to figure out her time

00:09:39.260 --> 00:09:40.250
babysitting.

00:09:40.250 --> 00:09:43.050
We know that the time
babysitting is equal to 19

00:09:43.050 --> 00:09:44.510
minus the time at the grocery.

00:09:44.510 --> 00:09:47.210
In this case, it's
19 minus 11.

00:09:47.210 --> 00:09:48.500
We just figured that out.

00:09:48.500 --> 00:09:50.790
19 minus 11 is 8.

00:09:50.790 --> 00:09:55.880
So her time babysitting, she
spent 8 hours babysitting.

00:09:55.880 --> 00:09:57.190
And we are done.

00:09:57.190 --> 00:09:59.490
They both add up to 19, and you
could make sure that they

00:09:59.490 --> 00:10:00.700
also verify.

00:10:00.700 --> 00:10:02.230
That also works out for
the second equation.


Systems and rate problems

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:04.080
Jason bicycled from home to
the train station at an

00:00:04.080 --> 00:00:07.120
average speed of 10
miles per hour.

00:00:07.120 --> 00:00:09.910
Then he boarded a train and
traveled into the city at an

00:00:09.910 --> 00:00:12.100
average speed of 50
miles per hour.

00:00:12.100 --> 00:00:14.360
The entire distance
was 30 miles; the

00:00:14.360 --> 00:00:16.440
entire trip took 1 hour.

00:00:16.440 --> 00:00:20.360
How many miles did Jason
travel by train?

00:00:20.360 --> 00:00:21.980
So let's write that down
as a variable.

00:00:21.980 --> 00:00:24.620
So let's say distance, and we'll
write a little small t

00:00:24.620 --> 00:00:26.060
right here, a little
subscript t.

00:00:26.060 --> 00:00:32.840
This is the distance by train,
so train distance.

00:00:32.840 --> 00:00:33.720
Right there.

00:00:33.720 --> 00:00:36.280
And let's have a d with
a little bit of a

00:00:36.280 --> 00:00:37.250
little b right there.

00:00:37.250 --> 00:00:38.570
That is the bike distance.

00:00:42.180 --> 00:00:44.990
Now, they give us one piece of
information, that the total

00:00:44.990 --> 00:00:48.920
distance, the entire distance,
was 30 miles.

00:00:48.920 --> 00:00:51.200
So that means that the distance
by train plus the

00:00:51.200 --> 00:00:54.030
distance by bike is 30 miles.

00:00:54.030 --> 00:00:55.140
So I could write that
right here.

00:00:55.140 --> 00:00:59.680
The distance by train plus
the distance by bike

00:00:59.680 --> 00:01:01.910
is equal to 30 miles.

00:01:01.910 --> 00:01:03.650
That's what that constraint
tells us.

00:01:03.650 --> 00:01:04.810
Now the next one, they
tell us that the

00:01:04.810 --> 00:01:07.150
entire trip took 1 hour.

00:01:07.150 --> 00:01:11.980
So the time by train plus the
time by bike took 1 hour.

00:01:11.980 --> 00:01:13.450
And you might be thinking,
hey, gee, that's going to

00:01:13.450 --> 00:01:15.040
introduce two new variables.

00:01:15.040 --> 00:01:17.760
But let's think a little bit
about whether we can express

00:01:17.760 --> 00:01:20.970
the time by train and the
time by bike in terms

00:01:20.970 --> 00:01:22.540
of these two variables.

00:01:22.540 --> 00:01:26.560
So just as a bit of review-- I
think this is review-- you're

00:01:26.560 --> 00:01:30.630
probably familiar with distance
is equal to rate

00:01:30.630 --> 00:01:31.960
times time.

00:01:31.960 --> 00:01:36.780
Or if you divide both sides of
this equation by rate, you get

00:01:36.780 --> 00:01:39.095
time is equal to distance
divided by rate.

00:01:42.890 --> 00:01:44.700
So let's think about
the situation for

00:01:44.700 --> 00:01:45.800
each of these guys.

00:01:45.800 --> 00:01:47.340
What is the time by train?

00:01:47.340 --> 00:01:48.200
I'll write it like this.

00:01:48.200 --> 00:01:52.130
The time by train is going to
be equal to the distance by

00:01:52.130 --> 00:01:56.220
train divided by the rate that
the train was going at.

00:01:56.220 --> 00:01:58.200
And they gave us that
information.

00:01:58.200 --> 00:02:01.140
They told us that the train
traveled into the city at an

00:02:01.140 --> 00:02:03.200
average speed of 50
miles per hour.

00:02:03.200 --> 00:02:05.350
That is the rate of the train.

00:02:05.350 --> 00:02:09.259
So the time of the train was
the distance of the train

00:02:09.259 --> 00:02:11.320
divided by 50.

00:02:11.320 --> 00:02:13.420
Same exact argument.

00:02:13.420 --> 00:02:16.990
The time of the bicycle, the
time traveled on the bicycle,

00:02:16.990 --> 00:02:19.950
will be the distance traveled on
the bicycle divided by the

00:02:19.950 --> 00:02:21.050
speed of the bicycle.

00:02:21.050 --> 00:02:22.340
And they give us the
speed right there,

00:02:22.340 --> 00:02:23.510
10 miles per hour.

00:02:23.510 --> 00:02:25.530
And everything in this problem
we're assuming is going to be

00:02:25.530 --> 00:02:27.050
in either miles or hours.

00:02:27.050 --> 00:02:29.350
There's not any major
unit conversion.

00:02:29.350 --> 00:02:30.610
So that's there right there.

00:02:30.610 --> 00:02:33.690
So the second statement
is that the

00:02:33.690 --> 00:02:36.110
entire trip took 1 hour.

00:02:36.110 --> 00:02:39.470
So the time by train plus
the time by bicycle is

00:02:39.470 --> 00:02:41.000
going to be 1 hour.

00:02:41.000 --> 00:02:43.765
Actually, let me do that
in a different color.

00:02:43.765 --> 00:02:46.570
The entire trip took 1 hour.

00:02:46.570 --> 00:02:49.210
So this time plus this
time is 1 hour.

00:02:49.210 --> 00:02:51.100
And I'm going to write it in
terms of the distances so I

00:02:51.100 --> 00:02:53.210
only have two unknowns.

00:02:53.210 --> 00:02:57.780
So the time by train is the
distance by train divided by

00:02:57.780 --> 00:03:03.050
50 plus the time by bicycle
is the distance by bicycle

00:03:03.050 --> 00:03:04.510
divided by 10.

00:03:04.510 --> 00:03:07.250
And then that is equal
to 1 hour.

00:03:07.250 --> 00:03:09.020
That's what this statement
right here is telling us,

00:03:09.020 --> 00:03:12.200
although we had to use that
information and that

00:03:12.200 --> 00:03:16.150
information to divide by the
10 and to divide by the 50.

00:03:16.150 --> 00:03:18.810
Now we have two equations
of two unknowns.

00:03:18.810 --> 00:03:21.570
And our whole goal, the whole
purpose of this problem, they

00:03:21.570 --> 00:03:25.510
want to know how many miles
did Jason travel by train.

00:03:25.510 --> 00:03:29.440
So we want to figure out that
variable, or we want to figure

00:03:29.440 --> 00:03:30.730
out that variable.

00:03:30.730 --> 00:03:32.710
Now, the easiest way to do
this is if we can just

00:03:32.710 --> 00:03:36.020
eliminate the distance by
bicycle and then solve for the

00:03:36.020 --> 00:03:37.050
distance by train.

00:03:37.050 --> 00:03:37.640
Then we're done.

00:03:37.640 --> 00:03:39.630
We would have solved
the problem.

00:03:39.630 --> 00:03:42.530
Now, the easiest way I can think
of doing that is this

00:03:42.530 --> 00:03:43.990
one is pretty simple already.

00:03:43.990 --> 00:03:46.020
Let me just rewrite it
to the right here.

00:03:46.020 --> 00:03:49.700
So the distance by train plus
the distance by bicycle is

00:03:49.700 --> 00:03:51.230
equal to 30.

00:03:51.230 --> 00:03:53.500
And I want to cancel out the
distance by bicycle.

00:03:53.500 --> 00:03:56.330
So if I could make this into
just a negative distance by

00:03:56.330 --> 00:03:58.030
bicycle, then I'm all set.

00:03:58.030 --> 00:03:59.960
And if I add the two equations,
they'll cancel out.

00:03:59.960 --> 00:04:01.950
So the easiest way to just make
this a negative distance

00:04:01.950 --> 00:04:05.210
by bicycle is multiply both
sides of this equation by

00:04:05.210 --> 00:04:10.100
negative 10.

00:04:10.100 --> 00:04:12.360
Because you multiply negative 10
times this, the 10's cancel

00:04:12.360 --> 00:04:14.920
out, this just becomes a
negative distance by bicycle.

00:04:14.920 --> 00:04:15.740
So let's write it over here.

00:04:15.740 --> 00:04:19.180
Negative 10 times the distance
by train over 50.

00:04:19.180 --> 00:04:21.420
10 divided by 50 is 1/5,
so it's negative

00:04:21.420 --> 00:04:27.050
1/5 distance by train.

00:04:27.050 --> 00:04:29.620
And then negative 10 times this
expression right here,

00:04:29.620 --> 00:04:30.470
the 10's cancel out.

00:04:30.470 --> 00:04:35.160
It's just negative distance by
bicycle is equal to-- forgot

00:04:35.160 --> 00:04:38.890
that parentheses-- is equal
to negative 10.

00:04:38.890 --> 00:04:41.000
The whole point here was so that
this becomes the negative

00:04:41.000 --> 00:04:41.700
version of that.

00:04:41.700 --> 00:04:44.460
So when I add these two
equations, they'll cancel out.

00:04:44.460 --> 00:04:45.830
So let's do that.

00:04:45.830 --> 00:04:48.410
Let's add these two equations.

00:04:48.410 --> 00:04:51.470
So if you add the left-hand
side, these guys cancel out.

00:04:51.470 --> 00:04:54.760
You have 1 dt minus 1/5 dt.

00:04:54.760 --> 00:04:57.910
1 dt you can view as 5/5 dt.

00:04:57.910 --> 00:05:06.140
So 5/5 minus 1/5, you have 4/5
distance by train is equal to

00:05:06.140 --> 00:05:09.990
30 plus negative 10,
or is equal to 20.

00:05:09.990 --> 00:05:13.270
Now to solve for distance by
train, we can just multiply

00:05:13.270 --> 00:05:16.060
both sides of this expression,
both sides of this equation,

00:05:16.060 --> 00:05:17.970
by the inverse of 4/5.

00:05:17.970 --> 00:05:22.630
So we'll multiply both
sides by 5/4.

00:05:22.630 --> 00:05:25.920
The whole point of that is that
this cancels out, and we

00:05:25.920 --> 00:05:31.530
are left with the distance by
train is equal to 20 times

00:05:31.530 --> 00:05:34.840
5/4, that's the same thing
as 20/1 times 5/4.

00:05:34.840 --> 00:05:38.850
20 divided by 4 is 5,
4 divided by 4 is 1.

00:05:38.850 --> 00:05:40.850
So it's just 5 times 5.

00:05:40.850 --> 00:05:45.210
So the distance traveled
by train is 25 miles.

00:05:45.210 --> 00:05:45.830
And we're done.

00:05:45.830 --> 00:05:47.620
And we could go back if we
wanted to figure out any of

00:05:47.620 --> 00:05:50.270
the times, or the distance
by bicycle.

00:05:50.270 --> 00:05:51.060
Actually, it's very easy.

00:05:51.060 --> 00:05:54.280
The distance by train is 25,
distance by bicycle has to be

00:05:54.280 --> 00:05:57.630
5 if they're going to add
up to be 30 miles.


Example 3: Solving systems by elimination

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.730
We're told to solve and graph
the solution for the system of

00:00:03.730 --> 00:00:05.050
equations right here.

00:00:05.050 --> 00:00:07.180
And the first thing that jumps
out at me, is that we might be

00:00:07.180 --> 00:00:08.930
able to eliminate one
of the variables.

00:00:08.930 --> 00:00:12.510
And if we just focus on the x,
we have a 4x here and we have

00:00:12.510 --> 00:00:13.530
a 2x right here.

00:00:13.530 --> 00:00:15.330
If we were to just add them
right now, we would get a 6x.

00:00:15.330 --> 00:00:16.740
So that wouldn't eliminate it.

00:00:16.740 --> 00:00:20.780
But if we can multiply this 2x
by negative 2, it'll become a

00:00:20.780 --> 00:00:24.400
negative 4x, and then when you
add it, they would cancel out.

00:00:24.400 --> 00:00:27.050
So let's multiply this
equation, this second

00:00:27.050 --> 00:00:29.510
equation, by negative 2.

00:00:29.510 --> 00:00:31.830
So I'm going to multiply both
sides of this equation by

00:00:31.830 --> 00:00:32.710
negative 2.

00:00:32.710 --> 00:00:35.350
And the whole motivation is
so that this 2x becomes a

00:00:35.350 --> 00:00:36.280
negative 4x.

00:00:36.280 --> 00:00:38.790
And, of course, I can't just
multiply only the 2x.

00:00:38.790 --> 00:00:41.010
Anything I do to the left-hand
side of the equation I have to

00:00:41.010 --> 00:00:42.740
do to every term, and
I have to do to both

00:00:42.740 --> 00:00:44.100
sides of the equation.

00:00:44.100 --> 00:00:47.760
So the second equation becomes
negative 4x-- that's negative

00:00:47.760 --> 00:00:51.520
2 times 2x-- plus-- we have
negative 2 times negative y--

00:00:51.520 --> 00:00:56.430
which is plus 2y is equal to 2.5
times negative 2, is equal

00:00:56.430 --> 00:00:57.630
to negative 5.

00:00:57.630 --> 00:00:59.880
I just rewrote the second
equation, multiplying both

00:00:59.880 --> 00:01:01.500
sides by negative 2.

00:01:01.500 --> 00:01:04.160
Now, this top equation-- I'll
write it on the bottom now--

00:01:04.160 --> 00:01:09.260
we have 4x minus 2y is
equal to positive 5.

00:01:09.260 --> 00:01:10.450
And now we can eliminate it.

00:01:10.450 --> 00:01:12.530
We can say, hey, look, the
negative 4x and the positive

00:01:12.530 --> 00:01:14.900
4x should cancel out, or
they will cancel out.

00:01:14.900 --> 00:01:16.780
So let's add these
two equations.

00:01:16.780 --> 00:01:18.940
Let's add the left side to the
left side, the right side to

00:01:18.940 --> 00:01:21.160
the right side, and we can
do that because these

00:01:21.160 --> 00:01:22.100
two things are equal.

00:01:22.100 --> 00:01:25.060
We're doing the same thing to
both sides of the equation.

00:01:25.060 --> 00:01:26.450
So what do we get?

00:01:26.450 --> 00:01:31.340
If we take our negative 4x
plus our 4x, well, those

00:01:31.340 --> 00:01:32.550
cancel out.

00:01:32.550 --> 00:01:33.810
So you're left with nothing.

00:01:33.810 --> 00:01:35.770
Maybe I could write a 0 there.

00:01:35.770 --> 00:01:37.410
0x if you want.

00:01:37.410 --> 00:01:40.160
And then you have your plus
2y and your negative 2y.

00:01:40.160 --> 00:01:41.370
Those also cancel out.

00:01:41.370 --> 00:01:43.530
So you're also left with 0y.

00:01:43.530 --> 00:01:47.460
And then that equals negative
5 plus 5 is equal to 0.

00:01:47.460 --> 00:01:54.360
So this just simplifies to 0
equals 0, which is true, but

00:01:54.360 --> 00:01:55.570
it's kind of bizarre.

00:01:55.570 --> 00:01:56.940
We had all these x's and y's.

00:01:56.940 --> 00:01:59.690
Everything canceled out.

00:01:59.690 --> 00:02:01.180
So let's explore this
a little bit more.

00:02:01.180 --> 00:02:04.950
Let's graph it and see what this
0 equals 0 is telling us

00:02:04.950 --> 00:02:07.710
when we try to solve this
system of equations.

00:02:07.710 --> 00:02:11.150
So let me graph this top guy.

00:02:11.150 --> 00:02:11.920
I'll do it in blue.

00:02:11.920 --> 00:02:13.870
So right now it's in
standard form.

00:02:13.870 --> 00:02:16.280
Let's put it in slope-intercept
form.

00:02:16.280 --> 00:02:21.130
So we have 4x minus
2y is equal to 5.

00:02:21.130 --> 00:02:25.000
Let's subtract 4x
from both sides.

00:02:25.000 --> 00:02:27.790
I want the x terms on
the right-hand side.

00:02:27.790 --> 00:02:31.950
So then I'm left with negative
2y is equal to

00:02:31.950 --> 00:02:34.990
negative 4x plus 5.

00:02:34.990 --> 00:02:37.140
Now we can divide both
sides by negative 2.

00:02:40.280 --> 00:02:47.170
And we are left with y is equal
to positive 2x, right,

00:02:47.170 --> 00:02:50.350
that's positive 2x, minus 2.5.

00:02:53.590 --> 00:02:54.950
So let's graph that.

00:02:54.950 --> 00:02:57.540
The y-intercept is
negative 2.5.

00:02:57.540 --> 00:03:01.330
So negative 2.5 right there, and
then it has a slope of 2.

00:03:01.330 --> 00:03:05.380
So if we move up 1, if we move
up in the x-direction, if we

00:03:05.380 --> 00:03:08.080
move to the right 1 in the
positive x-direction, we

00:03:08.080 --> 00:03:09.300
will move up 2.

00:03:09.300 --> 00:03:11.340
So 1, 2.

00:03:11.340 --> 00:03:12.370
Right there.

00:03:12.370 --> 00:03:15.920
And if we were to do it again,
we move up 1, 2.

00:03:15.920 --> 00:03:16.630
Just like that.

00:03:16.630 --> 00:03:18.590
So the line's going to look
something like this.

00:03:18.590 --> 00:03:20.920
I'll try my best to draw
a straight line.

00:03:20.920 --> 00:03:24.170
This is the hardest part
about a lot of these

00:03:24.170 --> 00:03:25.720
problems. There you go.

00:03:25.720 --> 00:03:26.900
So that's the top equation.

00:03:26.900 --> 00:03:29.370
Now, let me draw the
bottom equation.

00:03:29.370 --> 00:03:33.680
Let me draw and I'll do it
in this green color.

00:03:33.680 --> 00:03:40.980
So this bottom equation was 2x
minus y is equal to 2.5.

00:03:40.980 --> 00:03:43.240
And we can subtract 2x
from both sides.

00:03:46.540 --> 00:03:52.110
The left-hand side becomes
negative y is equal to 2x

00:03:52.110 --> 00:03:55.270
plus-- or is equal to negative
2x plus 2.5.

00:03:55.270 --> 00:03:58.460
Now let's multiply or divide
both sides by negative 1.

00:03:58.460 --> 00:04:04.090
And you get y is equal to
positive 2x minus 2.5.

00:04:04.090 --> 00:04:07.080
And let's try to graph this, and
you already might notice

00:04:07.080 --> 00:04:09.380
something interesting about
these two equations.

00:04:09.380 --> 00:04:11.480
You try to graph this, the
y-intercept is at negative

00:04:11.480 --> 00:04:13.120
2.5, right there.

00:04:13.120 --> 00:04:14.550
The slope is 2.

00:04:14.550 --> 00:04:16.400
So it's going to be this
exact same line.

00:04:20.190 --> 00:04:21.320
And you saw that
algebraically.

00:04:21.320 --> 00:04:22.670
I didn't have to graph it.

00:04:22.670 --> 00:04:25.370
These two lines have the exact
same equation when you put

00:04:25.370 --> 00:04:28.460
them in slope-intercept form.

00:04:28.460 --> 00:04:30.160
That's the first equation.

00:04:30.160 --> 00:04:31.850
That's the second equation.

00:04:31.850 --> 00:04:34.460
So what this 0 equals 0 is
telling us is actually that

00:04:34.460 --> 00:04:35.815
these are the same line.

00:04:35.815 --> 00:04:38.680
That these actually have an
infinite number of solutions.

00:04:38.680 --> 00:04:42.310
Any point on this line, which
is both of those lines, will

00:04:42.310 --> 00:04:44.550
satisfy both of these
equations.

00:04:44.550 --> 00:04:47.290
You give me an arbitrary y,
solve for x in the top

00:04:47.290 --> 00:04:49.930
equation, that x
and y will also

00:04:49.930 --> 00:04:51.720
satisfy the bottom equation.

00:04:51.720 --> 00:04:53.970
So this actually has an infinite
number of solutions.

00:04:53.970 --> 00:04:56.490
These are the same line.


Example 2: Solving systems by elimination

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.390
We have this system
of equations here.

00:00:02.390 --> 00:00:04.660
3t plus 4g is equal to 6.

00:00:04.660 --> 00:00:07.860
And we have negative 6t plus
g is also equal to 6.

00:00:07.860 --> 00:00:09.120
And there's a bunch
of ways you can

00:00:09.120 --> 00:00:10.570
solve systems of equations.

00:00:10.570 --> 00:00:11.510
You can graph them.

00:00:11.510 --> 00:00:13.510
You can solve by substitution.

00:00:13.510 --> 00:00:17.350
Whenever you see the
coefficients on one of the

00:00:17.350 --> 00:00:20.330
terms, in this case the
t term, they're almost

00:00:20.330 --> 00:00:20.890
cancelable.

00:00:20.890 --> 00:00:24.780
And when I say cancelable, if
I add 3t to negative 6t, you

00:00:24.780 --> 00:00:26.200
won't be able to cancel
them out.

00:00:26.200 --> 00:00:27.560
The t won't disappear.

00:00:27.560 --> 00:00:29.950
But if I were to scale the 3 up,
if I were to multiply it

00:00:29.950 --> 00:00:34.070
by 2 and it becomes 6t, and then
I were to add these two

00:00:34.070 --> 00:00:36.310
things, then they would
cancel out.

00:00:36.310 --> 00:00:38.420
And I would be left
just with g's.

00:00:38.420 --> 00:00:39.690
So let's try to do that.

00:00:39.690 --> 00:00:43.460
And now remember, I can't just
multiply this 3t by 2.

00:00:43.460 --> 00:00:46.240
In order for this equation to
hold true, anything I do to

00:00:46.240 --> 00:00:48.870
the left-hand side, I have to
do to the right-hand side as

00:00:48.870 --> 00:00:51.550
well, and I have to do it to
the entire left-hand side.

00:00:51.550 --> 00:00:55.570
So let me multiply this
equation by 2.

00:00:55.570 --> 00:00:56.940
So I'm multiplying it by 2.

00:00:56.940 --> 00:00:57.780
So let me just write it here.

00:00:57.780 --> 00:01:01.980
So I'm going to multiply
2 times 3t plus 4g is

00:01:01.980 --> 00:01:04.870
equal to 2 times 6.

00:01:04.870 --> 00:01:06.830
Anything I do to one side, I
have to do to the other side.

00:01:06.830 --> 00:01:08.610
The equality still holds true.

00:01:08.610 --> 00:01:15.840
So 2 times 3t is 6t plus 2 times
4g is 8g, is equal to 2

00:01:15.840 --> 00:01:17.680
times 6, which is 12.

00:01:17.680 --> 00:01:20.340
So I really just rewrote the
same information, the same

00:01:20.340 --> 00:01:21.740
constraint in this
first equation.

00:01:21.740 --> 00:01:24.450
I just multiplied
both sides by 2.

00:01:24.450 --> 00:01:26.640
Now, let me write the second
equation right below it.

00:01:26.640 --> 00:01:27.990
I'll write it in orange.

00:01:27.990 --> 00:01:33.450
So the second equation right
here is negative 6t plus g is

00:01:33.450 --> 00:01:35.450
equal to 6.

00:01:35.450 --> 00:01:37.510
Now, think about what
happens if I were to

00:01:37.510 --> 00:01:38.740
add these two equations.

00:01:38.740 --> 00:01:40.840
Remember, I can do that because
I'm essentially adding

00:01:40.840 --> 00:01:43.120
the same thing to both sides
of this top equation.

00:01:43.120 --> 00:01:44.720
Or you could say I'm adding the
same thing to both sides

00:01:44.720 --> 00:01:47.120
of this bottom equation because
the other equation is

00:01:47.120 --> 00:01:48.010
an equality.

00:01:48.010 --> 00:01:50.860
This negative 6t plus
g, it is 6.

00:01:50.860 --> 00:01:53.070
So if I'm adding 6 to 12, I'm
really adding the same

00:01:53.070 --> 00:01:54.800
quantity to the left-hand
side.

00:01:54.800 --> 00:01:56.180
That's why I can do it.

00:01:56.180 --> 00:01:58.640
So let's add the left-hand
sides together.

00:01:58.640 --> 00:02:00.870
When we do that, the
6t's cancel out.

00:02:00.870 --> 00:02:03.390
That was the whole point behind
multiplying this first

00:02:03.390 --> 00:02:07.480
character by 2, so that the 3t
becomes a 6t, and we're left

00:02:07.480 --> 00:02:14.440
with 8g plus g is 9g, is equal
to 12 plus 6, which is 18.

00:02:14.440 --> 00:02:19.040
Divide both sides by 9, and you
are left with g is equal

00:02:19.040 --> 00:02:21.780
to 18/9, or 2.

00:02:21.780 --> 00:02:23.590
So we've solved for g.

00:02:23.590 --> 00:02:26.310
Now we can substitute back and
solve for t, and we can use

00:02:26.310 --> 00:02:27.440
either of these equations.

00:02:27.440 --> 00:02:29.610
Let's use the second equation
right here.

00:02:29.610 --> 00:02:34.920
So we have negative 6t plus g--
we just solved for g; g is

00:02:34.920 --> 00:02:39.610
2-- is equal to 6.

00:02:39.610 --> 00:02:43.360
And we can subtract 2 from both
sides of this equation.

00:02:43.360 --> 00:02:47.190
Subtracting 2, the left-hand
side of the equation, that

00:02:47.190 --> 00:02:48.180
cancels out.

00:02:48.180 --> 00:02:54.460
You have negative 6t is equal
to 6 minus 2 is 4.

00:02:54.460 --> 00:02:59.920
Now I can divide both sides of
this equation by negative 6.

00:02:59.920 --> 00:03:04.300
And we're left with t is equal
to-- what is this?

00:03:04.300 --> 00:03:07.550
t is equal to negative 2/3.

00:03:07.550 --> 00:03:08.240
And we're done.

00:03:08.240 --> 00:03:11.280
We've solved for a t and a g
that satisfy both equations.

00:03:11.280 --> 00:03:13.340
We just saw that it satisfies
the bottom one.

00:03:13.340 --> 00:03:15.460
If you want to feel good that
it satisfies the top,

00:03:15.460 --> 00:03:18.390
substitute them back
into the top.

00:03:18.390 --> 00:03:18.970
Let's do that.

00:03:18.970 --> 00:03:25.310
3 times the t we got, negative
2/3, plus 4 times the g we

00:03:25.310 --> 00:03:26.890
got, so plus 4 times 2.

00:03:26.890 --> 00:03:28.050
Let's see what that is.

00:03:28.050 --> 00:03:30.650
3 times negative 2/3,
that's negative 2.

00:03:30.650 --> 00:03:32.040
The 3's cancel out.

00:03:32.040 --> 00:03:34.150
Plus 4 times 2 is 8.

00:03:34.150 --> 00:03:37.310
Negative 2 plus 8
is equal to 6.

00:03:37.310 --> 00:03:39.390
And that's exactly what that
first equation got us.

00:03:39.390 --> 00:03:44.010
So these two values definitely
satisfy both equations.


Systems of equations word problems example 1

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:04.620
An electronics warehouse ships
televisions and DVD players in

00:00:04.620 --> 00:00:06.640
certain combinations
to retailers

00:00:06.640 --> 00:00:08.310
throughout the country.

00:00:08.310 --> 00:00:11.660
They tell us that the weight
of 3 televisions and 5 DVD

00:00:11.660 --> 00:00:16.160
players is 62.5 pounds, and the
weight of 3 televisions

00:00:16.160 --> 00:00:18.040
and 2 DVD players-- so they're
giving us different

00:00:18.040 --> 00:00:20.940
combinations-- is 52 pounds.

00:00:20.940 --> 00:00:22.200
Create a system of
equations that

00:00:22.200 --> 00:00:23.810
represents this situation.

00:00:23.810 --> 00:00:27.770
Then solve it to find out how
much each television and DVD

00:00:27.770 --> 00:00:28.990
player weighs.

00:00:28.990 --> 00:00:31.280
Well, the two pieces of
information they gave us in

00:00:31.280 --> 00:00:34.760
each of these statements can be
converted into an equation.

00:00:34.760 --> 00:00:40.280
The first one is is that the
weight of 3 televisions and 5

00:00:40.280 --> 00:00:43.780
DVD players is 62.5 pounds.

00:00:43.780 --> 00:00:48.550
Then they told us that the
weight of 3 televisions and 2

00:00:48.550 --> 00:00:50.790
DVD players is 52 pounds.

00:00:50.790 --> 00:00:53.670
So we can translate these
directly into equations.

00:00:53.670 --> 00:00:57.810
If we let t to be the weight of
a television, and d to be

00:00:57.810 --> 00:01:00.910
the weight of a DVD player, this
first statement up here

00:01:00.910 --> 00:01:05.090
says that 3 times the weight
of a television, or 3

00:01:05.090 --> 00:01:09.860
televisions, plus 5 times the
weight of a DVD player, is

00:01:09.860 --> 00:01:13.430
going to be equal
to 62.5 pounds.

00:01:13.430 --> 00:01:16.380
That's exactly what this first
statement is telling us.

00:01:16.380 --> 00:01:20.390
The second statement, the weight
of 3 televisions and 2

00:01:20.390 --> 00:01:26.520
DVD players, so if I have 3
televisions and 2 DVD players,

00:01:26.520 --> 00:01:29.300
so the weight of 3 televisions
plus the weight of 2 DVD

00:01:29.300 --> 00:01:33.920
players, they're telling us
that that is 52 pounds.

00:01:33.920 --> 00:01:36.180
And so now we've set up the
system of equations.

00:01:36.180 --> 00:01:38.080
We've done the first part,
to create a system that

00:01:38.080 --> 00:01:39.420
represents the situation.

00:01:39.420 --> 00:01:41.070
Now we need to solve it.

00:01:41.070 --> 00:01:42.800
Now, one thing that's especially
tempting when you

00:01:42.800 --> 00:01:45.560
have two systems, and both of
them have something where, you

00:01:45.560 --> 00:01:49.510
know, you have a 3t here and
you have a 3t here, what we

00:01:49.510 --> 00:01:54.400
can do is we can multiply one of
the systems by some factor,

00:01:54.400 --> 00:01:57.910
so that if we were to add this
equation to that equation, we

00:01:57.910 --> 00:01:59.770
would get one of the terms
to cancel out.

00:01:59.770 --> 00:02:01.340
And that's what we're going
to do right here.

00:02:01.340 --> 00:02:03.750
And you can do this, you can
do this business of adding

00:02:03.750 --> 00:02:06.530
equations to each other, because
remember, when we

00:02:06.530 --> 00:02:08.870
learned this at the beginning of
algebra, anything you do to

00:02:08.870 --> 00:02:11.030
one side of an equation, if
I add 5 to one side of an

00:02:11.030 --> 00:02:13.970
equation, I have to add 5 to
another side of the equation.

00:02:13.970 --> 00:02:16.890
So if I add this business to
this side of the equation, if

00:02:16.890 --> 00:02:20.890
I add this blue stuff to the
left side of the equation, I

00:02:20.890 --> 00:02:23.320
can add this 52 to the
right-hand side, because this

00:02:23.320 --> 00:02:26.730
is saying that 52 is the same
thing as this thing over here.

00:02:26.730 --> 00:02:28.840
This thing is also 52.

00:02:28.840 --> 00:02:31.150
So if we're adding this to the
left-hand side, we're actually

00:02:31.150 --> 00:02:32.220
adding 52 to it.

00:02:32.220 --> 00:02:34.120
We're just writing it
a different way.

00:02:34.120 --> 00:02:37.150
Now, before we do that, what I
want to do is multiply the

00:02:37.150 --> 00:02:40.590
second, blue equation
by negative 1.

00:02:40.590 --> 00:02:43.100
And I want to multiply
it by negative 1.

00:02:43.100 --> 00:02:46.540
So negative 3t plus-- I could
write negative 2d is equal to

00:02:46.540 --> 00:02:47.390
negative 52.

00:02:47.390 --> 00:02:49.360
So I haven't changed the
information in this equation.

00:02:49.360 --> 00:02:51.890
I just multiplied everything
by negative 1.

00:02:51.890 --> 00:02:54.090
The reason why I did that is
because now if I add these two

00:02:54.090 --> 00:02:57.380
equations, these 3t terms
are going to cancel out.

00:02:57.380 --> 00:02:58.470
So let's do that.

00:02:58.470 --> 00:02:59.790
Let's add these two equations.

00:02:59.790 --> 00:03:02.170
And remember, all we're doing is
we're adding the same thing

00:03:02.170 --> 00:03:04.110
to both sides of this
top equation.

00:03:04.110 --> 00:03:06.700
We're adding essentially
negative 52 now, now that

00:03:06.700 --> 00:03:09.660
we've multiplied everything
by a negative 1.

00:03:09.660 --> 00:03:13.330
This negative 3t plus negative
2d is the same thing as

00:03:13.330 --> 00:03:14.680
negative 52.

00:03:14.680 --> 00:03:16.690
So let's add this left-hand
side over here.

00:03:16.690 --> 00:03:19.410
The 3t and the negative
3t will cancel out.

00:03:19.410 --> 00:03:20.660
That was the whole point.

00:03:20.660 --> 00:03:23.940
5d plus negative 2d is 3d.

00:03:23.940 --> 00:03:30.340
So you have a 3d is equal to
62.5 plus negative 52, or 62.5

00:03:30.340 --> 00:03:35.810
minus 52 is 10.5.

00:03:35.810 --> 00:03:38.445
And now we can divide both sides
of this equation by 3.

00:03:41.050 --> 00:03:44.940
And you get d is equal
to 10.5 divided by 3.

00:03:44.940 --> 00:03:46.670
So let's figure out
what that is.

00:03:46.670 --> 00:03:52.110
3 goes into 10.5-- it goes
into 10 three times.

00:03:52.110 --> 00:03:54.450
3 times 3 is 9.

00:03:54.450 --> 00:03:55.940
Subtract.

00:03:55.940 --> 00:03:56.650
Get 1.

00:03:56.650 --> 00:03:58.520
Bring down the 5.

00:03:58.520 --> 00:04:00.510
Of course, you have your decimal
point right here.

00:04:00.510 --> 00:04:03.010
3 goes into 15 five times.

00:04:03.010 --> 00:04:05.450
5 times 3 is 15.

00:04:05.450 --> 00:04:07.190
You've got to subtract,
and you get a 0.

00:04:07.190 --> 00:04:09.490
So it goes exactly 3.5 times.

00:04:09.490 --> 00:04:11.590
So the weight of a DVD player--
that's what d

00:04:11.590 --> 00:04:15.570
represents-- is 3.5 pounds.

00:04:15.570 --> 00:04:18.300
Now we can substitute back into
one of these equations up

00:04:18.300 --> 00:04:20.399
here to figure out the weight
of a television.

00:04:20.399 --> 00:04:22.520
We can just use that
top equation.

00:04:22.520 --> 00:04:28.790
So you get 3t plus 5 times the
weight of a DVD player, which

00:04:28.790 --> 00:04:31.010
we just figured out is 3.5.

00:04:31.010 --> 00:04:33.420
Remember, we're just looking for
values that satisfy both

00:04:33.420 --> 00:04:34.600
of these equations.

00:04:34.600 --> 00:04:41.920
So 5 times 3.5-- needs
to be equal to 62.5.

00:04:41.920 --> 00:04:45.380
So you get 3t plus-- what
is this going to be?

00:04:45.380 --> 00:04:48.850
This is going to be 15
plus 2.5, right?

00:04:48.850 --> 00:04:52.850
5 times 0.5 is 2.5,
5 times 3 is 15.

00:04:52.850 --> 00:04:58.610
So it's 17.5, is
equal to 62.5.

00:04:58.610 --> 00:05:01.815
Now we can subtract 17.5 from
both sides of this equation.

00:05:07.650 --> 00:05:08.520
And what do we get?

00:05:08.520 --> 00:05:10.960
The left-hand side is
just going to be 3t.

00:05:10.960 --> 00:05:13.690
This cancels out, that was
the whole point of it.

00:05:13.690 --> 00:05:16.240
3t is going to be equal
to-- let's see.

00:05:16.240 --> 00:05:19.060
The 0.5 minus 0.5,
that cancels out.

00:05:19.060 --> 00:05:22.830
So this is the same thing
as 62 minus 17.

00:05:22.830 --> 00:05:28.050
62 minus 7 would be 55.

00:05:28.050 --> 00:05:29.770
And so we're going to
subtract another 10.

00:05:29.770 --> 00:05:31.830
So it's going to be 45.

00:05:31.830 --> 00:05:33.980
So this is going to
be equal to 45.

00:05:33.980 --> 00:05:37.960
Now you can divide both sides
of this equation by 3.

00:05:37.960 --> 00:05:41.250
And we get t is equal to 15.

00:05:41.250 --> 00:05:42.580
So we've solved our system.

00:05:42.580 --> 00:05:46.070
The weight of a DVD player is
3.5 pounds, and the weight of

00:05:46.070 --> 00:05:49.060
a television is 15 pounds.

00:05:49.060 --> 00:05:50.630
And we're done.


Example 3: Solving systems by substitution

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.720
We're asked to solve and graph
this system of equations here.

00:00:03.720 --> 00:00:07.810
And just as a bit of a review,
solving a system of equations

00:00:07.810 --> 00:00:11.090
really just means figuring out
the x and y value that will

00:00:11.090 --> 00:00:13.910
satisfy both of these
equations.

00:00:13.910 --> 00:00:17.230
And one way to do it is to use
one of the equations to solve

00:00:17.230 --> 00:00:20.990
for either the x or the y, and
then substitute for that value

00:00:20.990 --> 00:00:21.820
in the other one.

00:00:21.820 --> 00:00:23.870
That makes sure that you're
making use of both

00:00:23.870 --> 00:00:24.780
constraints.

00:00:24.780 --> 00:00:26.830
So let's start with this bottom
equation right here.

00:00:26.830 --> 00:00:30.900
So we have y minus
x is equal to 5.

00:00:30.900 --> 00:00:33.100
It's pretty straightforward
to solve for y here.

00:00:33.100 --> 00:00:36.350
We just have to add x to both
sides of this equation.

00:00:36.350 --> 00:00:37.850
So add x.

00:00:37.850 --> 00:00:40.800
And so the left-hand side,
these x's cancel out, the

00:00:40.800 --> 00:00:43.980
negative x and the positive x,
and we're left with y is equal

00:00:43.980 --> 00:00:46.520
to 5 plus x.

00:00:49.780 --> 00:00:52.970
Now, the whole point of me doing
that, is now any time we

00:00:52.970 --> 00:00:56.340
see a y in the other equation,
we can replace it

00:00:56.340 --> 00:00:57.970
with a 5 plus x.

00:00:57.970 --> 00:01:01.010
So the other equation was-- let
me do it in this orange

00:01:01.010 --> 00:01:06.940
color-- 9x plus 3y
is equal to 15.

00:01:06.940 --> 00:01:10.020
This second equation told us, if
we just rearranged it, that

00:01:10.020 --> 00:01:14.840
y is equal to 5 plus x, so we
can replace y in the second

00:01:14.840 --> 00:01:16.740
equation with 5 plus x.

00:01:16.740 --> 00:01:19.460
That makes sure we're making
use of both constraints.

00:01:19.460 --> 00:01:20.260
So let's do that.

00:01:20.260 --> 00:01:23.100
We're going to replace
y with 5 plus x.

00:01:23.100 --> 00:01:30.380
So this 9x plus 3y equals 15
becomes 9x plus 3 times y.

00:01:30.380 --> 00:01:32.540
The second equation says
y is 5 plus x.

00:01:32.540 --> 00:01:35.510
So we're going to put 5 plus
x there instead of a y.

00:01:35.510 --> 00:01:42.020
3 times 5 plus x
is equal to 15.

00:01:42.020 --> 00:01:43.520
And now we can just
solve for x.

00:01:43.520 --> 00:01:49.970
We get 9x plus 3 times 5
is 15 plus 3 times x is

00:01:49.970 --> 00:01:53.720
3x is equal to 15.

00:01:53.720 --> 00:02:04.870
So we can add the 9x and the 3x,
so we get 12x, plus 15, is

00:02:04.870 --> 00:02:07.570
equal to 15.

00:02:07.570 --> 00:02:13.820
Now we can subtract 15 from both
sides, just so you get

00:02:13.820 --> 00:02:15.870
only x terms on the
left-hand side.

00:02:15.870 --> 00:02:19.270
These guys cancel each other
out, and you're left with 12x

00:02:19.270 --> 00:02:21.210
is equal to 0.

00:02:21.210 --> 00:02:25.480
Now you divide both sides by 12,
and you get x is equal to

00:02:25.480 --> 00:02:28.380
0/12, or x is equal to 0.

00:02:28.380 --> 00:02:30.160
So let me scroll down
a little bit.

00:02:30.160 --> 00:02:31.880
So x is equal to 0.

00:02:34.430 --> 00:02:36.520
Now if x equals 0, what is y?

00:02:36.520 --> 00:02:39.850
Well, we could substitute
into either one of these

00:02:39.850 --> 00:02:41.750
equations up here.

00:02:41.750 --> 00:02:45.610
If we substitute x equals 0 in
this first equation, you get 9

00:02:45.610 --> 00:02:51.340
times 0 plus 3y is
equal to 15.

00:02:51.340 --> 00:02:55.100
Or that's just a 0, so you
get 3y is equal to 15.

00:02:55.100 --> 00:03:00.730
Divide both sides by 3, you get
y is equal to 15/3 or 5.

00:03:00.730 --> 00:03:02.170
y is equal to 5.

00:03:02.170 --> 00:03:05.040
And we can test that that also
satisfies this equation.

00:03:05.040 --> 00:03:09.760
y, 5, minus 0 is also
equal to 5.

00:03:09.760 --> 00:03:14.720
So the value x is equal to--
I'll do this in green. x is

00:03:14.720 --> 00:03:18.690
equal to 0, y is equal
to 5 satisfies

00:03:18.690 --> 00:03:19.590
both of these equations.

00:03:19.590 --> 00:03:21.230
So we've done the first part.

00:03:21.230 --> 00:03:24.430
Let's do the second part, where
we're asked to graph it.

00:03:24.430 --> 00:03:26.510
The second equation is pretty
straightforward to graph.

00:03:26.510 --> 00:03:29.420
We actually ended up putting
it in mx plus

00:03:29.420 --> 00:03:30.650
b form right there.

00:03:30.650 --> 00:03:32.130
And actually, let
me rewrite it.

00:03:32.130 --> 00:03:34.330
Let me just switch the x and the
5, so it really is in that

00:03:34.330 --> 00:03:35.550
mx plus b form.

00:03:35.550 --> 00:03:39.710
So y is equal to x plus 5.

00:03:39.710 --> 00:03:45.185
So its y-intercept is 5--
1, 2, 3, 4, 5-- and

00:03:45.185 --> 00:03:46.690
its slope is 1, right?

00:03:46.690 --> 00:03:50.040
There's a 1 implicitly being
multiplied, or the x is being

00:03:50.040 --> 00:03:51.650
multiplied by 1.

00:03:51.650 --> 00:03:55.830
So it looks like Let me see
how well I can draw it.

00:03:55.830 --> 00:03:58.450
The line will look like that.

00:03:58.450 --> 00:03:59.640
It has a slope of 1.

00:03:59.640 --> 00:04:02.560
You move back 1,
you go down 1.

00:04:02.560 --> 00:04:04.960
You move forward
1, you go up 1.

00:04:04.960 --> 00:04:06.750
That's a pretty good job.

00:04:06.750 --> 00:04:09.670
So that right here
is this equation.

00:04:09.670 --> 00:04:11.760
Now let's graph that
top equation.

00:04:11.760 --> 00:04:14.530
And we just have to put it
in mx plus b form, or

00:04:14.530 --> 00:04:16.079
slope-intercept form.

00:04:16.079 --> 00:04:18.110
And I'll do that in green.

00:04:18.110 --> 00:04:22.620
So we have 9x plus 3y
is equal to 15.

00:04:22.620 --> 00:04:26.630
One simplification we can do
right from the get-go is every

00:04:26.630 --> 00:04:28.620
number here is divisible by
3, so let's just divide

00:04:28.620 --> 00:04:31.460
everything by 3 to make
things simpler.

00:04:31.460 --> 00:04:37.180
So we get 3x plus
y is equal to 5.

00:04:37.180 --> 00:04:39.430
Now we can subtract 3x
from both sides.

00:04:43.380 --> 00:04:48.900
We are left with y is equal
to negative 3x plus 5.

00:04:48.900 --> 00:04:51.955
So that's what this first
equation gets turned into, if

00:04:51.955 --> 00:04:54.740
you put it in slope-intercept
form. y is equal to

00:04:54.740 --> 00:04:56.850
negative 3x plus 5.

00:04:56.850 --> 00:04:59.100
So if you were to graph it,
the y-intercept is 5.

00:04:59.100 --> 00:05:00.900
0, 5.

00:05:00.900 --> 00:05:02.350
And then its slope
is negative 3.

00:05:02.350 --> 00:05:04.570
So you move 1 in the
x-direction, you move down 3

00:05:04.570 --> 00:05:05.970
in the y-direction.

00:05:05.970 --> 00:05:08.210
Move 2, you would move down 6.

00:05:08.210 --> 00:05:10.870
2, 4, 6.

00:05:10.870 --> 00:05:14.120
Move 2, you go 2, 4, 6.

00:05:14.120 --> 00:05:17.190
So this line is going to look
something like this.

00:05:17.190 --> 00:05:23.410
It's going to look something
like that right there.

00:05:23.410 --> 00:05:27.160
As you can see, the solution to
this system is the point of

00:05:27.160 --> 00:05:29.090
intersection of these
two lines.

00:05:29.090 --> 00:05:32.900
It's the combination of x and y
that satisfy both of these.

00:05:32.900 --> 00:05:36.090
Remember, this pink line, or
this red line, is all of the

00:05:36.090 --> 00:05:38.670
x's and y's that satisfy
this equation: y minus

00:05:38.670 --> 00:05:40.060
x is equal to 5.

00:05:40.060 --> 00:05:43.060
This green line is all of the
x's and y's, or all the

00:05:43.060 --> 00:05:46.810
combinations of them, that
satisfy this first equation.

00:05:46.810 --> 00:05:49.950
Now, the one x and y combination
that satisfies

00:05:49.950 --> 00:05:52.420
both is their point
of intersection.

00:05:52.420 --> 00:05:54.370
And we figured it out
algebraically using

00:05:54.370 --> 00:05:55.320
substitution.

00:05:55.320 --> 00:05:58.780
That happens at x is equal to 0,
y is equal to 5. x is equal

00:05:58.780 --> 00:06:03.600
to 0-- this is the x-axis-- y
is equal to 5 right there.


Example 2: Solving systems by substitution

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.890
We have this system of
equations, y is equal to 4x

00:00:03.890 --> 00:00:08.440
minus 17.5, and y plus
2x is equal to 6.5.

00:00:08.440 --> 00:00:10.150
And we have to solve
for x and y.

00:00:10.150 --> 00:00:12.950
So we're looking for x's
and y's that satisfy

00:00:12.950 --> 00:00:15.460
both of these equations.

00:00:15.460 --> 00:00:17.940
Now, the easiest way to think
about it is we've already

00:00:17.940 --> 00:00:19.750
solved for y in this
top equation.

00:00:19.750 --> 00:00:21.600
Let me write it again.

00:00:21.600 --> 00:00:22.800
I'll write it in pink.

00:00:22.800 --> 00:00:28.790
We have y is equal
to 4x minus 17.5.

00:00:28.790 --> 00:00:31.190
So this first equation is
telling us, literally, by this

00:00:31.190 --> 00:00:34.720
constraint, y should be
4 times x minus 17.5.

00:00:34.720 --> 00:00:37.940
Now, the second equation says
whatever y is, we had 2 times

00:00:37.940 --> 00:00:40.070
x, and that should be 6.5.

00:00:40.070 --> 00:00:43.760
Well, the y here also has to
meet this constraint up here.

00:00:43.760 --> 00:00:46.420
It also has to meet the
constraint that it has to be 4

00:00:46.420 --> 00:00:49.530
times x minus 17.5.

00:00:49.530 --> 00:00:53.390
So what we can do is, is we can
substitute this value for

00:00:53.390 --> 00:00:55.720
y into this equation.

00:00:55.720 --> 00:00:56.870
Let me be clear what
I'm doing.

00:00:56.870 --> 00:01:04.180
The second equation here is
y plus 2x is equal to 6.5.

00:01:04.180 --> 00:01:08.600
We know that y has to be equal
to this thing right here. y

00:01:08.600 --> 00:01:12.870
has to be equal to
4x minus 17.5.

00:01:12.870 --> 00:01:19.790
So let's take 4x minus 17.5,
and substitute y with that.

00:01:19.790 --> 00:01:21.320
So let's put that right there.

00:01:21.320 --> 00:01:24.240
So if we were to do that, if we
were to replace this y with

00:01:24.240 --> 00:01:27.940
4x minus 17.5, because that's
what the first equation is

00:01:27.940 --> 00:01:36.340
telling us, then we get
4x minus 17.5, plus

00:01:36.340 --> 00:01:39.810
2x is equal to 6.5.

00:01:39.810 --> 00:01:41.490
And now we have a single linear

00:01:41.490 --> 00:01:42.860
equation with one unknown.

00:01:42.860 --> 00:01:44.440
Let's solve for x.

00:01:44.440 --> 00:01:47.130
So first we have our x
terms. We have a 4x,

00:01:47.130 --> 00:01:48.320
and we have a 2x.

00:01:48.320 --> 00:01:50.210
We can group them or
add them together.

00:01:50.210 --> 00:01:53.400
4x plus 2x is 6x.

00:01:53.400 --> 00:02:03.510
And then we have 6x minus
17.5 is equal to 6.5.

00:02:03.510 --> 00:02:06.630
Then we can get the 17.5 out
of the way by adding it to

00:02:06.630 --> 00:02:07.850
both sides of the equation.

00:02:07.850 --> 00:02:13.020
So this is negative 17.5, so
let's add positive 17.5 to

00:02:13.020 --> 00:02:15.390
both sides of this equation.

00:02:15.390 --> 00:02:18.560
And we are left with the
left-hand side is just going

00:02:18.560 --> 00:02:22.020
to be 6x, because these
guys cancel out.

00:02:22.020 --> 00:02:27.480
6x is going to be equal to--
and 6.5-- see, 6 plus 17 is

00:02:27.480 --> 00:02:30.010
23, and then 0.5
plus 0.5 is 1.

00:02:30.010 --> 00:02:32.180
So this is going to be 24.

00:02:32.180 --> 00:02:34.670
And then we can divide both
sides of this equation by 6.

00:02:37.390 --> 00:02:42.250
And you are left with x is equal
to 24 over 6, which is

00:02:42.250 --> 00:02:43.720
the same thing as 4.

00:02:43.720 --> 00:02:47.480
So we figured out the x value
for the x and y pair that

00:02:47.480 --> 00:02:49.030
satisfy both of these
equations.

00:02:49.030 --> 00:02:51.860
Now we need to figure
out the y value.

00:02:51.860 --> 00:02:54.930
And we can do that by taking
this x and putting it back

00:02:54.930 --> 00:02:56.180
into one of these equations.

00:02:56.180 --> 00:02:57.320
We can do it in to either one.

00:02:57.320 --> 00:02:58.965
We should get the
same y value.

00:02:58.965 --> 00:03:01.450
So let's just do this
top one up here.

00:03:01.450 --> 00:03:04.270
So if we assume x is equal to 4,
this top equation tells us

00:03:04.270 --> 00:03:15.420
y is equal to 4 times x, which
in this case is 4, minus 17.5.

00:03:15.420 --> 00:03:21.830
Well, this is equal to 16 minus
17.5, which is equal to

00:03:21.830 --> 00:03:23.610
negative 1.5.

00:03:23.610 --> 00:03:25.880
So y is equal to negative 1.5.

00:03:25.880 --> 00:03:31.820
So the solution to this system
is x is equal to 4, y is equal

00:03:31.820 --> 00:03:34.620
to negative 1.5.

00:03:34.620 --> 00:03:37.540
And you can even verify that
these two, they definitely

00:03:37.540 --> 00:03:41.040
work for the top one if you put
4 times 4, minus 17.5, you

00:03:41.040 --> 00:03:42.320
get negative 1.5.

00:03:42.320 --> 00:03:44.180
But they also work for
the second one.

00:03:44.180 --> 00:03:45.290
And let's do that.

00:03:45.290 --> 00:03:49.900
In the second one, if you take
negative 1.5, plus 2 times x--

00:03:49.900 --> 00:03:54.070
plus 2 times 4-- what
does that equal?

00:03:54.070 --> 00:03:57.590
That's negative 1.5 plus 8.

00:03:57.590 --> 00:04:00.740
Well, negative 1.5
plus 8 is 6.5.

00:04:00.740 --> 00:04:04.970
So this x and y satisfy both
of these equations.


Example 3: Graphically solving systems

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.390
We're told to graph this
system of equations and

00:00:03.390 --> 00:00:06.040
identify the number of solutions
that it has.

00:00:06.040 --> 00:00:07.650
And they have the system
of equations here.

00:00:07.650 --> 00:00:10.480
So they want us to graph each of
these equations and think a

00:00:10.480 --> 00:00:12.260
little bit about
the solutions.

00:00:12.260 --> 00:00:15.560
So the first equation here--
I'll rewrite it, so I'll graph

00:00:15.560 --> 00:00:16.790
it in the same color
that I write it.

00:00:16.790 --> 00:00:21.000
This first equation's already in
slope-intercept form, y is

00:00:21.000 --> 00:00:23.930
equal to 3x plus 1.

00:00:23.930 --> 00:00:28.120
We see that the slope, or m, is
equal to 3, and we see that

00:00:28.120 --> 00:00:32.720
the y-intercept here
is equal to 1.

00:00:32.720 --> 00:00:35.180
So let me be clear, that
is also the slope.

00:00:35.180 --> 00:00:37.590
I just called it m because a lot
of times people say y is

00:00:37.590 --> 00:00:40.850
equal to mx plus b.

00:00:40.850 --> 00:00:41.950
So we can graph it.

00:00:41.950 --> 00:00:44.740
We can look at its y-intercept,
the point 0, 1

00:00:44.740 --> 00:00:46.000
must be on this graph.

00:00:46.000 --> 00:00:49.620
So that's the point
0, 1 right there.

00:00:49.620 --> 00:00:52.640
This is the y-axis, that
is the x-axis.

00:00:52.640 --> 00:00:54.380
And the slope is 3.

00:00:54.380 --> 00:00:57.170
That means if we move 1 in the
positive x-direction, we're

00:00:57.170 --> 00:00:59.785
going to move up 3 in the
positive y-direction.

00:00:59.785 --> 00:01:03.310
So we move 1 in the x-direction,
we move up 3.

00:01:03.310 --> 00:01:05.200
If we moved 2 in the
x-direction, we

00:01:05.200 --> 00:01:06.450
would move up 6.

00:01:09.210 --> 00:01:09.920
Just like that.

00:01:09.920 --> 00:01:12.640
Because 6 over 2 is still 3.

00:01:12.640 --> 00:01:16.200
Likewise, if we moved down 1, if
we went negative 1 in x, we

00:01:16.200 --> 00:01:18.100
would go negative 3 in y.

00:01:18.100 --> 00:01:19.700
So negative 1, negative 3.

00:01:19.700 --> 00:01:23.760
Because negative 3 divided
by negative 1 is still 3.

00:01:23.760 --> 00:01:29.950
If we went negative 2 in x, we
would go negative 6-- 1, 2, 3,

00:01:29.950 --> 00:01:33.770
4, 5, 6 in y.

00:01:33.770 --> 00:01:36.140
So these are all points along
the line, and I can connect

00:01:36.140 --> 00:01:36.920
the dots now.

00:01:36.920 --> 00:01:38.180
So let me do that.

00:01:38.180 --> 00:01:41.520
So let me connect the dots
as best as I can.

00:01:41.520 --> 00:01:44.030
This should be a line,
not a curve.

00:01:44.030 --> 00:01:47.050
My hand isn't 100% steady, but I
think you get-- let me do it

00:01:47.050 --> 00:01:47.960
a little bit better than that.

00:01:47.960 --> 00:01:50.160
I think I can do a better
job than that.

00:01:50.160 --> 00:01:52.950
Let me draw-- that's
even worse.

00:01:52.950 --> 00:01:53.350
All right.

00:01:53.350 --> 00:01:54.565
Last attempt.

00:01:54.565 --> 00:01:56.130
That's throwing me off.

00:01:56.130 --> 00:01:57.645
So last attempt right here.

00:02:01.540 --> 00:02:02.450
There you go.

00:02:02.450 --> 00:02:05.000
So that's that first line
right there. y is

00:02:05.000 --> 00:02:06.480
equal to 3x plus 1.

00:02:06.480 --> 00:02:08.650
So let me do the
second one now.

00:02:08.650 --> 00:02:13.630
So it's written in standard form
right now, 2y plus 4 is

00:02:13.630 --> 00:02:15.560
equal to 6x.

00:02:15.560 --> 00:02:19.110
We want to get this in
slope-intercept form, y is

00:02:19.110 --> 00:02:20.950
equal to mx plus b.

00:02:20.950 --> 00:02:25.070
So a good place to start could
be to subtract this 4 from

00:02:25.070 --> 00:02:25.630
both sides.

00:02:25.630 --> 00:02:26.710
So it goes on the other side.

00:02:26.710 --> 00:02:30.350
So let's subtract 4 from both
sides of this equation.

00:02:30.350 --> 00:02:34.110
The left-hand side, we're left
with just a 2y, and then the

00:02:34.110 --> 00:02:37.310
right-hand side becomes
6x minus 4.

00:02:37.310 --> 00:02:39.780
So 2y is equal to 6x minus 4.

00:02:39.780 --> 00:02:42.800
And then to get everything in
terms to solve for y, we just

00:02:42.800 --> 00:02:44.760
have to divide everything
by 2.

00:02:44.760 --> 00:02:49.980
So let's divide everything
by 2, and we get y is

00:02:49.980 --> 00:02:55.830
equal to 3x minus 2.

00:02:55.830 --> 00:02:59.290
So that's the second equation
in slope-intercept form.

00:02:59.290 --> 00:03:00.350
So same drill here.

00:03:00.350 --> 00:03:02.650
The y-intercept is negative 2.

00:03:02.650 --> 00:03:06.100
So we go-- that's negative 1,
negative 2 right there, and

00:03:06.100 --> 00:03:07.360
its slope is 3.

00:03:07.360 --> 00:03:10.450
And notice its slope is the
same as the other line.

00:03:10.450 --> 00:03:12.530
So it's going to have the
same inclination.

00:03:12.530 --> 00:03:15.970
If we move 1 in the x-direction,
we move up 3 in

00:03:15.970 --> 00:03:17.450
the y-direction.

00:03:17.450 --> 00:03:20.610
1, up 3.

00:03:20.610 --> 00:03:21.060
Just like that.

00:03:21.060 --> 00:03:23.630
If we go back 1 in
x, we go down 3.

00:03:23.630 --> 00:03:26.240
Back 1 in x, we go down 3.

00:03:26.240 --> 00:03:28.660
Just like that.

00:03:28.660 --> 00:03:30.980
So if we connect the dots
here, it'll look

00:03:30.980 --> 00:03:32.940
something like this.

00:03:32.940 --> 00:03:36.820
I'll do my best to draw
a straight line.

00:03:36.820 --> 00:03:40.530
So the second graph, 2y plus
4 equals 6, we put it into

00:03:40.530 --> 00:03:42.660
slope-intercept form
and we graphed it.

00:03:42.660 --> 00:03:46.060
Now, the whole point of this
question was to identify the

00:03:46.060 --> 00:03:48.820
number of solutions that
it has, the system.

00:03:48.820 --> 00:03:52.660
A solution to a system of
equations is an x and y value

00:03:52.660 --> 00:03:55.790
that satisfy both of
these equations.

00:03:55.790 --> 00:03:58.860
Now, if there were such an x
and y value that satisfied

00:03:58.860 --> 00:04:01.630
both of these equations, then
that x and y value would have

00:04:01.630 --> 00:04:03.820
to lie on both of
these graphs.

00:04:03.820 --> 00:04:07.800
Because this blue line is all of
the pairs of x and y's that

00:04:07.800 --> 00:04:09.690
satisfy the first equation.

00:04:09.690 --> 00:04:13.050
The red line is all of the
pairs of x's and y's that

00:04:13.050 --> 00:04:14.480
satisfy the second equation.

00:04:14.480 --> 00:04:17.390
So if something's going to
satisfy both, it's got to be

00:04:17.390 --> 00:04:18.810
on both lines.

00:04:18.810 --> 00:04:20.610
When you look here, are
there any points

00:04:20.610 --> 00:04:22.710
that are on both lines?

00:04:22.710 --> 00:04:23.130
Well, no.

00:04:23.130 --> 00:04:25.560
These two lines never
intersect.

00:04:25.560 --> 00:04:28.210
A point of intersection is a
point that is common to both

00:04:28.210 --> 00:04:29.260
of these lines.

00:04:29.260 --> 00:04:31.050
No, they don't intersect.

00:04:31.050 --> 00:04:32.300
No intersection.

00:04:34.480 --> 00:04:39.700
So there is not a solution to
this system of equations.

00:04:39.700 --> 00:04:41.520
There is no solution.

00:04:41.520 --> 00:04:46.320
We know that because these two
lines don't intersect.

00:04:46.320 --> 00:04:48.790
And you didn't even
have to graph it.

00:04:48.790 --> 00:04:51.370
The kind of giveaway was that
these are two different lines.

00:04:51.370 --> 00:04:54.230
They have different
y-intercepts, but their slopes

00:04:54.230 --> 00:04:55.410
are identical.

00:04:55.410 --> 00:04:57.650
So if you have two different
y-intercepts and your slopes

00:04:57.650 --> 00:05:00.140
are identical, then you have two
different lines that will

00:05:00.140 --> 00:05:01.140
never intersect.

00:05:01.140 --> 00:05:02.860
And if they represent a system,
or if they're the

00:05:02.860 --> 00:05:04.790
graphs of a system of
equations, that

00:05:04.790 --> 00:05:06.880
system has no solution.


Graphing linear inequalities in two variables 2 (old and redone in newer video)

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.110
We're told to graph the
inequality y is less

00:00:04.110 --> 00:00:07.040
than 3x plus 5.

00:00:07.040 --> 00:00:10.330
So you give me any x, y is going
to be less than 3 times

00:00:10.330 --> 00:00:12.040
that x plus 5.

00:00:12.040 --> 00:00:14.520
So the way to think about it is
you want to think about the

00:00:14.520 --> 00:00:18.930
line y equals 3x plus 5, but
since we're going to be less

00:00:18.930 --> 00:00:20.920
than that, instead of drawing
the line, we're going to draw

00:00:20.920 --> 00:00:23.980
it as a dotted line and then
shade all the y's that are

00:00:23.980 --> 00:00:24.760
less than that.

00:00:24.760 --> 00:00:25.730
So let me show you
what I mean.

00:00:25.730 --> 00:00:28.990
So if we wanted to just draw 3x
plus 5, if this was just an

00:00:28.990 --> 00:00:32.330
equality, if we wanted to draw
that graph-- I'll do it in

00:00:32.330 --> 00:00:36.020
this dark green color-- we'd say
OK, the y-intercept here

00:00:36.020 --> 00:00:37.220
is positive 5.

00:00:37.220 --> 00:00:40.220
So we'd go 1, 2, 3, 4, 5.

00:00:40.220 --> 00:00:43.400
And we'd say the slope is 3,
which means if we move 1 in

00:00:43.400 --> 00:00:47.770
the x direction, we move up 3.

00:00:47.770 --> 00:00:49.840
If we move back in the x
direction, we're going

00:00:49.840 --> 00:00:51.150
to move down 3.

00:00:53.730 --> 00:00:56.660
If we move back 2, we're
going to go down 6.

00:00:56.660 --> 00:01:00.590
2, 4, 6, just like that.

00:01:00.590 --> 00:01:04.970
Now, if this was graphing the
line y is equal to 3x plus 5,

00:01:04.970 --> 00:01:07.070
we would just connect
all of these dots.

00:01:07.070 --> 00:01:09.500
But because we're not less than
or equal to, we're just

00:01:09.500 --> 00:01:11.410
less than, we're going to do a
dotted line here to show that

00:01:11.410 --> 00:01:14.640
the line is actually not
going to be included.

00:01:14.640 --> 00:01:18.740
So let me draw a dotted line
here, which is actually

00:01:18.740 --> 00:01:21.650
surprisingly easier to draw
than a regular line.

00:01:24.240 --> 00:01:26.040
So that is a dotted line.

00:01:26.040 --> 00:01:27.700
We're not going to include
that line, we're going to

00:01:27.700 --> 00:01:30.540
include all the y's that are
less than 3x plus 5.

00:01:30.540 --> 00:01:32.220
So let's think about
it for second.

00:01:32.220 --> 00:01:37.180
If you give me some arbitrary
x, so let's say that x is 0.

00:01:37.180 --> 00:01:40.470
If x is 0, then what's
3x plus 5?

00:01:40.470 --> 00:01:43.170
Well, 3 times 0 plus 5 is 5.

00:01:43.170 --> 00:01:45.290
So y is going to be
less than that.

00:01:45.290 --> 00:01:48.500
If x is 0, y has to
be less than 5.

00:01:48.500 --> 00:01:52.020
So y is going to be all of
these values down here.

00:01:52.020 --> 00:01:54.390
And it would not include-- and
if you were doing it as a

00:01:54.390 --> 00:01:56.050
number line, you would do a
circle over there-- you would

00:01:56.050 --> 00:01:57.530
not include 5.

00:01:57.530 --> 00:02:02.710
If x was equal to 1,
what is 3x plus 5?

00:02:02.710 --> 00:02:06.320
3 times 1 is 3, plus 5 is 8.

00:02:06.320 --> 00:02:08.919
So y would have to be less
than 8, 8 is over here.

00:02:08.919 --> 00:02:12.230
So it's all the values less
than eight over here.

00:02:12.230 --> 00:02:15.500
So in general, for any x over
here, the y's that are less

00:02:15.500 --> 00:02:20.210
than 3x plus five are
all of the values

00:02:20.210 --> 00:02:22.800
below the dotted line.

00:02:22.800 --> 00:02:28.570
All of these values below the
dotted line is the graph of

00:02:28.570 --> 00:02:31.250
the inequality y is less
than 3x plus 5.

00:02:31.250 --> 00:02:33.360
If it was less than or equal
to, you would make that a

00:02:33.360 --> 00:02:33.970
solid line.

00:02:33.970 --> 00:02:36.670
But it's not, so it's just all
of this area down here.

00:02:36.670 --> 00:02:39.730
All of this area below this
line, that's what that less

00:02:39.730 --> 00:02:41.530
than tells you.


Solving and graphing linear inequalities in two variables 1

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.960
We're asked to represent the
inequality y is greater than 5

00:00:04.960 --> 00:00:07.790
on a number line and on
the coordinate plane.

00:00:07.790 --> 00:00:09.300
Let's do the number
line first.

00:00:09.300 --> 00:00:11.970
Let me just draw out
a number line.

00:00:11.970 --> 00:00:15.420
That's my number line, all
the possible values of y.

00:00:15.420 --> 00:00:17.560
Let's make that 0 on
the number line.

00:00:17.560 --> 00:00:21.000
We could obviously go into
negative numbers, but we're

00:00:21.000 --> 00:00:22.750
going to be greater than
5, so I'll focus on

00:00:22.750 --> 00:00:23.620
the positive side.

00:00:23.620 --> 00:00:31.940
So let's say that's 1, 2, 3,
4, 5, and then 6, 7, so

00:00:31.940 --> 00:00:34.000
forth and so on.

00:00:34.000 --> 00:00:37.350
This number line represents y,
and y is going to be greater

00:00:37.350 --> 00:00:40.160
than 5, not greater
than or equal to.

00:00:40.160 --> 00:00:44.790
So we're not going to be
including 5 in the numbers

00:00:44.790 --> 00:00:46.240
that can be y.

00:00:46.240 --> 00:00:48.790
So we're not going to include
5, so we're going to do an

00:00:48.790 --> 00:00:52.530
open circle around 5, and all
of the other values greater

00:00:52.530 --> 00:00:55.470
than 5 will be included.

00:00:55.470 --> 00:00:58.990
So if there was a greater than
or equal to sign, we would

00:00:58.990 --> 00:01:00.830
have filled it in, but since
it's just greater than, we're

00:01:00.830 --> 00:01:02.330
not including the 5.

00:01:02.330 --> 00:01:04.360
So we've represented it
on the number line.

00:01:04.360 --> 00:01:06.900
Let's do the same thing on
the coordinate plane.

00:01:06.900 --> 00:01:10.010
Let me draw a coordinate
plane here.

00:01:10.010 --> 00:01:11.620
I'm just using the standard
convention.

00:01:11.620 --> 00:01:14.760
That is my y-axis right there.

00:01:14.760 --> 00:01:17.540
And then the horizontal axis,
I'll just assume is my x-axis.

00:01:21.700 --> 00:01:24.740
Let me draw some y values,
positive y values.

00:01:24.740 --> 00:01:27.960
1, 2, 3, 4, 5.

00:01:27.960 --> 00:01:30.910
That is 5 right there, and you
go 6, 7, you can just keep

00:01:30.910 --> 00:01:33.180
going into larger and
larger numbers.

00:01:33.180 --> 00:01:36.250
And we want y to be greater than
5, so it's not going to

00:01:36.250 --> 00:01:37.770
be greater than or equal to.

00:01:37.770 --> 00:01:39.920
So we're not going
to include 5.

00:01:39.920 --> 00:01:44.170
So at 5, at y is equal to 5,
we will draw a dotted line.

00:01:44.170 --> 00:01:51.310
That shows that we're not
including y is equal to 5, but

00:01:51.310 --> 00:01:54.750
we want include all of the other
values greater than 5.

00:01:54.750 --> 00:01:56.580
So that we will shade in.

00:01:56.580 --> 00:02:02.540
So here we have shaded in all of
the values greater than 5.

00:02:02.540 --> 00:02:05.070
If it was greater than or equal
to 5, we would have

00:02:05.070 --> 00:02:07.590
drawn a bold line over here.

00:02:07.590 --> 00:02:11.610
So no matter what x is, no
matter what x we pick, y is

00:02:11.610 --> 00:02:14.310
going to be greater than 5.


Constructing, solving two-step inequality example

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:04.230
A popular R&B band recently
returned from a successful

00:00:04.230 --> 00:00:06.860
three-city tour where
they played to at

00:00:06.860 --> 00:00:11.500
least 120,000 people.

00:00:11.500 --> 00:00:15.140
My brain immediately says that's
greater than or equal

00:00:15.140 --> 00:00:17.180
to 120,000.

00:00:17.180 --> 00:00:20.600
If they had an audience of
45,000 in Mesa and another

00:00:20.600 --> 00:00:23.950
33,000 in Denver, how
many people attended

00:00:23.950 --> 00:00:26.090
their show in Las Vegas?

00:00:26.090 --> 00:00:29.170
So let's say Las Vegas, I'll
just use l for Las Vegas.

00:00:29.170 --> 00:00:31.330
So the number of people who
attended their show in Las

00:00:31.330 --> 00:00:35.080
Vegas plus the number that
attended their show in Mesa,

00:00:35.080 --> 00:00:40.100
which is 45,000, plus the number
of people that attended

00:00:40.100 --> 00:00:43.020
their show in Denver, which is
33,000-- those are three

00:00:43.020 --> 00:00:47.210
cities right there, Las Vegas,
Mesa, and Denver-- that has to

00:00:47.210 --> 00:00:49.390
be at least 120,000 people.

00:00:49.390 --> 00:00:52.410
Or another way of interpreting
that is greater than or equal

00:00:52.410 --> 00:00:55.890
to 120,000.

00:00:55.890 --> 00:00:58.040
So to figure out how many people
attended their show in

00:00:58.040 --> 00:01:01.910
Las Vegas, we just solve for
l on this inequalty.

00:01:01.910 --> 00:01:06.960
So if we simplify this left-hand
side, we get the

00:01:06.960 --> 00:01:11.060
number of people in Las Vegas
plus-- what's 45,000 plus

00:01:11.060 --> 00:01:17.630
33,000, that is 78,000-- 78,000
is going to be greater

00:01:17.630 --> 00:01:21.520
than or equal to 120,000.

00:01:21.520 --> 00:01:23.830
Now to isolate the l on the
left-hand side of the

00:01:23.830 --> 00:01:27.630
inequality, we can subtract
78,000 from both sides.

00:01:27.630 --> 00:01:34.790
So minus 78,000, minus 78,000
on the left-hand side, these

00:01:34.790 --> 00:01:35.920
cancel out.

00:01:35.920 --> 00:01:37.870
And we're just left with the
number of people who attended

00:01:37.870 --> 00:01:42.440
the show in Las Vegas is going
to be greater than or equal to

00:01:42.440 --> 00:01:45.990
120,000 minus 78,000.

00:01:45.990 --> 00:01:49.810
So 120,000 minus 80,000 is
40,000, and it's going to be

00:01:49.810 --> 00:01:50.880
another 2,000.

00:01:50.880 --> 00:01:52.690
So the number of people who
attended Las Vegas is going to

00:01:52.690 --> 00:01:57.880
be greater than or equal
to 42,000 people.

00:01:57.880 --> 00:02:00.310
And we're done, that's it