Interpreting linear graphs word problems example 2

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.680
Jordan is a cat enthusiast.

00:00:03.680 --> 00:00:06.190
She currently cares
for several cats,

00:00:06.190 --> 00:00:09.020
but she just can't
say no to adopting

00:00:09.020 --> 00:00:11.500
another cute little face.

00:00:11.500 --> 00:00:13.710
After moving into
a bigger house,

00:00:13.710 --> 00:00:17.190
Jordan started adopting
cats at a constant rate.

00:00:17.190 --> 00:00:20.180
The number of cats
in Jordan's house, C,

00:00:20.180 --> 00:00:23.410
as a function of time
since she moved in days, T,

00:00:23.410 --> 00:00:24.650
is plotted below.

00:00:24.650 --> 00:00:27.717
So this Jordan, she just
keeps adopting cats.

00:00:27.717 --> 00:00:30.300
Her house is just getting more
and more and more full of cats.

00:00:30.300 --> 00:00:32.040
I guess that's got to
stop at some point.

00:00:32.040 --> 00:00:33.539
But for this question,
they just say

00:00:33.539 --> 00:00:36.270
she keeps adopting them
at a constant rate.

00:00:36.270 --> 00:00:38.160
So let's look at
the choices here

00:00:38.160 --> 00:00:40.290
and see which of these
statements is true.

00:00:40.290 --> 00:00:42.850
Jordan had 10 cats
when she moved.

00:00:42.850 --> 00:00:45.970
So let's see this is time
from when she moved in.

00:00:45.970 --> 00:00:49.220
So time equals 0 is right
when she moved in her house.

00:00:49.220 --> 00:00:52.429
And it does look like,
yes, she had 10 cats right

00:00:52.429 --> 00:00:53.220
where she moved in.

00:00:53.220 --> 00:00:55.090
So that first part of
the statement is true.

00:00:55.090 --> 00:00:58.730
And after moving, she
adopted 2 cats each day.

00:00:58.730 --> 00:01:01.170
So let's see the rate at
which she's adopting cats.

00:01:01.170 --> 00:01:03.470
And I'm going to find
another point on here.

00:01:03.470 --> 00:01:06.830
So we know that the point
0 comma 10 is on here.

00:01:06.830 --> 00:01:10.220
So on day 0, when time
equals 0, she already

00:01:10.220 --> 00:01:14.292
had 10 cats, which seems to
be more than enough for me.

00:01:14.292 --> 00:01:15.750
But let's see how
many she gets to.

00:01:15.750 --> 00:01:17.860
So all of these other
ones, it's hard to read.

00:01:17.860 --> 00:01:21.740
Well, it looks like here,
this is definitely on day 15.

00:01:21.740 --> 00:01:24.520
On day 15, she had 40 cats.

00:01:24.520 --> 00:01:25.120
So let's see.

00:01:25.120 --> 00:01:33.770
After 15 days, when her
change in time was 15,

00:01:33.770 --> 00:01:36.210
what was the change
in her number of cats?

00:01:36.210 --> 00:01:39.340
Well, she went from
10 cats to 40 cats.

00:01:39.340 --> 00:01:43.660
So she went up to her change
in the number of cats--

00:01:43.660 --> 00:01:46.110
and we could call that
delta C-- is equal

00:01:46.110 --> 00:01:48.180
to-- she went from 10 to 40.

00:01:48.180 --> 00:01:50.490
Her change in cats is 30.

00:01:50.490 --> 00:01:57.500
So her change in cats over
change in time is 30 over 15.

00:01:57.500 --> 00:02:02.720
Or you could say, 2 cats
per unit time, per day.

00:02:02.720 --> 00:02:04.700
So it does look like
she got 2 cats per day.

00:02:04.700 --> 00:02:08.779
After 15 days at 2 cats a
day, she had 30 more cats.

00:02:08.779 --> 00:02:10.550
She went from 10 to 40.

00:02:10.550 --> 00:02:11.927
So this statement looks right.

00:02:11.927 --> 00:02:13.260
And this one is multiple choice.

00:02:13.260 --> 00:02:14.884
So these other ones
are probably wrong.

00:02:14.884 --> 00:02:16.390
But let's verify that.

00:02:16.390 --> 00:02:18.014
Jordan had 3 cats
when she moved.

00:02:18.014 --> 00:02:18.930
Well, that's not true.

00:02:18.930 --> 00:02:20.430
3 would have been
right around here.

00:02:20.430 --> 00:02:21.750
So that's not right.

00:02:21.750 --> 00:02:23.500
Jordan had 10 cats
when she moved.

00:02:23.500 --> 00:02:27.920
And after moving, she
adopted 1 cat every 2 days.

00:02:27.920 --> 00:02:30.020
Well, that would
be 1/2 a cat a day.

00:02:30.020 --> 00:02:31.620
And we know that
that's not the case.

00:02:31.620 --> 00:02:33.700
She adopted 2 cats a day.

00:02:33.700 --> 00:02:36.130
After 15 days, she had 30 cats.

00:02:36.130 --> 00:02:37.744
So this isn't right either.

00:02:37.744 --> 00:02:39.160
Jordan had 3 cats
where she moved.

00:02:39.160 --> 00:02:41.118
Well, we already know
that that's not the case.

00:02:41.118 --> 00:02:43.430
So we know it's
the first choice.


Constructing and interpreting a linear function

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.210
Entrance to the paint
ball court costs $5,

00:00:03.210 --> 00:00:05.600
and paint balls are
paid for separately.

00:00:05.600 --> 00:00:09.830
A ticket for entrance with 5
balls, for example, costs $8.

00:00:09.830 --> 00:00:11.330
So let's define
some variables here.

00:00:11.330 --> 00:00:15.600
Let's let p equal
the price that I'm

00:00:15.600 --> 00:00:18.570
paying for admission
and the balls.

00:00:18.570 --> 00:00:23.002
And let's let b equal the number
of balls-- number of balls.

00:00:23.002 --> 00:00:24.460
And we know that
the price is going

00:00:24.460 --> 00:00:26.126
to be a function of
the number of balls.

00:00:26.126 --> 00:00:27.790
We could make a
little table here.

00:00:27.790 --> 00:00:31.080
So let's make a table where,
for a given number of balls,

00:00:31.080 --> 00:00:33.320
let's figure out what
the price is going to be.

00:00:33.320 --> 00:00:35.300
So let's just start at 0 balls.

00:00:35.300 --> 00:00:37.420
If we have 0 balls,
we have to pay just

00:00:37.420 --> 00:00:41.430
to get into the paint ball
court, and that costs us $5.

00:00:41.430 --> 00:00:43.910
Now they tell us a ticket
for entrance with 5 balls--

00:00:43.910 --> 00:00:46.300
so now we have
entrance and 5 balls--

00:00:46.300 --> 00:00:49.340
this is going to cost $8.

00:00:49.340 --> 00:00:52.010
So let's think about what the
incremental price per ball

00:00:52.010 --> 00:00:52.510
here.

00:00:52.510 --> 00:00:54.760
And we're going to assume
that the price per ball

00:00:54.760 --> 00:00:57.500
is constant after you pay
for admission, after you

00:00:57.500 --> 00:00:58.610
pay this first $5.

00:00:58.610 --> 00:00:59.140
So What's.

00:00:59.140 --> 00:01:01.100
The price per ball?

00:01:01.100 --> 00:01:05.209
Well, when our number
of balls increase by 5,

00:01:05.209 --> 00:01:12.060
our price increases by
$3-- by $3, so plus 3.

00:01:12.060 --> 00:01:14.380
So we could say
our change in price

00:01:14.380 --> 00:01:18.460
over the change in the number
of balls is equal to our change

00:01:18.460 --> 00:01:22.800
in price is $3, when our
change in the balls are 5.

00:01:22.800 --> 00:01:25.000
So this right over
here, you could

00:01:25.000 --> 00:01:27.440
view this as your
unit price per ball--

00:01:27.440 --> 00:01:31.030
your unit rate of change
of price per ball.

00:01:31.030 --> 00:01:36.500
This is going to be $0.60
per ball-- $0.60 per ball.

00:01:36.500 --> 00:01:38.810
So let's think about
constructing an equation that

00:01:38.810 --> 00:01:42.249
represents how much we will
pay as a function of balls.

00:01:42.249 --> 00:01:44.540
And we could write it in
function notation, if we like.

00:01:44.540 --> 00:01:47.460
We could write price
as a function of balls

00:01:47.460 --> 00:01:49.219
is going to be equal to.

00:01:49.219 --> 00:01:51.010
Well, you're going to
start off just paying

00:01:51.010 --> 00:01:53.030
$5 just for getting
into the place.

00:01:53.030 --> 00:01:54.825
So you're going to
start off getting $5,

00:01:54.825 --> 00:01:56.200
and then you're
going to pay more

00:01:56.200 --> 00:01:58.140
depending on the number
of balls you get.

00:01:58.140 --> 00:02:00.552
And you're going to
pay $0.60 per ball.

00:02:00.552 --> 00:02:01.760
You see that right over here.

00:02:01.760 --> 00:02:04.400
When you've got 5
balls, you paid $3.

00:02:04.400 --> 00:02:08.259
$3 for 5 balls is the same
thing as $0.60 per ball.

00:02:08.259 --> 00:02:14.290
So you're then going to pay
$0.60 per ball after that.

00:02:14.290 --> 00:02:17.650
So this right over here is
the equation that represents--

00:02:17.650 --> 00:02:20.210
or that's a way of defining
price as a function of balls.

00:02:20.210 --> 00:02:21.560
And we can also try to graph it.

00:02:21.560 --> 00:02:22.940
So let's do that.

00:02:22.940 --> 00:02:25.830
So let's actually try
to graph it, as well.

00:02:25.830 --> 00:02:29.050
So that's our price
axis, our vertical axis.

00:02:29.050 --> 00:02:31.930
And let's say this is
the number of balls axis.

00:02:31.930 --> 00:02:36.010
So this is balls, and this
is price as a function--

00:02:36.010 --> 00:02:37.760
as a function of balls.

00:02:37.760 --> 00:02:41.189
So when the balls are 0--
let me do this in a color

00:02:41.189 --> 00:02:41.980
I haven't used yet.

00:02:41.980 --> 00:02:47.480
When the balls are 0, the price,
we've already seen, is $5.

00:02:47.480 --> 00:02:52.960
Let's say this is $5,
this is $10, this is $15.

00:02:52.960 --> 00:02:57.000
So when we have 0 balls,
we're going to pay $5.

00:02:57.000 --> 00:02:59.440
And then when you see when
we have 5 balls-- so let's

00:02:59.440 --> 00:03:02.570
say that this is 5 right over
here-- when you have 5 balls,

00:03:02.570 --> 00:03:05.260
we paid $8, which gets
us right around here.

00:03:05.260 --> 00:03:09.500
And two points define a
line, so this function

00:03:09.500 --> 00:03:13.812
is a linear function, and it
would look something like this.

00:03:13.812 --> 00:03:15.770
And if you ask, what is
the slope of this line?

00:03:15.770 --> 00:03:17.650
Well, the slope is
the rate of change

00:03:17.650 --> 00:03:21.640
of the vertical axis with
respect to the horizontal axis.

00:03:21.640 --> 00:03:22.680
So what is that?

00:03:22.680 --> 00:03:25.060
So in the horizontal
axis-- and frankly,

00:03:25.060 --> 00:03:27.140
we've already calculated this.

00:03:27.140 --> 00:03:32.070
When the horizontal axis changed
by 5, when your change in b

00:03:32.070 --> 00:03:39.450
is equal to 5, your vertical
axis-- so your vertical axis.

00:03:39.450 --> 00:03:42.020
So your change in
price was equal to $3.

00:03:42.020 --> 00:03:45.170
So your change in
price over your change

00:03:45.170 --> 00:03:47.090
in the number of
balls, which would

00:03:47.090 --> 00:03:49.150
be the change in
the vertical axis

00:03:49.150 --> 00:03:51.030
over the change in the
horizontal axis, which

00:03:51.030 --> 00:03:52.620
is the definition of slope.

00:03:52.620 --> 00:03:55.430
It's a way of defining the
inclination of this line.

00:03:55.430 --> 00:04:02.830
That is going to be equal
to 3 over 5-- 3 over 5,

00:04:02.830 --> 00:04:05.586
which is exactly what we
calculated right over here,

00:04:05.586 --> 00:04:07.710
just thinking about it in
a slightly different way.

00:04:07.710 --> 00:04:09.570
Now we've thought about
this in multiple ways.

00:04:09.570 --> 00:04:11.445
Let's actually try to
answer their questions.

00:04:11.445 --> 00:04:14.190
Let's figure out which
of these actually apply.

00:04:14.190 --> 00:04:16.870
So the first is that the
relationship is proportional.

00:04:16.870 --> 00:04:19.649
So if we had a proportional
relationship between the two,

00:04:19.649 --> 00:04:21.200
this function would
look like this.

00:04:21.200 --> 00:04:24.030
It would be p is equal
to some constant times

00:04:24.030 --> 00:04:25.260
the number of balls.

00:04:25.260 --> 00:04:27.790
And by definition, in a
proportional relationship--

00:04:27.790 --> 00:04:31.340
in a proportional relationship--
when your number of balls is 0,

00:04:31.340 --> 00:04:32.262
price would be 0.

00:04:32.262 --> 00:04:33.970
And that's not the
scenario we have here.

00:04:33.970 --> 00:04:37.030
When your balls are 0,
you're still paying $5.

00:04:37.030 --> 00:04:40.030
So this, this is
not-- this is not--

00:04:40.030 --> 00:04:43.060
a proportional relationship.

00:04:43.060 --> 00:04:47.500
Next statement-- entrance
with 10 balls cost $16.

00:04:47.500 --> 00:04:48.860
Well, we can test that out.

00:04:48.860 --> 00:04:53.470
When b equals 10-- So the price
for 10 balls is going to be

00:04:53.470 --> 00:04:58.070
equal to 5 plus 0.60 times 10.

00:04:58.070 --> 00:05:01.670
Well, this is 5 plus 6,
which is equal to 11, not 10.

00:05:01.670 --> 00:05:03.310
So this isn't right, either.

00:05:03.310 --> 00:05:05.930
When the number of
balls increases by 11,

00:05:05.930 --> 00:05:10.950
the price increases by $6.60.

00:05:10.950 --> 00:05:13.890
So remember, the change
in price over the change

00:05:13.890 --> 00:05:16.110
in the number of
balls is going to be

00:05:16.110 --> 00:05:18.311
equal to-- let we get it
right in a good color here.

00:05:18.311 --> 00:05:19.310
So let me write it here.

00:05:19.310 --> 00:05:22.900
The change in price over the
change in the number of balls

00:05:22.900 --> 00:05:25.880
is always going to
be equal to 3/5.

00:05:25.880 --> 00:05:31.880
And so, they say number
of balls increases by 11.

00:05:31.880 --> 00:05:36.730
So 3/5 is equal to what over 11?

00:05:36.730 --> 00:05:38.820
So let's think about
that a little bit.

00:05:38.820 --> 00:05:42.570
So 3/5 is equal to-- well,
let's write it this way.

00:05:42.570 --> 00:05:44.940
3/5 is the same
thing as 0.6, so let

00:05:44.940 --> 00:05:49.176
me write this-- 0.6 is equal
to the price change over 11.

00:05:49.176 --> 00:05:51.050
And I'll just write that
as a question mark--

00:05:51.050 --> 00:05:53.060
as a question mark over 11.

00:05:53.060 --> 00:05:55.430
You multiply both
sides by 11, you

00:05:55.430 --> 00:05:58.550
get 6.6 is equal
to question mark.

00:05:58.550 --> 00:05:59.860
So this is absolutely true.

00:05:59.860 --> 00:06:04.420
The price increases by $6.60
when the number of balls

00:06:04.420 --> 00:06:05.540
increases by 11.

00:06:05.540 --> 00:06:08.240
So this one right over
here is absolutely true.

00:06:08.240 --> 00:06:10.430
And then finally, when
the x-axis represents

00:06:10.430 --> 00:06:12.330
the number of paint
balls, the slope

00:06:12.330 --> 00:06:16.020
of the graph of the
relationship is 5/3.

00:06:16.020 --> 00:06:17.980
Well here, we did
represent the x-axis

00:06:17.980 --> 00:06:20.380
as the number of paint
balls-- or the horizontal axis

00:06:20.380 --> 00:06:23.100
as the number of paint balls--
but our slope wasn't 5/3.

00:06:23.100 --> 00:06:24.600
It was 3/5.

00:06:24.600 --> 00:06:26.980
So this last statement
is absolutely not right.

00:06:26.980 --> 00:06:30.880
So this is the only one
that we can say applies.


Constructing a linear function word problem

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.100
Archimedes is
draining his bathtub.

00:00:03.100 --> 00:00:06.945
Every 2 minutes that pass, 7
gallons of water are drained.

00:00:06.945 --> 00:00:09.320
Which of the following functions
can represent the number

00:00:09.320 --> 00:00:11.910
of gallons of water
in the tub, W,

00:00:11.910 --> 00:00:14.690
as a function of the
minutes that have passed, T?

00:00:14.690 --> 00:00:16.430
So let's think
about this visually.

00:00:16.430 --> 00:00:20.540
So let's plot the number of
gallons of water we have.

00:00:20.540 --> 00:00:22.080
Let's put that in
the vertical axis.

00:00:22.080 --> 00:00:23.320
That is a function of time.

00:00:23.320 --> 00:00:27.210
As time passes, we will
lose water at a fixed rate.

00:00:27.210 --> 00:00:30.490
So this is our time
axis right over here.

00:00:30.490 --> 00:00:33.980
So we're going to start with
some initial volume of water

00:00:33.980 --> 00:00:34.600
in the tub.

00:00:34.600 --> 00:00:36.099
And they actually
don't tell us what

00:00:36.099 --> 00:00:37.670
our initial volume of water is.

00:00:37.670 --> 00:00:43.770
But let's just call
that capital W sub 0.

00:00:43.770 --> 00:00:47.424
So this is just the
initial volume of water.

00:00:47.424 --> 00:00:48.840
We're doing
everything in gallons,

00:00:48.840 --> 00:00:55.500
so let's say the initial
number of gallons

00:00:55.500 --> 00:00:57.860
or the number of gallons
at time equals 0.

00:00:57.860 --> 00:01:00.210
This is T equals
zero right over here.

00:01:00.210 --> 00:01:04.050
Then they tell us it every 2
minutes that pass-- so time

00:01:04.050 --> 00:01:07.230
is going to be given in minutes.

00:01:07.230 --> 00:01:16.345
Time is in minutes, and the W is
in gallons-- 7 gallons of water

00:01:16.345 --> 00:01:17.010
are drained.

00:01:17.010 --> 00:01:21.820
So let's say 2 minutes
pass from time equals 0.

00:01:21.820 --> 00:01:26.740
So this is 1 minute, and this
is 2 minutes right over here.

00:01:26.740 --> 00:01:28.459
So we have 2 minutes passing.

00:01:28.459 --> 00:01:30.250
And how much water is
going to get drained?

00:01:30.250 --> 00:01:33.270
Well they tell 7 gallons
of water are drained.

00:01:33.270 --> 00:01:38.130
So after 2 minutes, we're
going to get to W0 minus 7.

00:01:38.130 --> 00:01:39.830
However much water
we had before,

00:01:39.830 --> 00:01:42.010
we're going to have
7 gallons less.

00:01:42.010 --> 00:01:45.390
W0 minus 7 is where we go.

00:01:45.390 --> 00:01:48.590
And so if we were to plot
the amount of gallons

00:01:48.590 --> 00:01:51.270
or the number of gallons we
have as a function of time,

00:01:51.270 --> 00:01:52.810
it would look
something like this.

00:01:52.810 --> 00:02:00.114
It would look like a line
that goes something like that.

00:02:00.114 --> 00:02:01.780
So one thing that we
could ask ourselves

00:02:01.780 --> 00:02:06.110
is, what is the rate of
change of W with respect to T?

00:02:06.110 --> 00:02:08.690
What is the rate of change of
our vertical axis with respect

00:02:08.690 --> 00:02:09.800
to our horizontal axis?

00:02:09.800 --> 00:02:11.970
And that's also known as
the slope of our line.

00:02:14.890 --> 00:02:18.580
So we could write that
as the rate of change--

00:02:18.580 --> 00:02:23.510
this Delta symbol, this
Greek letter, this triangle,

00:02:23.510 --> 00:02:26.830
is just shorthand for change
in-- so our change in W

00:02:26.830 --> 00:02:32.530
for a given change
in time, well we

00:02:32.530 --> 00:02:36.120
know that when time increases
by 2 minutes-- and time

00:02:36.120 --> 00:02:40.540
is in minutes-- W
decreases by 7 gallons.

00:02:40.540 --> 00:02:42.884
Because it decreases,
it's going down.

00:02:42.884 --> 00:02:43.550
And we see that.

00:02:43.550 --> 00:02:45.810
This is a downward sloping
line right over here.

00:02:45.810 --> 00:02:47.900
And we see it graphically
right over here.

00:02:47.900 --> 00:02:56.300
When our change in time
is equal to positive 2,

00:02:56.300 --> 00:03:02.790
our change in volume-- so
this is this distance, which

00:03:02.790 --> 00:03:05.670
is the same thing as this
distance right over here--

00:03:05.670 --> 00:03:11.200
our change in the number of
gallons is equal to negative 7.

00:03:11.200 --> 00:03:13.430
So this right over here
is the slope of our line

00:03:13.430 --> 00:03:15.850
or the rate of change
of the number of gallons

00:03:15.850 --> 00:03:17.900
with respect to time.

00:03:17.900 --> 00:03:20.470
Now, we don't know the
exact starting position

00:03:20.470 --> 00:03:21.870
or the exact W intercept.

00:03:21.870 --> 00:03:23.640
But we just called it W sub 0.

00:03:23.640 --> 00:03:25.970
So we could write a
general equation here.

00:03:25.970 --> 00:03:29.750
We could say that
the number of gallons

00:03:29.750 --> 00:03:31.830
is equal to the
initial gallons that we

00:03:31.830 --> 00:03:38.710
have plus negative
7/2 times time.

00:03:38.710 --> 00:03:46.230
Or we could just
call this minus 7/2--

00:03:46.230 --> 00:03:51.890
that's just the negative 7/2
right over there-- times time.

00:03:51.890 --> 00:03:53.560
And this should make
complete sense now.

00:03:53.560 --> 00:03:57.990
We have our vertical
axis intercept, often

00:03:57.990 --> 00:03:59.050
called the y-intercept.

00:03:59.050 --> 00:04:02.620
But this isn't called
the y-axis anymore.

00:04:02.620 --> 00:04:05.710
We have our vertical intercept.

00:04:05.710 --> 00:04:08.550
And we also have the
slope of our line here.

00:04:08.550 --> 00:04:11.590
If you're familiar with
slope-intercept form,

00:04:11.590 --> 00:04:15.760
we'd normally write y is
equal to mx plus b, where this

00:04:15.760 --> 00:04:18.241
is the slope and this
is the y-intercept.

00:04:18.241 --> 00:04:19.740
We've written it
the exact same way.

00:04:19.740 --> 00:04:24.980
We've said W is equal to
negative 7/2 T plus W0.

00:04:24.980 --> 00:04:26.980
I just swapped these
two things around.

00:04:26.980 --> 00:04:30.780
And notice, this is the slope,
and this is the y-intercept.

00:04:30.780 --> 00:04:33.610
So which of these
match that pattern?

00:04:33.610 --> 00:04:36.271
Well let's see, I should start
with some positive number

00:04:36.271 --> 00:04:36.770
of gallons.

00:04:36.770 --> 00:04:39.980
And I'm subtracting
7/2 T. So the only one

00:04:39.980 --> 00:04:42.430
that matches this is
this one right over here,

00:04:42.430 --> 00:04:47.159
is 50 minus 7/2 T. Now even
if you didn't think anything

00:04:47.159 --> 00:04:49.700
about slopes and you were just
looking at these choices here,

00:04:49.700 --> 00:04:51.116
you would say,
well look, I should

00:04:51.116 --> 00:04:53.240
start with a certain
number of gallons of water,

00:04:53.240 --> 00:04:55.110
and then it should
only decreases

00:04:55.110 --> 00:04:57.020
as the amount of time passes.

00:04:57.020 --> 00:04:58.440
This one wouldn't satisfy this.

00:04:58.440 --> 00:05:00.246
At time equals 0, I
have no gallons here.

00:05:00.246 --> 00:05:01.870
That doesn't meet
the constraints here.

00:05:01.870 --> 00:05:03.286
I have a certain
number of gallons

00:05:03.286 --> 00:05:05.620
here, so this
doesn't make sense.

00:05:05.620 --> 00:05:08.530
Here at time equals
0, I have 100 gallons.

00:05:08.530 --> 00:05:11.460
But then as time passes,
that will only increase.

00:05:11.460 --> 00:05:13.332
So this is the faucet's on here.

00:05:13.332 --> 00:05:14.290
It's not being drained.

00:05:14.290 --> 00:05:15.920
So that doesn't work.

00:05:15.920 --> 00:05:18.850
Here, once again, I'm
starting with a certain number

00:05:18.850 --> 00:05:21.030
of gallons when time equals 0.

00:05:21.030 --> 00:05:22.905
And then as time
passes, it's increasing

00:05:22.905 --> 00:05:23.780
my number of gallons.

00:05:23.780 --> 00:05:25.906
So once again, the faucet
is on right over here.

00:05:25.906 --> 00:05:27.280
So once again,
the only one where

00:05:27.280 --> 00:05:29.154
I'm starting with a
certain number of gallons

00:05:29.154 --> 00:05:31.500
and then as time passes
my number of gallons

00:05:31.500 --> 00:05:33.910
are reducing-- because
this is a negative value--

00:05:33.910 --> 00:05:37.820
is this choice right over here.


Multiple rates word problem

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:02.610
Starting at home,
Umaima traveled

00:00:02.610 --> 00:00:04.850
uphill to the gift
store for 45 minutes

00:00:04.850 --> 00:00:07.030
at just 8 miles per hour.

00:00:07.030 --> 00:00:10.130
She then traveled back
home along the same path

00:00:10.130 --> 00:00:13.490
downhill at a speed
of 24 miles per hour.

00:00:13.490 --> 00:00:16.700
What is her average speed
for the entire trip from home

00:00:16.700 --> 00:00:18.600
to the gift store and back?

00:00:18.600 --> 00:00:21.510
So we're trying to figure
out her average speed

00:00:21.510 --> 00:00:23.390
for the entire trip.

00:00:23.390 --> 00:00:27.780
That's going to be equal
to the total distance

00:00:27.780 --> 00:00:31.025
that she traveled
over the total time.

00:00:37.350 --> 00:00:39.586
Well, what's the total
distance going to be?

00:00:39.586 --> 00:00:40.960
Well, the total
distance is going

00:00:40.960 --> 00:00:43.200
to be the distance
to the gift store

00:00:43.200 --> 00:00:45.630
and then the distance
back from the gift store,

00:00:45.630 --> 00:00:47.415
which are the same distances.

00:00:47.415 --> 00:00:50.040
So it's really you could say two
times the distance to the gift

00:00:50.040 --> 00:00:50.540
store.

00:01:01.200 --> 00:01:04.230
And then what is going
to be her total time?

00:01:04.230 --> 00:01:22.020
Well, it's time to gift store
plus the time coming back

00:01:22.020 --> 00:01:23.910
from the gift store.

00:01:23.910 --> 00:01:26.410
Now, we know that the
distance to the gift store

00:01:26.410 --> 00:01:28.700
and the distance back from
the gift store is the same.

00:01:28.700 --> 00:01:30.580
So that's why I just said that
the total distance is just

00:01:30.580 --> 00:01:32.999
going to be two times the
distance to the gift store.

00:01:32.999 --> 00:01:34.540
We don't know-- in
fact we know we're

00:01:34.540 --> 00:01:37.080
going to have different times
in terms of times to the gift

00:01:37.080 --> 00:01:38.550
store and times coming back.

00:01:38.550 --> 00:01:39.630
How do I know that?

00:01:39.630 --> 00:01:41.160
Well, she went at
different speeds.

00:01:41.160 --> 00:01:43.160
So it's going to
take her-- actually,

00:01:43.160 --> 00:01:45.304
she went there much
slower than she came back.

00:01:45.304 --> 00:01:46.970
So it would take her
longer to get there

00:01:46.970 --> 00:01:48.590
than it took her to get back.

00:01:48.590 --> 00:01:52.190
So let's see which of these we
can actually-- we already know.

00:01:52.190 --> 00:01:54.730
So how do we figure out the
distance to the gift store?

00:01:54.730 --> 00:01:56.850
At not point here do they
say, hey, the gift store

00:01:56.850 --> 00:01:58.360
is this far away.

00:01:58.360 --> 00:02:01.470
But they do tell us, this
first sentence right over here,

00:02:01.470 --> 00:02:04.740
"Umaima traveled uphill to
the gift store for 45 minutes

00:02:04.740 --> 00:02:10.270
at just 8 miles per hour."

00:02:10.270 --> 00:02:12.220
So we're given a time.

00:02:12.220 --> 00:02:14.900
And we are given a speed.

00:02:14.900 --> 00:02:17.480
We should be able to
figure out a distance.

00:02:17.480 --> 00:02:19.900
So let's just do a
little bit of aside here.

00:02:19.900 --> 00:02:24.330
We should be able to figure out
the distance to the-- actually

00:02:24.330 --> 00:02:26.630
let me write it this way.

00:02:26.630 --> 00:02:34.500
The distance to
the store will be

00:02:34.500 --> 00:02:37.170
equal to-- now we've got to make
sure we have our units right.

00:02:37.170 --> 00:02:38.378
Here they gave it in minutes.

00:02:38.378 --> 00:02:40.900
Here they have 8 miles per hour.

00:02:40.900 --> 00:02:44.320
So let's convert
this into hours.

00:02:44.320 --> 00:02:49.630
So 45 minutes in hours,
so it's 45 minutes

00:02:49.630 --> 00:02:51.860
out of 60 minutes per hour.

00:02:51.860 --> 00:02:54.820
So that's going
to give us 45/60.

00:02:54.820 --> 00:02:56.120
Divide both by 15.

00:02:56.120 --> 00:02:58.990
That's the same thing as 3/4.

00:02:58.990 --> 00:03:06.150
So it's going to be 3/4 hours is
the time times an average speed

00:03:06.150 --> 00:03:07.305
of 8 miles per hour.

00:03:13.880 --> 00:03:15.930
So what is the
distance to the store?

00:03:15.930 --> 00:03:17.370
Well, 3/4 times 8.

00:03:17.370 --> 00:03:19.810
Or you could view it
as 3/4 times 8 times

00:03:19.810 --> 00:03:24.280
1, is going to be-- well,
it's going to be 24 over 4.

00:03:24.280 --> 00:03:25.680
Let me just write that.

00:03:25.680 --> 00:03:30.140
That's going to
be 24 over 4 which

00:03:30.140 --> 00:03:31.900
is equal to-- did I get it?

00:03:31.900 --> 00:03:32.400
Yeah.

00:03:32.400 --> 00:03:35.650
24 over 4, which is equal to 6.

00:03:35.650 --> 00:03:39.460
And units-wise, we're
just left with miles.

00:03:39.460 --> 00:03:42.250
So the distance to
the store is 6 miles.

00:03:42.250 --> 00:03:44.080
2 times the distance
to the gift store,

00:03:44.080 --> 00:03:46.760
well, this whole thing
is going to be 12 miles.

00:03:46.760 --> 00:03:50.510
12 miles is the total
distance she traveled.

00:03:50.510 --> 00:03:52.740
Now, what is the time
to the gift store?

00:03:52.740 --> 00:03:54.360
Well, they already
told that to us.

00:03:54.360 --> 00:03:58.110
They already told us
that it's 45 minutes.

00:03:58.110 --> 00:03:59.949
Now, I want to put
everything in hours.

00:03:59.949 --> 00:04:02.240
I'm assuming that they want
our average speed in hours.

00:04:02.240 --> 00:04:04.090
So I'm going to put
everything in hours.

00:04:04.090 --> 00:04:07.650
So the time to the gift
store was 3/4 of an hour.

00:04:12.200 --> 00:04:15.020
And what's the time coming
back from the gift store?

00:04:15.020 --> 00:04:16.810
Well, we know her speed.

00:04:16.810 --> 00:04:19.668
We know her speed coming back.

00:04:19.668 --> 00:04:21.709
We already know the distance
from the gift store.

00:04:21.709 --> 00:04:23.740
It's the same as the
distance to the gift store.

00:04:23.740 --> 00:04:27.850
So we can take this distance,
we can take 6 miles, that's

00:04:27.850 --> 00:04:31.620
the distance to the gift
store, 6 miles divided

00:04:31.620 --> 00:04:35.490
by her speed coming back,
which is 24 miles per hour,

00:04:35.490 --> 00:04:40.890
so divided by 24 miles per hour.

00:04:40.890 --> 00:04:42.990
It gives us-- well, let's see.

00:04:42.990 --> 00:04:47.170
We're going to have 6 over
24 is the same thing as 1/4.

00:04:47.170 --> 00:04:49.440
It's going to be 1/4.

00:04:49.440 --> 00:04:51.590
And then miles divided
by miles per hour

00:04:51.590 --> 00:04:54.070
is the same thing as miles
times hours per mile.

00:04:54.070 --> 00:04:55.530
The miles cancel out.

00:04:55.530 --> 00:04:57.620
And you'll have 1/4 of an hour.

00:04:57.620 --> 00:04:59.720
So it takes her 1/4 of
an hour to get back.

00:04:59.720 --> 00:05:01.495
And that fits our intuition.

00:05:01.495 --> 00:05:03.620
Actually, let me write that
in the same green color

00:05:03.620 --> 00:05:07.090
since I'm writing all
the times in green color.

00:05:07.090 --> 00:05:08.920
So 1/4 of an hour.

00:05:08.920 --> 00:05:10.280
So she went there.

00:05:10.280 --> 00:05:12.040
Going to the gift
store was slow.

00:05:12.040 --> 00:05:13.260
It took her 3/4 of an hour.

00:05:13.260 --> 00:05:17.570
Coming back only took
her 1/4 of an hour.

00:05:17.570 --> 00:05:21.100
So now we're ready to
calculate her average speed

00:05:21.100 --> 00:05:22.770
for the entire trip.

00:05:22.770 --> 00:05:24.890
Average speed for
the entire trip

00:05:24.890 --> 00:05:27.010
is going to be equal to
the total distance, which

00:05:27.010 --> 00:05:33.440
is 12 miles, divided
by her total time.

00:05:33.440 --> 00:05:38.280
3/4 hours plus 1/4
hour is exactly 1 hour.

00:05:38.280 --> 00:05:41.290
So her average
speed is 12 over 1,

00:05:41.290 --> 00:05:45.880
which is just 12 miles per hour.

00:05:45.880 --> 00:05:50.249
Her average speed is
just 12 miles per hour.

00:05:50.249 --> 00:05:52.290
And you might have been
tempted to say hey, wait.

00:05:52.290 --> 00:05:53.974
Why don't I just
average 24 and 8.

00:05:53.974 --> 00:05:55.390
But that wouldn't
have been right,

00:05:55.390 --> 00:05:58.669
because she's traveling those
for different amounts of time.

00:05:58.669 --> 00:06:00.210
So what you really
have to do is just

00:06:00.210 --> 00:06:02.710
think in terms of go back to
your basics-- total distance,

00:06:02.710 --> 00:06:03.820
total time.

00:06:03.820 --> 00:06:05.160
Figure out the total distance.

00:06:05.160 --> 00:06:06.630
This first sentence
right over here

00:06:06.630 --> 00:06:08.130
gives us half of
the total distance,

00:06:08.130 --> 00:06:09.250
the time to the store.

00:06:09.250 --> 00:06:11.310
We just doubled that
to get the time back.

00:06:11.310 --> 00:06:13.460
And then our total
time we can figure out.

00:06:13.460 --> 00:06:15.140
They tell us the
time to the store.

00:06:15.140 --> 00:06:17.390
And then we can figure out
the distance from the store

00:06:17.390 --> 00:06:21.970
and using that and the speed
to figure out her time back.

00:06:21.970 --> 00:06:23.840
And then we get total
distance divided

00:06:23.840 --> 00:06:27.490
by total time--
12 miles per hour.


Systems of equations word problems example 2

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.900
You've gone to a fruit stand
to get some fresh produce.

00:00:03.900 --> 00:00:06.250
You notice that the
person in front of you

00:00:06.250 --> 00:00:11.530
gets 5 apples and
4 oranges for $10.

00:00:11.530 --> 00:00:18.170
You get 5 apples and
5 oranges for $11.

00:00:18.170 --> 00:00:20.590
Can we solve for the price
of an apple and an orange

00:00:20.590 --> 00:00:22.170
using this information
in a system

00:00:22.170 --> 00:00:24.220
of linear equations
in two variables?

00:00:24.220 --> 00:00:26.090
If yes, what is the solution?

00:00:26.090 --> 00:00:28.525
If no, what is the
reason we cannot?

00:00:28.525 --> 00:00:30.650
So we're trying to figure
out the price of an apple

00:00:30.650 --> 00:00:32.770
and the price of an orange.

00:00:32.770 --> 00:00:35.760
So I would use a for apple, but
I don't like using o for orange

00:00:35.760 --> 00:00:37.760
because o looks too
much like a zero.

00:00:37.760 --> 00:00:40.540
So I'll just say x for apples.

00:00:40.540 --> 00:00:45.333
Let's let x equal
the price of apples.

00:00:49.710 --> 00:00:54.410
And let's let y equal
the price of oranges.

00:00:58.670 --> 00:01:01.200
So let's describe what
happened to the person in line

00:01:01.200 --> 00:01:02.220
in front of us.

00:01:02.220 --> 00:01:04.080
They bought 5 apples.

00:01:04.080 --> 00:01:06.100
So how much did they
spend on apples?

00:01:06.100 --> 00:01:10.560
Well, they bought 5 apples
times x dollars per apple,

00:01:10.560 --> 00:01:13.660
so they spent 5x dollars
on their 5 apples.

00:01:13.660 --> 00:01:17.250
And they bought 4 oranges.

00:01:17.250 --> 00:01:21.420
They bought 4 oranges
times y dollars per orange.

00:01:21.420 --> 00:01:23.980
So they spent 4y on oranges.

00:01:23.980 --> 00:01:27.440
So the total amount that
they spent is 5x plus 4y.

00:01:27.440 --> 00:01:29.540
And they tell us
that this is $10.

00:01:29.540 --> 00:01:31.450
This is equal to $10.

00:01:31.450 --> 00:01:34.120
Now, you get in line,
and you buy 5 apples.

00:01:34.120 --> 00:01:37.440
So you buy 5 apples, just
like the guy in front of you.

00:01:37.440 --> 00:01:39.980
And you paid x
dollars per apple.

00:01:39.980 --> 00:01:44.060
So you're going to pay 5 apples
times the price per apple.

00:01:44.060 --> 00:01:46.300
This is the amount that
you spend on apples.

00:01:46.300 --> 00:01:49.800
And then you buy 5 oranges.

00:01:49.800 --> 00:01:53.980
So you're going to pay 5 oranges
times the price per orange,

00:01:53.980 --> 00:01:54.860
which is y.

00:01:54.860 --> 00:01:56.730
So this is how much
you spend on oranges.

00:01:56.730 --> 00:01:59.730
This is how much you spend on
apples and oranges, the sum.

00:01:59.730 --> 00:02:03.510
And they tell us
that this is $11.

00:02:03.510 --> 00:02:07.490
So can we solve
for an x and a y?

00:02:07.490 --> 00:02:09.020
And it looks like we can.

00:02:09.020 --> 00:02:11.590
And a big giveaway
right over here

00:02:11.590 --> 00:02:14.817
is the ratio between the x's and
the y's in these two equations

00:02:14.817 --> 00:02:15.400
are different.

00:02:15.400 --> 00:02:17.520
So we're getting some
information here.

00:02:17.520 --> 00:02:19.050
If the ratios were
exactly the same,

00:02:19.050 --> 00:02:21.142
if this was 5x plus
4y right over here,

00:02:21.142 --> 00:02:23.600
and we got a different number,
then we would be in trouble.

00:02:23.600 --> 00:02:24.980
Because we bought
the same combination,

00:02:24.980 --> 00:02:26.269
but we got a different price.

00:02:26.269 --> 00:02:28.810
But the good thing is is that
we have a different combination

00:02:28.810 --> 00:02:29.730
here.

00:02:29.730 --> 00:02:31.520
So let's see if we
can work it out.

00:02:31.520 --> 00:02:33.930
Now, the most obvious
thing that jumps out at me

00:02:33.930 --> 00:02:37.820
is that I have a 5x here, and
I have a 5x right over here.

00:02:37.820 --> 00:02:40.240
So if I could subtract
this 5x from that 5x,

00:02:40.240 --> 00:02:42.090
then I would cancel
out all of the x terms.

00:02:42.090 --> 00:02:43.590
So what I'm going
to do is I'm going

00:02:43.590 --> 00:02:46.190
to multiply this bottom
equation by negative 1.

00:02:46.190 --> 00:02:50.750
So it becomes negative
5x plus negative 5y

00:02:50.750 --> 00:02:52.700
is equal to negative 11.

00:02:52.700 --> 00:02:54.980
And then I'm going
to essentially add

00:02:54.980 --> 00:02:56.230
both of these equations.

00:02:56.230 --> 00:02:57.190
And I could do that
because I'm doing

00:02:57.190 --> 00:02:58.470
the same thing to both sides.

00:02:58.470 --> 00:03:01.190
I already know that this
thing is equal to this thing.

00:03:01.190 --> 00:03:04.100
So I'm just adding those
things to either side.

00:03:04.100 --> 00:03:09.620
So on the left hand
side, I have 5x minus 5x.

00:03:09.620 --> 00:03:11.280
Well, those cancel out.

00:03:11.280 --> 00:03:15.470
And then I have 4y minus 5y.

00:03:15.470 --> 00:03:17.100
Well, that's negative y.

00:03:17.100 --> 00:03:21.080
And that's going to be
equal to 10 minus 11,

00:03:21.080 --> 00:03:23.100
which is negative 1.

00:03:23.100 --> 00:03:25.370
And then if we multiply
both sides of this times

00:03:25.370 --> 00:03:27.870
negative 1, or divide
both sides by negative 1,

00:03:27.870 --> 00:03:30.046
we're going to get
y is equal to 1.

00:03:30.046 --> 00:03:31.920
So just like that, we
were able to figure out

00:03:31.920 --> 00:03:32.820
the price of oranges.

00:03:32.820 --> 00:03:35.780
It's $1 per orange.

00:03:35.780 --> 00:03:37.130
So this is equal to 1.

00:03:37.130 --> 00:03:38.940
Now let's figure out
the price of apples.

00:03:38.940 --> 00:03:41.240
So we can go back into
either 1 of these equations.

00:03:41.240 --> 00:03:43.230
I'll go back into this first 1.

00:03:43.230 --> 00:03:46.720
So 5 times-- so let's go to the
person in line in front of us.

00:03:46.720 --> 00:03:49.960
They bought 5 apples
at x dollars per apple,

00:03:49.960 --> 00:03:56.135
plus 4 oranges at $1 per orange,
and they spent a total of $10.

00:03:59.490 --> 00:04:01.310
So this of course is just 4.

00:04:01.310 --> 00:04:04.590
Let's subtract 4
from both sides,

00:04:04.590 --> 00:04:09.420
and we get-- well, 4 times 1
minus 4, that just cancels out.

00:04:09.420 --> 00:04:12.070
We're just going to be left
with 5x on the left hand side.

00:04:15.170 --> 00:04:17.920
And on the right hand
side, we have 10 minus 4,

00:04:17.920 --> 00:04:19.540
which is equal to 6.

00:04:19.540 --> 00:04:22.490
And we can just divide both
sides by 6 now in order

00:04:22.490 --> 00:04:23.790
to solve for x.

00:04:26.990 --> 00:04:27.490
Oh, sorry.

00:04:27.490 --> 00:04:31.850
We can divide both sides by
5 in order to solve for x.

00:04:31.850 --> 00:04:32.800
It's late in the day.

00:04:32.800 --> 00:04:33.634
Brain isn't working.

00:04:33.634 --> 00:04:35.383
Dividing by 6 wouldn't
have done anything.

00:04:35.383 --> 00:04:36.470
We would have gotten 5/6x.

00:04:36.470 --> 00:04:38.350
We just want to get an x here.

00:04:38.350 --> 00:04:44.240
So dividing both sides by 5, we
get x is equal to 6/5 dollars.

00:04:44.240 --> 00:04:49.520
Or you could say that
x is equal to 6/5,

00:04:49.520 --> 00:04:51.900
which is the same
thing as 1 and 1/5,

00:04:51.900 --> 00:04:54.540
which is the same
thing as $1.20.

00:04:54.540 --> 00:04:59.410
So it's $1 per orange,
and $1.20 per apple.

00:04:59.410 --> 00:05:01.500
So we absolutely
could figure out

00:05:01.500 --> 00:05:05.980
the prices of apples and oranges
using the information given.


Solving systems of two linear equations example

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.210
You are solving a system
of two linear equations

00:00:03.210 --> 00:00:04.520
in two variables.

00:00:04.520 --> 00:00:06.530
You have found more
than one solution

00:00:06.530 --> 00:00:08.270
that satisfies the system.

00:00:08.270 --> 00:00:10.184
Which of the following
statements is true?

00:00:10.184 --> 00:00:11.850
So before even reading
these statements,

00:00:11.850 --> 00:00:13.475
let's just think
about what's going on.

00:00:13.475 --> 00:00:15.580
So let me draw my axes here.

00:00:15.580 --> 00:00:16.820
Let's draw my axes.

00:00:16.820 --> 00:00:18.510
So this is going to
be my vertical axis.

00:00:18.510 --> 00:00:20.290
That could be one
of the variables.

00:00:20.290 --> 00:00:23.984
And then this is
my horizontal axis.

00:00:23.984 --> 00:00:25.400
That's one of the
other variables.

00:00:25.400 --> 00:00:27.890
And maybe, for sake of
convention, this could be x,

00:00:27.890 --> 00:00:30.900
and this could be y, but they're
whatever our two variables are.

00:00:30.900 --> 00:00:33.430
So it's a system of
two linear equations.

00:00:33.430 --> 00:00:35.920
So if we're graphing them,
each of the linear equations

00:00:35.920 --> 00:00:38.850
in two variables can be
represented by a line.

00:00:38.850 --> 00:00:41.290
Now, there's only
three scenarios here.

00:00:41.290 --> 00:00:44.856
One scenario is where the
lines don't intersect at all.

00:00:44.856 --> 00:00:46.230
So the only way
that you're going

00:00:46.230 --> 00:00:48.890
to have two lines in two
dimensions that don't intersect

00:00:48.890 --> 00:00:51.200
is if they have the
same slope and they

00:00:51.200 --> 00:00:52.634
have different y-intercepts.

00:00:52.634 --> 00:00:54.050
So that's one
scenario, but that's

00:00:54.050 --> 00:00:56.280
not the scenario that's
being described here.

00:00:56.280 --> 00:00:58.350
They say, you have found
more than one solution

00:00:58.350 --> 00:00:59.460
that satisfies the system.

00:00:59.460 --> 00:01:01.280
Here there are no solutions.

00:01:01.280 --> 00:01:03.870
So that's not the scenario
that we're talking about.

00:01:03.870 --> 00:01:05.340
There's another
scenario where they

00:01:05.340 --> 00:01:08.110
intersect in exactly one place.

00:01:08.110 --> 00:01:10.420
So they intersect in
exactly one place.

00:01:10.420 --> 00:01:13.110
There's one point, one
xy-coordinate right over there

00:01:13.110 --> 00:01:16.970
that satisfies both
of these constraints,

00:01:16.970 --> 00:01:20.330
but this also is not the
scenario they're talking about.

00:01:20.330 --> 00:01:23.640
They're telling us that you have
found more than one solution

00:01:23.640 --> 00:01:24.840
that satisfies the system.

00:01:24.840 --> 00:01:26.740
So this isn't the
scenario either.

00:01:26.740 --> 00:01:28.790
So the only other scenario
that we can have-- we

00:01:28.790 --> 00:01:30.760
don't have parallel lines
that don't intersect.

00:01:30.760 --> 00:01:33.010
We don't have lines that
only intersect in one place.

00:01:33.010 --> 00:01:36.150
The only other scenario is that
we're dealing with a situation

00:01:36.150 --> 00:01:38.430
where both linear
equations are essentially

00:01:38.430 --> 00:01:39.740
the same constraint.

00:01:39.740 --> 00:01:41.640
They both are
essentially representing

00:01:41.640 --> 00:01:44.190
the same xy-relationship.

00:01:44.190 --> 00:01:46.850
That's the only way that
I can have two lines,

00:01:46.850 --> 00:01:50.620
and this only applies to
linear relationship and lines.

00:01:50.620 --> 00:01:53.010
But the only way that
two lines can intersect

00:01:53.010 --> 00:01:58.880
more than one place is if
they intersect everywhere.

00:01:58.880 --> 00:02:00.550
So in this situation,
we know that we

00:02:00.550 --> 00:02:03.610
must have an infinite
number of solutions.

00:02:03.610 --> 00:02:05.500
So which of these
choices say that?

00:02:05.500 --> 00:02:08.250
This one right here-- "there are
infinitely many more solutions

00:02:08.250 --> 00:02:11.580
to the system"--
right over there.


Systems of equations word problems example 4

WEBVTT
Kind: captions
Language: en

00:00:01.050 --> 00:00:03.620
You are at a Parisian
cafe with a friend.

00:00:03.620 --> 00:00:07.310
A local in front of you buys a
cup of coffee and a croissant

00:00:07.310 --> 00:00:10.860
for $5 or 5.30 Euro.

00:00:10.860 --> 00:00:13.650
When you and your friend get
two cups of coffee and two

00:00:13.650 --> 00:00:17.150
croissants, you are
charged 14 Euro.

00:00:17.150 --> 00:00:20.310
Can we solve for the price of
a cup of coffee and croissant

00:00:20.310 --> 00:00:22.390
using the information
into a system

00:00:22.390 --> 00:00:24.450
of linear equations
in two variables?

00:00:24.450 --> 00:00:26.490
If yes, what is the solution?

00:00:26.490 --> 00:00:28.725
If no, what is the
reason we cannot?

00:00:28.725 --> 00:00:30.100
So we're looking
for two things--

00:00:30.100 --> 00:00:33.100
the price of a cup of coffee
and the price of a croissant.

00:00:33.100 --> 00:00:36.700
So let's define
two variables here.

00:00:36.700 --> 00:00:38.120
Since we have all
these C's here,

00:00:38.120 --> 00:00:39.536
I'm just going to
use x's and y's.

00:00:39.536 --> 00:00:46.965
So let's let x be equal to the
price of the cup of coffee.

00:00:50.160 --> 00:00:54.290
And let's let y be equal to
the price of a croissant.

00:00:59.460 --> 00:01:01.519
So we first have
this information

00:01:01.519 --> 00:01:03.060
of what the local
in front of us did.

00:01:03.060 --> 00:01:06.050
The local in front of us buys
one cup of coffee and one

00:01:06.050 --> 00:01:11.920
croissant for 5.30 Euro.

00:01:11.920 --> 00:01:14.610
So how would we set
that up as an equation?

00:01:14.610 --> 00:01:16.200
Well, we got one cup of coffee.

00:01:16.200 --> 00:01:19.050
So that's going to be one
x, or we could just write x,

00:01:19.050 --> 00:01:22.180
plus one y because
he got one croissant,

00:01:22.180 --> 00:01:32.240
and it cost 5.30 Euro.

00:01:32.240 --> 00:01:35.380
So this equation describes
what happened to the local--

00:01:35.380 --> 00:01:39.546
bought one cup of coffee, one
croissant, paid 5.30 Euro.

00:01:39.546 --> 00:01:40.920
Now, when you and
your friend get

00:01:40.920 --> 00:01:42.850
two cups of coffee
and two croissants,

00:01:42.850 --> 00:01:45.100
you are charged 14 Euro.

00:01:45.100 --> 00:01:47.530
So what's an equation
to describe this?

00:01:47.530 --> 00:01:50.240
So we should be charged
two times the price

00:01:50.240 --> 00:01:51.650
of a cup of coffee.

00:01:51.650 --> 00:01:54.020
So it should be 2x.

00:01:54.020 --> 00:01:56.450
And then we should be
charged two times the price

00:01:56.450 --> 00:01:58.652
of a croissant, so plus 2y.

00:01:58.652 --> 00:02:00.610
And the sum of these
should be the total amount

00:02:00.610 --> 00:02:01.401
that we're charged.

00:02:01.401 --> 00:02:03.374
So we've been charged 14 Euro.

00:02:07.810 --> 00:02:10.630
So let's see if we can solve
this system of equations.

00:02:10.630 --> 00:02:12.890
And there's many, many,
many ways to solve this.

00:02:12.890 --> 00:02:14.684
But the most
obvious way at least

00:02:14.684 --> 00:02:17.100
looking at this right over
here is you have x, we have 2x,

00:02:17.100 --> 00:02:18.810
we have y, we have 2y.

00:02:18.810 --> 00:02:21.660
Let's take this first
equation that described local

00:02:21.660 --> 00:02:23.870
and multiply it by two.

00:02:23.870 --> 00:02:25.290
So let's just
multiply it by two.

00:02:25.290 --> 00:02:26.873
So we're going to
multiply both sides,

00:02:26.873 --> 00:02:28.970
otherwise equality
won't hold anymore.

00:02:28.970 --> 00:02:37.940
So we would get 2x plus 2y is
equal to 2 times 5.30 is 10

00:02:37.940 --> 00:02:38.575
euro 60.

00:02:41.950 --> 00:02:45.580
Now, something very
interesting is going on here.

00:02:45.580 --> 00:02:51.660
If the local had bought twice
as many cups of coffee and twice

00:02:51.660 --> 00:02:54.145
as many croissants, he
would have paid 10.60.

00:02:54.145 --> 00:02:56.270
And that would have been
the exact amount of coffee

00:02:56.270 --> 00:02:58.710
and croissants you
got and you paid 14.

00:02:58.710 --> 00:03:00.160
So it looks pretty
clear that you

00:03:00.160 --> 00:03:02.020
got charged a different amount.

00:03:02.020 --> 00:03:04.960
You got the tourist rate for
the cup of coffee and croissant,

00:03:04.960 --> 00:03:07.130
while he got the local rate.

00:03:07.130 --> 00:03:09.280
And we can verify that
there's no x and y that's

00:03:09.280 --> 00:03:10.250
going to satisfy this.

00:03:10.250 --> 00:03:12.420
And even logically
it makes sense here.

00:03:12.420 --> 00:03:14.640
2x plus 2y is 14.

00:03:14.640 --> 00:03:18.692
Here, 2x plus 2y is 10 euro 60.

00:03:18.692 --> 00:03:20.650
And we could even show
that mathematically this

00:03:20.650 --> 00:03:22.040
doesn't make sense.

00:03:22.040 --> 00:03:25.460
So if we were to subtract this
bottom equation from this top,

00:03:25.460 --> 00:03:29.660
so essentially you could
imagine multiplying

00:03:29.660 --> 00:03:32.500
the entire bottom
equation times negative 1.

00:03:32.500 --> 00:03:35.750
So let's multiply the entire
bottom equation by negative 1

00:03:35.750 --> 00:03:37.326
and then we add
these two equations.

00:03:37.326 --> 00:03:38.700
Remember all we're
doing is we're

00:03:38.700 --> 00:03:40.620
starting with say this
equation, and we're

00:03:40.620 --> 00:03:42.530
adding the same
thing to both sides.

00:03:42.530 --> 00:03:44.360
We're going to add
this to this side.

00:03:44.360 --> 00:03:46.700
And we already know that
negative 10.60 is the same

00:03:46.700 --> 00:03:48.940
thing as this, we're
going to add to that side.

00:03:48.940 --> 00:03:51.090
So on the left hand side,
this cancels with this,

00:03:51.090 --> 00:03:53.140
this cancels with this,
we're left with 0.

00:03:53.140 --> 00:03:57.405
And on the right hand side,
14 minus 10.60 will get you

00:03:57.405 --> 00:03:59.000
to 3.40.

00:03:59.000 --> 00:04:03.820
And there's no x and y that
you can think of that can all

00:04:03.820 --> 00:04:06.430
of a sudden make 0 equal 3.40.

00:04:06.430 --> 00:04:08.220
So there is no solution.

00:04:08.220 --> 00:04:10.320
And the only
explanation over here

00:04:10.320 --> 00:04:14.268
is that the local was
charged a cheaper rate.


Graphing proportional relationships example

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:03.730
Graph the line that represents
a proportional relationship

00:00:03.730 --> 00:00:07.130
between y and x with
a unit rate 0.4.

00:00:07.130 --> 00:00:10.390
That is, a change
of one unit in x

00:00:10.390 --> 00:00:13.780
corresponds to a change
of 0.4 units in y.

00:00:13.780 --> 00:00:15.410
And they also ask
us to figure out

00:00:15.410 --> 00:00:18.100
what the equation of
this line actually is.

00:00:18.100 --> 00:00:20.650
So let me get my scratch pad
out and we could think about it.

00:00:20.650 --> 00:00:24.350
So let's just think about some
potential x and y values here.

00:00:24.350 --> 00:00:27.077
So let's think about some
potential x and y values.

00:00:27.077 --> 00:00:29.410
So when we're thinking about
proportional relationships,

00:00:29.410 --> 00:00:34.750
that means that y is going to be
equal to some constant times x.

00:00:34.750 --> 00:00:36.500
So if we have a
proportional relationship,

00:00:36.500 --> 00:00:39.570
if you have zero x's, it doesn't
matter what your constant here

00:00:39.570 --> 00:00:41.670
is, you're going
to have zero y's.

00:00:41.670 --> 00:00:44.740
So the point 0, 0
should be on your line.

00:00:44.740 --> 00:00:48.280
So if this is the point 0, 0,
this should be on my line right

00:00:48.280 --> 00:00:49.340
over there.

00:00:49.340 --> 00:00:52.330
Now, let's think about what
happens as we increase x.

00:00:52.330 --> 00:00:55.620
So if x goes from 0
to 1, we already know

00:00:55.620 --> 00:00:57.560
that a change of 1
unit in x corresponds

00:00:57.560 --> 00:01:00.610
to a change of 0.4 units in y.

00:01:00.610 --> 00:01:05.470
So if x increases by 1, then
y is going to increase by 0.4.

00:01:05.470 --> 00:01:08.932
It's not so easy to
graph this 1 comma 0.4.

00:01:08.932 --> 00:01:12.560
The 0.4 is hard to graph on this
little tool right over here.

00:01:12.560 --> 00:01:14.730
So let's try to get this
to be a whole number.

00:01:14.730 --> 00:01:16.850
So then when x
increases another 1,

00:01:16.850 --> 00:01:18.900
y is going to
increase by 0.4 again.

00:01:18.900 --> 00:01:20.930
It's going to get to 0.8.

00:01:20.930 --> 00:01:23.580
When x increases
again by 1, then y

00:01:23.580 --> 00:01:25.370
is going to increase
by 0.4 again.

00:01:25.370 --> 00:01:27.540
It's going to get to 1.2.

00:01:27.540 --> 00:01:32.010
If x increases again, y is
going to increase by 0.4 again.

00:01:32.010 --> 00:01:32.920
So just to 1.6.

00:01:32.920 --> 00:01:36.400
Notice, every time x
is increasing by 1,

00:01:36.400 --> 00:01:38.940
y is increasing by 0.4.

00:01:38.940 --> 00:01:40.950
That's exactly what
they told us here.

00:01:40.950 --> 00:01:43.630
Now, if x increases
by 1 again to 5,

00:01:43.630 --> 00:01:46.350
then y is going to
increase 0.4 to 2.

00:01:46.350 --> 00:01:49.840
And I like this point because
this is nice and easy to graph.

00:01:49.840 --> 00:01:55.130
So we see that the point 0,
0 and the point 5 comma 2

00:01:55.130 --> 00:01:56.357
should be on this graph.

00:01:56.357 --> 00:01:57.190
And I could draw it.

00:01:57.190 --> 00:02:00.060
And I'm going to do it on
the tool in a second as well.

00:02:00.060 --> 00:02:04.760
So it'll look
something like this.

00:02:04.760 --> 00:02:08.070
And notice the slope of
this actual graph over here.

00:02:08.070 --> 00:02:10.509
Notice the slope of
this actual graph.

00:02:10.509 --> 00:02:15.480
If our change in x is 5.

00:02:15.480 --> 00:02:17.810
So notice, here our
change in x is 5.

00:02:17.810 --> 00:02:19.285
Our change in x is 5.

00:02:19.285 --> 00:02:20.160
You see that as well.

00:02:20.160 --> 00:02:24.220
When you go from 0 to 5,
this change in x is 5.

00:02:24.220 --> 00:02:26.500
Change in x is equal to 5.

00:02:26.500 --> 00:02:29.340
What was our
corresponding change in y?

00:02:29.340 --> 00:02:31.780
Well, our corresponding change
in y when our change in x

00:02:31.780 --> 00:02:36.040
was 5, our change
in y was equal to 2.

00:02:36.040 --> 00:02:38.610
And you see that here,
when x went from 0 to 5,

00:02:38.610 --> 00:02:41.760
y went from 0 to 2.

00:02:41.760 --> 00:02:45.920
So our change in y in this
circumstance is equal to 2.

00:02:45.920 --> 00:02:48.890
So our slope, which
is change in y

00:02:48.890 --> 00:02:53.320
over change in x, is
the rate of change

00:02:53.320 --> 00:02:54.960
of your vertical
axis with respect

00:02:54.960 --> 00:03:02.230
to your horizontal axis, is
going to be equal to 2 over 5,

00:03:02.230 --> 00:03:03.450
or 2/5.

00:03:03.450 --> 00:03:07.460
Which if you wrote it as
a decimal is equal to 0.4.

00:03:07.460 --> 00:03:10.807
So this right over
here is your slope.

00:03:10.807 --> 00:03:12.390
So I'm going to do
this with the tool.

00:03:12.390 --> 00:03:14.140
But first, let's
also think about what

00:03:14.140 --> 00:03:17.560
the equation of this
line is going to be.

00:03:17.560 --> 00:03:21.460
Well, we know that y is equal
to some constant times x.

00:03:21.460 --> 00:03:23.700
And we know that
the point 5, 2 is

00:03:23.700 --> 00:03:25.780
on this line right over here.

00:03:25.780 --> 00:03:30.050
So we could say, well, when x
is equal to 5, y is equal to 2.

00:03:30.050 --> 00:03:35.960
Or, when y equals 2, we
have k times 5, or k is

00:03:35.960 --> 00:03:42.660
equal to-- dividing both sides
by 5, you can't see that.

00:03:42.660 --> 00:03:49.410
If I divide both sides by 5,
I'm left with k is equal to 2/5.

00:03:49.410 --> 00:03:50.187
Which makes sense.

00:03:50.187 --> 00:03:51.270
We're used to seeing this.

00:03:51.270 --> 00:03:54.130
When we have y is equal
to something times x,

00:03:54.130 --> 00:03:58.140
this something right over
here is going to be our slope.

00:03:58.140 --> 00:04:04.490
So the equation of the
line is y is equal to 0.4x.

00:04:04.490 --> 00:04:05.560
So let's fill this in.

00:04:05.560 --> 00:04:08.270
Let's actually do
the exercise now.

00:04:08.270 --> 00:04:11.490
So we had two points,
one was the point 0, 0.

00:04:11.490 --> 00:04:13.340
When x is 0, y is 0.

00:04:13.340 --> 00:04:18.339
And when x is 5, y
is 0.4 times that.

00:04:18.339 --> 00:04:20.490
So it's y is equal to 2.

00:04:20.490 --> 00:04:27.470
And we said the equation is
y is equal to 0.4 times x.

00:04:27.470 --> 00:04:29.930
So let's check our answer.


Graphing proportional relationships example 3

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:04.612
We're asked to graph y
is equal to 2.5 times x.

00:00:04.612 --> 00:00:06.820
So we really just have to
think about two points that

00:00:06.820 --> 00:00:09.100
satisfy this equation here,
and the most obvious one

00:00:09.100 --> 00:00:10.580
is what happens when x equals 0.

00:00:10.580 --> 00:00:13.990
When x equals 0, 2.5
times 0 is going to be 0.

00:00:13.990 --> 00:00:18.416
So when x is 0, y is
going to be equal to 0.

00:00:18.416 --> 00:00:20.040
And then let's just
pick another x that

00:00:20.040 --> 00:00:23.640
will give us a y that
is a whole number.

00:00:23.640 --> 00:00:27.874
So if x increases by 1, y
is going to increase by 2.5.

00:00:27.874 --> 00:00:29.290
It's going to go
right over there,

00:00:29.290 --> 00:00:31.110
and I could graph
it just like that.

00:00:31.110 --> 00:00:33.320
And we see just by
what I just said

00:00:33.320 --> 00:00:37.990
that the unit rate of change
of y with respect to x is 2.5.

00:00:37.990 --> 00:00:42.820
A unit increase in x, an
increase of 1 and x, results

00:00:42.820 --> 00:00:44.750
in a 2.5 increase in y.

00:00:44.750 --> 00:00:47.540
You see that right over
here. x goes from 0 to 1,

00:00:47.540 --> 00:00:49.710
and y goes from 0 to 2.5.

00:00:49.710 --> 00:00:51.930
But let's increase
x by another 1,

00:00:51.930 --> 00:00:55.840
and then y is going to increase
by 2.5 again to get to 5.

00:00:55.840 --> 00:00:59.150
Or you could say, hey, look,
if x is equal to 2, 2.5 times

00:00:59.150 --> 00:01:01.390
2 is equal to 5.

00:01:01.390 --> 00:01:05.080
So this is a legitimate
graph for this equation,

00:01:05.080 --> 00:01:06.970
but then they also
tell us to select

00:01:06.970 --> 00:01:09.620
the statements that are true.

00:01:09.620 --> 00:01:11.960
So the first one is
the equation does not

00:01:11.960 --> 00:01:14.050
represent a proportional
relationship.

00:01:14.050 --> 00:01:15.800
Well, this is a
proportional relationship.

00:01:15.800 --> 00:01:17.629
A proportional
relationship is one

00:01:17.629 --> 00:01:19.670
where, first of all, if
you have zero x's, you're

00:01:19.670 --> 00:01:23.690
going to have zero y's, where y
is equal to some constant times

00:01:23.690 --> 00:01:24.190
x.

00:01:24.190 --> 00:01:26.890
And here, y is equal
to 2.5 times x.

00:01:26.890 --> 00:01:29.200
So this is definitely a
proportional relationship,

00:01:29.200 --> 00:01:30.870
so I'm not going to check that.

00:01:30.870 --> 00:01:33.834
The unit rate of the
relationship is 2/5.

00:01:33.834 --> 00:01:35.250
So I'm assuming--
this is a little

00:01:35.250 --> 00:01:36.920
ambiguous the way
they stated it.

00:01:36.920 --> 00:01:39.760
I'm assuming they're saying
the unit rate of change of y

00:01:39.760 --> 00:01:41.950
with respect to x.

00:01:41.950 --> 00:01:44.640
And the unit rate of change
of y with respect to x

00:01:44.640 --> 00:01:49.260
is, when x increases
1, y changed 2.5.

00:01:49.260 --> 00:01:52.900
So here they're saying when x
changes by 1, y changes by 0.4,

00:01:52.900 --> 00:01:55.680
2/5 is the same thing as 0.4.

00:01:55.680 --> 00:01:57.010
This should be 5/2.

00:01:57.010 --> 00:01:58.620
5/2 would be 2.4.

00:01:58.620 --> 00:02:00.440
So this isn't right as well.

00:02:00.440 --> 00:02:02.462
The slope of the line is 2.5.

00:02:02.462 --> 00:02:03.420
Well, this looks right.

00:02:03.420 --> 00:02:06.230
Slope is change in
y over change in x.

00:02:06.230 --> 00:02:09.830
When x changes 1, y changes 2.5.

00:02:09.830 --> 00:02:13.710
So change in y, 2.5,
over change in x, 1.

00:02:13.710 --> 00:02:15.986
2.5 over 1 is 2.5.

00:02:15.986 --> 00:02:17.360
And you could also
see it looking

00:02:17.360 --> 00:02:18.930
at the form of
this equation. y is

00:02:18.930 --> 00:02:22.370
equal to-- this is
the slope times x.

00:02:22.370 --> 00:02:24.070
So that's right.

00:02:24.070 --> 00:02:28.760
A change of 5 units in x results
in a change of 2 units in y.

00:02:28.760 --> 00:02:30.670
Well, let's test that idea.

00:02:30.670 --> 00:02:32.770
We know when x is 0, y is 0.

00:02:32.770 --> 00:02:37.750
So if x goes from 0 to 5,
what's going to happen to y?

00:02:37.750 --> 00:02:42.080
Well, y is going
to be 2.5 times 5.

00:02:42.080 --> 00:02:46.640
2.5 times 5 is 12.5.

00:02:46.640 --> 00:02:48.870
So y would not just change 2.

00:02:48.870 --> 00:02:50.520
It actually would change 12.5.

00:02:50.520 --> 00:02:52.030
So this isn't right.

00:02:52.030 --> 00:02:56.170
A change of 2 units in x results
in a change of 5 units in y.

00:02:56.170 --> 00:02:57.270
Well, we see that.

00:02:57.270 --> 00:03:00.992
A change in 2 units of x results
in a change of 5 units in y.

00:03:00.992 --> 00:03:02.950
That's exactly what we
graphed right over here.

00:03:02.950 --> 00:03:04.690
These two points show that.

00:03:04.690 --> 00:03:07.430
So this is definitely true.


Graphing proportional relationships example 2

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.330
We're asked to graph the
proportional relationship shown

00:00:03.330 --> 00:00:04.150
in the table below.

00:00:04.150 --> 00:00:06.160
And they give us a
table of x values,

00:00:06.160 --> 00:00:08.189
and the corresponding y values.

00:00:08.189 --> 00:00:11.671
So we see when x is equal
to 0, y is equal to 0.

00:00:11.671 --> 00:00:13.420
Then they give us a
bunch of other points.

00:00:13.420 --> 00:00:15.730
When x is 3, y is 0.5.

00:00:15.730 --> 00:00:19.632
When x is 6, y is 1,
so on and so forth.

00:00:19.632 --> 00:00:21.340
So let's graph one of
these that actually

00:00:21.340 --> 00:00:23.690
have integer values
for both coordinates.

00:00:23.690 --> 00:00:29.660
So when x is 6, y is 1.

00:00:29.660 --> 00:00:31.660
So we only need two points
to specify the line.

00:00:31.660 --> 00:00:33.180
So we've actually graphed it.

00:00:33.180 --> 00:00:36.309
But then they also ask us,
what is the slope of this line?

00:00:36.309 --> 00:00:37.850
And we just have to
remind ourselves.

00:00:37.850 --> 00:00:43.160
Our slope is what is our change
in y for a given change in x.

00:00:43.160 --> 00:00:45.290
So, for example, or another
way to think about it,

00:00:45.290 --> 00:00:48.100
what is your change in
y over your change in x?

00:00:48.100 --> 00:00:51.210
So here, our x changed by 6.

00:00:51.210 --> 00:00:52.800
It went from 0 to 6.

00:00:52.800 --> 00:00:55.190
And our y changed by 1.

00:00:55.190 --> 00:00:57.289
So our change in
y over our change

00:00:57.289 --> 00:00:59.580
in x, which is the definition
of slope, our change in y

00:00:59.580 --> 00:01:03.528
is 1, when our change in x is 6.

00:01:03.528 --> 00:01:04.569
See that right over here.

00:01:04.569 --> 00:01:07.700
Change in y, 1 when
our change in x is 6.

00:01:07.700 --> 00:01:09.240
Let's check our answer.

00:01:09.240 --> 00:01:11.270
We got it right.


Interpreting linear expressions, 2

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.970
Martin likes to
make flower bouquets

00:00:02.970 --> 00:00:07.670
that each have 3
violets and 4 tulips.

00:00:07.670 --> 00:00:11.680
If the price of a violet is V
and the price of a tulip is T,

00:00:11.680 --> 00:00:14.370
match the expressions
to their meanings.

00:00:14.370 --> 00:00:18.140
So let's see-- the price
of 1 of Martin's bouquets.

00:00:18.140 --> 00:00:23.770
So one of Martin's bouquets
has 3 violets and 4 tulips.

00:00:23.770 --> 00:00:26.290
So the 3 violets are
going to cost 3 times

00:00:26.290 --> 00:00:28.190
the price of the
violet, which is V. So

00:00:28.190 --> 00:00:30.325
that's the cost of
the violets, 3V.

00:00:30.325 --> 00:00:32.330
And then the 4 tulips
are going to cost

00:00:32.330 --> 00:00:33.990
4 times the price of a tulip.

00:00:33.990 --> 00:00:35.350
So that's 4T.

00:00:35.350 --> 00:00:36.980
So it's 3V plus 4T.

00:00:36.980 --> 00:00:38.151
So it's not this one.

00:00:38.151 --> 00:00:38.650
Let's see.

00:00:38.650 --> 00:00:41.530
This one right over
here, this is 4T plus 3V.

00:00:41.530 --> 00:00:44.250
So this is the price of 4
tulips, 4 times the price

00:00:44.250 --> 00:00:47.400
of a tulip, plus 3 times
the price of a violet.

00:00:47.400 --> 00:00:50.850
The price of 3 of
Martin's bouquets,

00:00:50.850 --> 00:00:53.730
so it's essentially going
to be 3 times this quantity

00:00:53.730 --> 00:00:54.470
right over here.

00:00:54.470 --> 00:00:55.761
This is the price of 1 bouquet.

00:00:55.761 --> 00:00:56.950
We want 3 of them.

00:00:56.950 --> 00:01:01.430
So it's going to be 3 times
the quantity 4T plus 3V.

00:01:01.430 --> 00:01:02.590
And let's see.

00:01:02.590 --> 00:01:07.020
If I were to actually multiply
this out, 3 times 4T is 12T,

00:01:07.020 --> 00:01:09.452
and then plus 3 times 3V is 9V.

00:01:09.452 --> 00:01:10.910
So this is the same
thing if I were

00:01:10.910 --> 00:01:14.714
to distribute the
3-- is 12T plus 9V.

00:01:14.714 --> 00:01:16.130
Well, that's this
right over here.

00:01:16.130 --> 00:01:18.827
These two are
equivalent expressions.

00:01:18.827 --> 00:01:19.410
And let's see.

00:01:19.410 --> 00:01:21.118
Are any of these other
things equivalent?

00:01:21.118 --> 00:01:23.740
No, this says 3
times 3T plus 4V.

00:01:23.740 --> 00:01:27.310
So I'm going to put this
in the not used bucket.

00:01:27.310 --> 00:01:30.739
And then I have--
let's see-- 3V plus 2T.

00:01:30.739 --> 00:01:32.530
I'm going to put it in
the not used bucket.

00:01:35.377 --> 00:01:35.960
And let's see.

00:01:35.960 --> 00:01:41.410
This has 2V plus 4T plus V.
So if I were to simplify this,

00:01:41.410 --> 00:01:44.276
if I were to combine the V
terms-- if I have two V's and I

00:01:44.276 --> 00:01:46.460
add another V, that's
three V's plus 4T.

00:01:46.460 --> 00:01:48.975
So this is actually the
same thing as the price of 1

00:01:48.975 --> 00:01:49.850
of Martin's bouquets.

00:01:49.850 --> 00:01:51.850
So you could view this
as the price of 2 violets

00:01:51.850 --> 00:01:55.450
plus the price of 4 tulips
plus another violet.

00:01:55.450 --> 00:01:58.030
So it's really the price
of 3 violets and 4 tulips.

00:01:58.030 --> 00:01:59.540
So let's check our answer.

00:01:59.540 --> 00:02:01.490
We got it right.


Interpreting linear expressions, 1

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:04.880
Handsome Jack is buying
a pony made of diamonds.

00:00:04.880 --> 00:00:07.870
The price of the
pony is P dollars.

00:00:07.870 --> 00:00:13.410
And Jack also has to pay
a 25% diamond pony tax.

00:00:13.410 --> 00:00:16.650
Match the expressions to their
meaning for Handsome Jack.

00:00:16.650 --> 00:00:19.050
And they say
multiple expressions

00:00:19.050 --> 00:00:21.520
may fit the same description.

00:00:21.520 --> 00:00:24.570
So in this bucket, we have
the price of the diamond

00:00:24.570 --> 00:00:25.659
pony before tax.

00:00:25.659 --> 00:00:28.200
Well, they already tell us that
the price of the diamond pony

00:00:28.200 --> 00:00:29.680
is P dollars.

00:00:29.680 --> 00:00:32.689
So that's P right over here.

00:00:32.689 --> 00:00:35.990
Now over here, they say the
amount of tax Handsome Jack

00:00:35.990 --> 00:00:37.030
pays.

00:00:37.030 --> 00:00:39.940
Well, he pays a 25%
diamond pony tax.

00:00:39.940 --> 00:00:42.880
So whatever the price is,
he's going to pay 25% of that.

00:00:42.880 --> 00:00:44.380
Or another way of
thinking about it,

00:00:44.380 --> 00:00:48.120
he's going to pay 25% is
the same thing as 0.25.

00:00:48.120 --> 00:00:52.010
So 0.25 times P is
the amount of tax

00:00:52.010 --> 00:00:55.550
he's going to pay on
this diamond pony.

00:00:55.550 --> 00:00:58.570
Now, they say Handsome
Jack's total bill

00:00:58.570 --> 00:01:00.260
for the diamond pony.

00:01:00.260 --> 00:01:05.459
Well, he's going to pay P for
the pony plus 0.25P in taxes.

00:01:05.459 --> 00:01:07.290
So that is this one
right over here.

00:01:07.290 --> 00:01:11.750
P for the pony plus
0.25P for the taxes.

00:01:11.750 --> 00:01:13.690
So that's that one over there.

00:01:13.690 --> 00:01:17.240
But if we look at this, you
could view this literally as 1P

00:01:17.240 --> 00:01:19.720
plus 0.25P's.

00:01:19.720 --> 00:01:22.480
Well, that's the
same thing as 1.25P.

00:01:22.480 --> 00:01:25.680
So that's the same thing
as this right over here.

00:01:25.680 --> 00:01:27.610
So it's 1.25P.

00:01:27.610 --> 00:01:29.410
And these other three
don't seem to fit

00:01:29.410 --> 00:01:30.640
in any of these categories.

00:01:30.640 --> 00:01:33.050
So I'm going to put
it into the not used.

00:01:33.050 --> 00:01:34.920
We're required to
categorize everything.

00:01:34.920 --> 00:01:38.420
So let me put this
in the not used.

00:01:38.420 --> 00:01:40.280
It's falling off the
screen, I realize.

00:01:40.280 --> 00:01:43.030
Let me put this in the not used.

00:01:43.030 --> 00:01:48.010
And then let me put
this in the not used.

00:01:48.010 --> 00:01:50.070
Let's check our answer.

00:01:50.070 --> 00:01:51.930
We got it right.


Adding expressions

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:02.940
Add the blue and
pink expressions

00:00:02.940 --> 00:00:04.600
to create another expression.

00:00:04.600 --> 00:00:06.810
And be sure to
combine like terms.

00:00:06.810 --> 00:00:09.030
So let me get this
scratch pad out here.

00:00:09.030 --> 00:00:12.980
So we need to add this
6y plus 2 in blue.

00:00:12.980 --> 00:00:15.740
So we're going to add 6y plus 2.

00:00:15.740 --> 00:00:18.360
And we're going to add it
to negative 2/3 minus y.

00:00:22.850 --> 00:00:24.590
So first let's add
all the y terms.

00:00:24.590 --> 00:00:27.610
If we have six of something,
and I'm going to subtract one

00:00:27.610 --> 00:00:30.930
of that something,
or 6y's minus 1y--

00:00:30.930 --> 00:00:32.860
you could view this
literally as 1y--

00:00:32.860 --> 00:00:35.310
I'm going to be left with 5y's.

00:00:35.310 --> 00:00:38.829
And then I have 2 minus 2/3.

00:00:38.829 --> 00:00:40.370
And let's solve that
right over here.

00:00:40.370 --> 00:00:46.720
So 2 minus 2/3 is
the same thing as-- 2

00:00:46.720 --> 00:00:53.140
can be rewritten as 6/3-- 6
divided by 3 is 2-- minus 2/3,

00:00:53.140 --> 00:00:55.090
which equals 4/3.

00:00:55.090 --> 00:00:58.220
So 2 minus 2/3 is 4/3.

00:00:58.220 --> 00:01:01.840
So this is equal to 5y plus 4/3.

00:01:01.840 --> 00:01:04.780
Now let's actually input that
in to make sure we got it right.

00:01:04.780 --> 00:01:08.960
5y plus 4/3.

00:01:08.960 --> 00:01:11.960
And you see it
interpreted it properly.

00:01:11.960 --> 00:01:13.910
And we got it right.


How to expand an expressions using the distributive property

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.440
We're asked to use the
distributive property

00:00:02.440 --> 00:00:04.314
to expand this expression.

00:00:04.314 --> 00:00:05.980
So let me get my
little scratch pad out.

00:00:05.980 --> 00:00:08.320
So this is the same
question right over here.

00:00:08.320 --> 00:00:10.960
So we're going to multiply 6
times this entire quantity.

00:00:10.960 --> 00:00:17.390
So let me just rewrite
this as 6 times negative 4w

00:00:17.390 --> 00:00:20.210
plus one half.

00:00:20.210 --> 00:00:22.390
So we're going to
distribute the 6.

00:00:22.390 --> 00:00:26.490
So this is going to be 6
times negative 4w times 6,

00:00:26.490 --> 00:00:28.900
plus 6 times one
half, so this is

00:00:28.900 --> 00:00:32.320
going to be-- let me
just write it this way--

00:00:32.320 --> 00:00:45.060
6 times negative 4w
plus 6 times one half.

00:00:45.060 --> 00:00:48.100
And this is going to be equal
to 6 times negative 4w is

00:00:48.100 --> 00:00:53.570
negative 24w, and then
6 times one half is 3.

00:00:53.570 --> 00:00:56.950
So we get negative 24w plus 3.

00:00:56.950 --> 00:00:58.520
So let me write that down.

00:00:58.520 --> 00:01:01.980
Negative 24w plus 3.

00:01:01.980 --> 00:01:04.143
And let's check our answer.


Dividing whole numbers and fractions: studying

WEBVTT
Kind: captions
Language: en

00:00:00.790 --> 00:00:03.600
Tommy is studying for
final exams this weekend.

00:00:03.600 --> 00:00:06.390
He will spend 1/5 of
the weekend studying.

00:00:06.390 --> 00:00:08.660
What fraction of
the weekend will he

00:00:08.660 --> 00:00:13.260
spend studying for
each of his 4 subjects

00:00:13.260 --> 00:00:17.040
if he spends the same amount of
time studying for each subject?

00:00:17.040 --> 00:00:19.050
So the total amount
of time he's going

00:00:19.050 --> 00:00:23.730
to spend studying this
weekend is 1/5 of the weekend.

00:00:23.730 --> 00:00:27.300
And he has to divide that
into 4 equal sections.

00:00:27.300 --> 00:00:30.190
And he's going to spend that
much time on each subject.

00:00:30.190 --> 00:00:35.160
So he's going to
divide this by 4.

00:00:35.160 --> 00:00:38.380
Now, we've already seen
that dividing by a number

00:00:38.380 --> 00:00:41.052
is the same thing as
multiplying by its reciprocal.

00:00:41.052 --> 00:00:43.260
You might say, hey, well,
what's the reciprocal of 4?

00:00:43.260 --> 00:00:45.250
You just have to
remind yourself that 4

00:00:45.250 --> 00:00:47.980
is the same thing as 4/1.

00:00:47.980 --> 00:00:58.100
So 1/5 divided by 4/1 is the
same thing as 1/5 times 1/4.

00:01:00.790 --> 00:01:04.080
And you could also view
this as 1/4 of 1/5 or 1/5

00:01:04.080 --> 00:01:05.760
of 1/4, either way.

00:01:05.760 --> 00:01:09.570
But here we multiply
our numerators to get 1.

00:01:09.570 --> 00:01:13.500
And then we multiply our
denominators, 4 times 5 is 20.

00:01:13.500 --> 00:01:15.980
So you get 1/20 of
the weekend will

00:01:15.980 --> 00:01:18.890
be spent studying
for each subject.

00:01:18.890 --> 00:01:21.660
Now, let's also try to
think about this visually.

00:01:21.660 --> 00:01:25.050
Let's imagine that this
is his entire weekend.

00:01:25.050 --> 00:01:27.790
And I've divided it
into 5 equal sections.

00:01:27.790 --> 00:01:29.960
And so we already know
that the total amount

00:01:29.960 --> 00:01:32.350
of his weekend spent
studying is 1/5.

00:01:32.350 --> 00:01:34.600
So that's the total
amount studying

00:01:34.600 --> 00:01:36.660
for the weekend is 1/5.

00:01:36.660 --> 00:01:39.370
Now, he has to divide this
into 4 equals section.

00:01:39.370 --> 00:01:40.977
So let's do that.

00:01:40.977 --> 00:01:42.560
He's got four subjects,
and he's going

00:01:42.560 --> 00:01:45.840
to spend the same amount of
time on each of the 4 subjects.

00:01:45.840 --> 00:01:49.590
So he's going to divide
this into 4 equal sections.

00:01:49.590 --> 00:01:52.870
So how much time does
he spend on one subject?

00:01:52.870 --> 00:01:55.860
Well, in each subject, that
would be this little area

00:01:55.860 --> 00:01:57.880
that I'm doing in
yellow right over here.

00:01:57.880 --> 00:01:58.880
And what is that?

00:01:58.880 --> 00:02:03.425
Well, that's 1 over-- how
many equal sections are there

00:02:03.425 --> 00:02:05.080
of that size in the weekend?

00:02:05.080 --> 00:02:06.760
Well, I've just
drawn out the grid.

00:02:06.760 --> 00:02:10.310
You had 5 rows, and
now you have 4 columns.

00:02:10.310 --> 00:02:15.720
So 5 rows times 4 columns,
you have 20 equal sections.

00:02:15.720 --> 00:02:17.730
So once again, looking
at it visually,

00:02:17.730 --> 00:02:19.820
he's spending 1/20
of his weekend

00:02:19.820 --> 00:02:22.760
on each of the 4 subjects.

00:02:22.760 --> 00:02:25.630
And then if you do this
for 4 subjects, that

00:02:25.630 --> 00:02:27.300
means that in this
whole weekend,

00:02:27.300 --> 00:02:28.769
1/5 will be spent studying.

00:02:28.769 --> 00:02:30.310
But the question
that they're asking,

00:02:30.310 --> 00:02:34.180
he's spending 1/20 of the
weekend on each subject.


Dividing whole numbers and fractions: potpourri

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.290
Tracy is putting out
decorative bowls of potpourri

00:00:03.290 --> 00:00:05.970
in each room of the
hotel where she works.

00:00:05.970 --> 00:00:11.210
She wants to fill each bowl
with 1/5 of a can of potpourri.

00:00:11.210 --> 00:00:15.460
If Tracy has 4 cans of
potpourri, in how many rooms

00:00:15.460 --> 00:00:18.440
can she place a
bowl of potpourri?

00:00:18.440 --> 00:00:22.960
So she has 4 cans, and
she wants to divide this 4

00:00:22.960 --> 00:00:25.544
cans into groups
of 1/5 of a can.

00:00:25.544 --> 00:00:27.210
So if you have 4 of
something and you're

00:00:27.210 --> 00:00:29.376
trying to divide it into
groups of a certain amount,

00:00:29.376 --> 00:00:32.280
you would divide by
that amount per group.

00:00:32.280 --> 00:00:37.780
So you want to divide 4 by 1/5.

00:00:37.780 --> 00:00:42.650
You want to divide 4 cans of
potpourri into groups of 1/5.

00:00:42.650 --> 00:00:46.480
So let's visualize this.

00:00:46.480 --> 00:00:50.430
Let me draw one can of
potpourri right over here.

00:00:50.430 --> 00:00:55.565
So one can of potpourri can
clearly be cut up into 5/5.

00:00:55.565 --> 00:00:56.690
We have it right over here.

00:00:56.690 --> 00:01:00.640
1, 2, 3, 4, 5.

00:01:00.640 --> 00:01:05.660
So 1 can of potpourri
can fill 5 bowls

00:01:05.660 --> 00:01:07.740
if you put 1/5 in each bowl.

00:01:07.740 --> 00:01:10.570
Now, we have 4 cans.

00:01:10.570 --> 00:01:12.280
So let me paste these.

00:01:12.280 --> 00:01:18.050
So 2, 3, and 4.

00:01:18.050 --> 00:01:23.020
So how many total bowls of
potpourri can Tracy fill?

00:01:23.020 --> 00:01:25.370
Well, she's got 4 cans.

00:01:25.370 --> 00:01:27.635
So this is going to
be equal to-- let

00:01:27.635 --> 00:01:29.580
me do this is the right
color-- this is going

00:01:29.580 --> 00:01:33.500
to be equal to, once
again, she has 4 cans.

00:01:33.500 --> 00:01:35.070
And then for each
of those cans, she

00:01:35.070 --> 00:01:37.920
can fill 5 bowls of potpourri
because each bowl only

00:01:37.920 --> 00:01:40.080
requires a 1/5 of those cans.

00:01:40.080 --> 00:01:45.440
So this is going to be the
same thing as 4 times 5.

00:01:45.440 --> 00:01:48.570
Or we can even write
this as 4 times 5 over 1.

00:01:48.570 --> 00:01:51.720
5 is the same
thing as 5/1, which

00:01:51.720 --> 00:01:58.030
is the same thing
as 4 times 5, which,

00:01:58.030 --> 00:02:02.220
of course, is equal to 20.

00:02:02.220 --> 00:02:06.800
She can fill 20 cans-- or I
should say, with her 4 cans,

00:02:06.800 --> 00:02:10.440
she can fill 20
bowls of potpourri.

00:02:10.440 --> 00:02:13.190
Now, just as a
review here, we've

00:02:13.190 --> 00:02:15.420
already seen that
dividing by a number

00:02:15.420 --> 00:02:18.910
is equal to multiplying
by its reciprocal.

00:02:18.910 --> 00:02:20.410
And we see that right over here.

00:02:20.410 --> 00:02:22.670
Dividing by 1/5
is the same thing

00:02:22.670 --> 00:02:27.130
as multiplying by the
reciprocal of 1/5, which is 5/1.

00:02:27.130 --> 00:02:30.662
So she could fill up
20 bowls of potpourri.


Example of subtracting fractions with unlike denominators word problem

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.000
Silvia is growing tomato plants
and studying their heights.

00:00:04.000 --> 00:00:05.120
Here is her data.

00:00:05.120 --> 00:00:10.440
So the beefsteak tomato type
had a height of 3 and 1/4 feet,

00:00:10.440 --> 00:00:14.960
Roma 2 and 7/8 feet,
cherry 3 and 1/2 feet.

00:00:14.960 --> 00:00:16.620
And they ask us, what
is the difference

00:00:16.620 --> 00:00:21.150
between the heights
of the beefsteak,

00:00:21.150 --> 00:00:23.070
so that's that one
right over there,

00:00:23.070 --> 00:00:27.750
and the Roma tomato plants?

00:00:27.750 --> 00:00:30.041
So they want us to find the
distance between these two

00:00:30.041 --> 00:00:30.540
heights.

00:00:30.540 --> 00:00:33.010
So this cherry tomato height
was really unnecessary

00:00:33.010 --> 00:00:34.990
for the sake of this problem.

00:00:34.990 --> 00:00:37.050
So we want to find the
difference in these two

00:00:37.050 --> 00:00:37.550
heights.

00:00:37.550 --> 00:00:39.829
So we want to subtract
the smaller of these

00:00:39.829 --> 00:00:40.870
from the larger of these.

00:00:40.870 --> 00:00:45.760
So we're really trying to
evaluate what 3 and 1/4

00:00:45.760 --> 00:00:48.030
minus 2 and 7/8 is.

00:00:53.310 --> 00:00:56.790
Now, the first thing that I like
to do is convert both of these

00:00:56.790 --> 00:00:59.727
into mixed numbers.

00:00:59.727 --> 00:01:01.310
Sorry, they're already
mixed numbers--

00:01:01.310 --> 00:01:04.870
to convert both of these
into improper fractions.

00:01:04.870 --> 00:01:07.400
So 3 and 1/4 is
the same thing as 3

00:01:07.400 --> 00:01:14.840
plus 1/4, which is the same
thing as 12/4 plus 1/4.

00:01:14.840 --> 00:01:17.130
That's the same
thing as 3 and 1/4.

00:01:17.130 --> 00:01:20.210
And from that, we're going
to subtract 2 and 7/8.

00:01:20.210 --> 00:01:23.420
2 and 7/8 is the same
thing as 2 plus 7/8.

00:01:23.420 --> 00:01:29.130
2 is the same thing
as 16/8 plus 7/8.

00:01:29.130 --> 00:01:32.630
So this is essentially what
we are trying to figure out.

00:01:32.630 --> 00:01:34.820
Now, what is 12/4 plus 1/4?

00:01:34.820 --> 00:01:39.370
It's 13/4, 13 over 4.

00:01:39.370 --> 00:01:43.220
And then, what's 16/8 plus 7/8?

00:01:43.220 --> 00:01:45.850
Well, that's 23/8.

00:01:45.850 --> 00:01:51.160
So this is going to be
minus 23 over eight.

00:01:51.160 --> 00:01:54.697
Now, we're subtracting
one fraction from another.

00:01:54.697 --> 00:01:56.155
But we have different
denominators.

00:01:56.155 --> 00:01:58.010
So we can't make
sense of this until we

00:01:58.010 --> 00:02:00.240
have the same denominator.

00:02:00.240 --> 00:02:03.040
And so what is the least
common multiple of these two

00:02:03.040 --> 00:02:04.920
denominators, of both 4 and 8?

00:02:04.920 --> 00:02:09.740
What is the smallest number that
is divisible by both 4 and 8?

00:02:09.740 --> 00:02:11.410
Well, 8 is divisible by 8.

00:02:11.410 --> 00:02:14.360
And 8 is also divisible by 4.

00:02:14.360 --> 00:02:18.640
So if we can rewrite 13/4 as
having 8 as a denominator,

00:02:18.640 --> 00:02:19.710
then we are all set.

00:02:19.710 --> 00:02:21.570
So let's try to do that.

00:02:21.570 --> 00:02:23.489
So we're going to write
both of these with 8

00:02:23.489 --> 00:02:24.280
as the denominator.

00:02:24.280 --> 00:02:25.670
This one already has it.

00:02:25.670 --> 00:02:29.370
So 13/4, I'm going to have
it with 8 as the denominator.

00:02:29.370 --> 00:02:34.310
So to get from 4 to 8, we have
to multiply the denominator

00:02:34.310 --> 00:02:34.924
by 2.

00:02:34.924 --> 00:02:37.090
So in order to not change
the value of the fraction,

00:02:37.090 --> 00:02:38.465
we have to multiply
the numerator

00:02:38.465 --> 00:02:40.370
by that exact same value.

00:02:40.370 --> 00:02:41.750
We've got to multiply it by 2.

00:02:41.750 --> 00:02:44.610
So this becomes 26/8.

00:02:44.610 --> 00:02:49.350
And from 26/8, we're
going to subtract 23/8.

00:02:49.350 --> 00:02:59.600
And so this is going to be
over 8-- 26 minus 23, which

00:02:59.600 --> 00:03:04.770
is equal to, and we deserve
our drum roll now, is 3.

00:03:04.770 --> 00:03:06.820
So it's 3/8.

00:03:06.820 --> 00:03:09.330
So the difference between
the heights of the beefsteak

00:03:09.330 --> 00:03:12.136
and Roma tomato plants-- 3/8.

00:03:12.136 --> 00:03:13.760
And everything we've
been doing insofar

00:03:13.760 --> 00:03:18.440
has been in feet,
so 3/8 of a foot.


Example of adding fractions with unlike denominators word problem

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.530
Cindy and Michael need
1 gallon of orange paint

00:00:03.530 --> 00:00:06.970
for the giant cardboard pumpkin
they are making for Halloween.

00:00:06.970 --> 00:00:10.200
Cindy has 2/5 of a
gallon of red paint.

00:00:10.200 --> 00:00:13.630
Michael has got 1/2 a
gallon of yellow paint.

00:00:13.630 --> 00:00:15.440
If they mix their
paints together,

00:00:15.440 --> 00:00:19.207
will they have the
1 gallon they need?

00:00:19.207 --> 00:00:20.290
So let's think about that.

00:00:20.290 --> 00:00:27.340
We're going to add the 2/5
of a gallon of red paint,

00:00:27.340 --> 00:00:30.700
and we're going to add that to
1/2 a gallon of yellow paint.

00:00:30.700 --> 00:00:34.850
And we want to see if this
gets to being 1 whole gallon.

00:00:34.850 --> 00:00:37.090
So whenever we add
fractions, right over here

00:00:37.090 --> 00:00:38.423
we're not adding the same thing.

00:00:38.423 --> 00:00:39.460
Here we're adding 2/5.

00:00:39.460 --> 00:00:41.400
Here we're adding 1/2.

00:00:41.400 --> 00:00:43.620
So in order to be able
to add these two things,

00:00:43.620 --> 00:00:46.840
we need to get to a
common denominator.

00:00:46.840 --> 00:00:49.840
And the common denominator,
or the best common denominator

00:00:49.840 --> 00:00:53.880
to use, is the number that is
the smallest multiple of both 5

00:00:53.880 --> 00:00:55.410
and 2.

00:00:55.410 --> 00:00:58.342
And since 5 and 2 are
both prime numbers,

00:00:58.342 --> 00:01:00.550
the smallest number's just
going to be their product.

00:01:00.550 --> 00:01:02.100
10 is the smallest
number that we

00:01:02.100 --> 00:01:06.020
can think of that is
divisible by both 5 and 2.

00:01:06.020 --> 00:01:09.644
So let's rewrite each of
these fractions with 10

00:01:09.644 --> 00:01:10.435
as the denominator.

00:01:13.050 --> 00:01:15.420
So 2/5 is going to
be something over 10,

00:01:15.420 --> 00:01:20.430
and 1/2 is going to
be something over 10.

00:01:20.430 --> 00:01:24.080
And to help us visualize
this, let me draw a grid.

00:01:24.080 --> 00:01:26.640
Let me draw a grid
with tenths in it.

00:01:26.640 --> 00:01:30.710
So, that's that, and that's
that right over here.

00:01:30.710 --> 00:01:32.830
So each of these are in tenths.

00:01:32.830 --> 00:01:36.570
These are 10 equal segments
this bar is divided into.

00:01:36.570 --> 00:01:40.250
So let's try to visualize what
2/5 looks like on this bar.

00:01:40.250 --> 00:01:42.880
Well, right now it's
divided into tenths.

00:01:42.880 --> 00:01:45.980
If we were to divide
this bar into fifths,

00:01:45.980 --> 00:01:47.890
then we're going
to have-- actually,

00:01:47.890 --> 00:01:49.340
let me do it in that same color.

00:01:49.340 --> 00:01:56.850
So it's going to be, this
is 1 division, 2, 3, 4.

00:01:56.850 --> 00:02:01.030
So notice if you go
between the red marks,

00:02:01.030 --> 00:02:03.410
these are each a
fifth of the bar.

00:02:06.040 --> 00:02:12.360
And we have two of them, so
we're going to go 1 and 2.

00:02:12.360 --> 00:02:15.420
This right over here,
this part of the bar,

00:02:15.420 --> 00:02:17.000
represents 2/5 of it.

00:02:17.000 --> 00:02:19.610
Now let's do the
same thing for 1/2.

00:02:19.610 --> 00:02:23.570
So let's divide this
bar exactly in half.

00:02:23.570 --> 00:02:25.570
So, let me do that.

00:02:25.570 --> 00:02:28.030
I'm going to divide
it exactly in half.

00:02:28.030 --> 00:02:31.860
And 1/2 literally represents
1 of the 2 equal sections.

00:02:31.860 --> 00:02:36.330
So this is one 1/2.

00:02:36.330 --> 00:02:39.380
Now, to go from
fifths to tenths,

00:02:39.380 --> 00:02:41.680
you're essentially taking
each of the equal sections

00:02:41.680 --> 00:02:43.500
and you're multiplying by 2.

00:02:48.660 --> 00:02:49.900
You had 5 equals sections.

00:02:49.900 --> 00:02:53.130
You split each of those into
2, so you have twice as many.

00:02:53.130 --> 00:02:55.250
You now have 10 equal sections.

00:02:55.250 --> 00:02:57.290
So those 2 sections
that were shaded in,

00:02:57.290 --> 00:02:59.760
well, you are going to
multiply by 2 the same way.

00:02:59.760 --> 00:03:04.140
Those 2 are going
to turn into 4/10.

00:03:07.702 --> 00:03:10.160
And you see it right over here
when we shaded it initially.

00:03:10.160 --> 00:03:11.870
If you Look at the
tenths, you have

00:03:11.870 --> 00:03:17.430
1/10, 2/10, 3/10, and 4/10.

00:03:17.430 --> 00:03:18.970
Let's do the same
logic over here.

00:03:18.970 --> 00:03:21.680
If you have 2
halves and you want

00:03:21.680 --> 00:03:25.550
to make them into 10 tenths, you
have to take each of the halves

00:03:25.550 --> 00:03:27.390
and split them into 5 sections.

00:03:27.390 --> 00:03:29.960
You're going to have 5
times as many sections.

00:03:29.960 --> 00:03:35.440
So to go from 2 to
10, we multiply by 5.

00:03:35.440 --> 00:03:38.110
So, similarly, that
one shaded-in section

00:03:38.110 --> 00:03:43.990
in yellow, that 1/2 is
going to turn into 5/10.

00:03:43.990 --> 00:03:45.520
So we're going to multiply by 5.

00:03:45.520 --> 00:03:46.770
Another way to think about it.

00:03:46.770 --> 00:03:48.320
Whatever we did to
the denominator,

00:03:48.320 --> 00:03:49.445
we had to do the numerator.

00:03:52.260 --> 00:03:54.240
Otherwise, somehow
we're changing

00:03:54.240 --> 00:03:55.810
the value of the fraction.

00:03:55.810 --> 00:03:59.230
So, 1 times 5 is
going to get you to 5.

00:03:59.230 --> 00:04:01.600
And you see that over
here when we shaded it in,

00:04:01.600 --> 00:04:03.590
that 1/2, if you
look at the tenths,

00:04:03.590 --> 00:04:09.920
is equal to 1, 2,
3, 4, 5 tenths.

00:04:09.920 --> 00:04:11.700
And now we are ready to add.

00:04:11.700 --> 00:04:14.260
Now we are ready to
add these two things.

00:04:14.260 --> 00:04:17.519
4/10 plus 5/10,
well, this is going

00:04:17.519 --> 00:04:19.652
to be equal to a certain
number of tenths.

00:04:19.652 --> 00:04:21.860
It's going to be equal to
a certain number of tenths.

00:04:21.860 --> 00:04:23.730
It's going to be equal
to 4 plus 5 tenths.

00:04:28.670 --> 00:04:30.620
And we can once
again visualize that.

00:04:30.620 --> 00:04:33.540
Let me draw our grid again.

00:04:33.540 --> 00:04:35.690
So 4 plus 5/10,
I'll do it actually

00:04:35.690 --> 00:04:38.300
on top of the paint
can right over here.

00:04:38.300 --> 00:04:40.260
So let me color in 4/10.

00:04:40.260 --> 00:04:46.520
So 1, 2, 3, 4.

00:04:46.520 --> 00:04:48.280
And then let me
color in the 5/10.

00:04:48.280 --> 00:04:51.050
And notice that was
exactly the 4/10 here,

00:04:51.050 --> 00:04:52.980
which is exactly the 2/5.

00:04:52.980 --> 00:05:01.330
Let me color in the
5/10-- 1, 2, 3, 4, and 5.

00:05:01.330 --> 00:05:03.716
And so how many total
tenths do we have?

00:05:03.716 --> 00:05:10.130
We have a total of 1,
2, 3, 4, 5, 6, 7, 8, 9.

00:05:10.130 --> 00:05:12.410
9 of the tenths
are now shaded in.

00:05:12.410 --> 00:05:14.770
We had 9/10 of a
gallon of paint.

00:05:14.770 --> 00:05:16.930
So now to answer their
question, will they

00:05:16.930 --> 00:05:18.430
have the gallon they need?

00:05:18.430 --> 00:05:19.990
No, they have less than a whole.

00:05:19.990 --> 00:05:21.790
A gallon would be 10 tenths.

00:05:21.790 --> 00:05:23.210
They only have 9 tenths.

00:05:23.210 --> 00:05:27.240
So no, they do not have
enough of a gallon.

00:05:27.240 --> 00:05:29.860
Now, another way you could
have thought about this,

00:05:29.860 --> 00:05:34.240
you could have said, hey,
look, 2/5 is less than 1/2,

00:05:34.240 --> 00:05:36.630
and you could even visualize
that right over here.

00:05:36.630 --> 00:05:39.120
So if I have something
less than 1/2 plus 1/2,

00:05:39.120 --> 00:05:40.340
I'm not going to get a whole.

00:05:40.340 --> 00:05:42.881
So either way you could think
about it, but this way at least

00:05:42.881 --> 00:05:46.720
we can think it through with
actually adding the fractions.


Mixed numbers on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.960
In the previous video, splitting
this three-pack of soap

00:00:03.960 --> 00:00:06.620
with my wife, she cut it
right down the middle.

00:00:06.620 --> 00:00:10.820
So we each got half of all
three bars, or we got 3/2,

00:00:10.820 --> 00:00:16.409
or each of us got
3/2 bars of soap.

00:00:16.409 --> 00:00:17.950
Now, just to make
sure that we really

00:00:17.950 --> 00:00:20.570
understand what 3/2
is-- and remember,

00:00:20.570 --> 00:00:23.140
3/2 is really the same
thing as 3 divided by 2.

00:00:23.140 --> 00:00:24.950
If someone says 3
divided by 2, you

00:00:24.950 --> 00:00:27.600
could say, hey, I'm going
to represent that as 3/2.

00:00:27.600 --> 00:00:29.720
Or someone says 3/2,
you could say, well,

00:00:29.720 --> 00:00:33.565
that's the same thing
as 3 divided by 2.

00:00:37.430 --> 00:00:39.589
But just to really
understand this number,

00:00:39.589 --> 00:00:41.630
let's actually think about
where we would plot it

00:00:41.630 --> 00:00:43.170
on a number line, as well.

00:00:43.170 --> 00:00:45.170
So let me draw a
number line here.

00:00:45.170 --> 00:00:48.240
I'm going to start that as 0.

00:00:48.240 --> 00:00:52.620
Let me draw a line,
as straight a line

00:00:52.620 --> 00:00:55.820
as I am capable of drawing.

00:00:55.820 --> 00:01:00.410
And let's say that this
right over here represents 1.

00:01:00.410 --> 00:01:03.490
So that would be 1 bar
of soap in this example.

00:01:03.490 --> 00:01:06.770
And let's say that this
right over here is 1/2 bar.

00:01:06.770 --> 00:01:08.690
This is 1/2 bar.

00:01:08.690 --> 00:01:14.560
And let's say that
this is 1 and 1/2 bars.

00:01:14.560 --> 00:01:18.890
And let's say that this right
over here is 2 bars of soap.

00:01:18.890 --> 00:01:20.857
So we have 3/2.

00:01:20.857 --> 00:01:21.940
So let's think about this.

00:01:21.940 --> 00:01:25.800
Remember, 3/2 is 3 times 1/2.

00:01:25.800 --> 00:01:28.230
Or, since we already
split this into halves,

00:01:28.230 --> 00:01:30.260
we just have to go
3 of these halves.

00:01:30.260 --> 00:01:31.650
So let's do that.

00:01:31.650 --> 00:01:33.640
So this is 1/2.

00:01:33.640 --> 00:01:35.280
This is 1/2.

00:01:35.280 --> 00:01:38.240
Now we're going to have 2/2.

00:01:38.240 --> 00:01:39.220
That takes us to 1.

00:01:39.220 --> 00:01:42.870
And we know that because
2/2 is equal to 1.

00:01:42.870 --> 00:01:44.820
And now let's do our third half.

00:01:44.820 --> 00:01:47.380
Our third half is
right over here.

00:01:47.380 --> 00:01:49.160
And so that is 3/2.

00:01:49.160 --> 00:01:54.105
And we see 3/2 is the exact
same thing as 1 and 1/2.

00:01:54.105 --> 00:01:55.480
Now, just to hit
that point home,

00:01:55.480 --> 00:02:00.140
we already saw that 3/2 is the
same thing as 3 divided by 2.

00:02:00.140 --> 00:02:05.280
If you were to divide 3 by
2, you would get-- well,

00:02:05.280 --> 00:02:09.750
2 goes into 3 exactly one time.

00:02:09.750 --> 00:02:11.680
And then what would
you have left over?

00:02:11.680 --> 00:02:13.615
Well, you would
have 1/2 left over.

00:02:16.300 --> 00:02:18.834
So 3/2 is the same
thing as 1 and 1/2,

00:02:18.834 --> 00:02:20.250
which is exactly
what we saw here.

00:02:20.250 --> 00:02:22.639
You could view this as we
have 1 and 1/2 bars of soap.

00:02:22.639 --> 00:02:24.930
And this is where you would
plot it on the number line.

00:02:24.930 --> 00:02:29.941
3/2 is halfway in
between 1 and 2.