Linear models example 1

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.827
- [Voiceover] Naoya read
a book cover to cover

00:00:02.827 --> 00:00:06.356
in a single session, at a
rate of 55 pages per hour.

00:00:06.356 --> 00:00:08.713
55 pages per hour.

00:00:08.713 --> 00:00:12.684
After reading for four hours,
after reading for four hours,

00:00:12.684 --> 00:00:16.050
he had 330 pages left to read.

00:00:16.050 --> 00:00:17.954
How long is the book in pages and how long

00:00:17.954 --> 00:00:21.252
did it take Naoya to read the entire book?

00:00:21.252 --> 00:00:23.698
Well there's a bunch of ways
that we could approach this.

00:00:23.698 --> 00:00:26.656
Like always, pause the video
and try to figure it out.

00:00:26.656 --> 00:00:28.955
But I will go with one
way first, and then maybe

00:00:28.955 --> 00:00:30.685
I'll tackle it another way.

00:00:30.685 --> 00:00:33.355
So one way is, let's just
set up a little table here.

00:00:33.355 --> 00:00:36.791
And on one column, let's think about,

00:00:36.791 --> 00:00:39.856
let's think about how long Naoya has read

00:00:39.856 --> 00:00:41.888
in hours, so we'll just call that "time."

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"T" for time in hours.

00:00:43.850 --> 00:00:46.718
This is how long he has
read, and in this column,

00:00:46.718 --> 00:00:48.884
this is how much he has left to read.

00:00:48.884 --> 00:00:51.961
So let's just write
that "L," left to read.

00:00:51.961 --> 00:00:54.654
And then this is going
to be given in pages.

00:00:54.654 --> 00:00:57.580
This is going to be given in pages.

00:00:57.580 --> 00:00:59.856
So lets see, we don't know at zero hours,

00:00:59.856 --> 00:01:02.526
we don't know how many
pages he has left to read,

00:01:02.526 --> 00:01:05.584
at, if we knew that, at zero
hours, before he's read,

00:01:05.584 --> 00:01:07.750
the pages left to read, that's
gonna be the entire book.

00:01:07.750 --> 00:01:10.526
So that would answer this
question, "How long is the book?"

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So this is gonna be interesting.

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We don't know what happens at one hour,

00:01:14.288 --> 00:01:17.121
after two hours, after three hours.

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But they do tell us, that
after reading for four hours,

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after reading for four
hours, he had 330 pages

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left to read.

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So after four hours, he had 330 pages,

00:01:30.658 --> 00:01:34.211
330 pages left to read.

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So, based on this, and the fact that

00:01:37.647 --> 00:01:40.480
he's reading at 55 pages per hour,

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can we back up to figure out all of this?

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Well remember, every hour that goes by,

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every hour that goes by,
he has read 55 pages.

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So if he's read 55
pages, remember this 330,

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this isn't how many pages he's read,

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this is how many he has left to read.

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So after every hour he has
55 fewer pages to read.

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So at three hours, he should
have 55 more pages to read.

00:02:04.756 --> 00:02:08.181
So this should be, at
three hours, 330 plus 55

00:02:08.181 --> 00:02:10.863
would be 385.

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Now let's see whether this makes sense.

00:02:12.721 --> 00:02:15.275
So when our change in time, this triangle

00:02:15.275 --> 00:02:17.725
is just the Greek letter
Delta, means "change in."

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When our change in time is
plus one, plus one hour,

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our change in pages left
to read is going to be

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equal to negative 55 pages.

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And that makes sense,
the pages left to read

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goes down every hour.

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We're measuring not how much he's read,

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we're measuring how much
he has left to read.

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So that should go down
by 55 pages every hour.

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Or if we were to go
backwards through time,

00:02:44.056 --> 00:02:45.287
it should go up.

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So at two hours he should
have 55 more pages to read.

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So what's 385 plus 55?

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We'll let's see, 385 plus
5 is 390, plus 50 is 440.

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So he'd have 440 pages, and
all I did is I added 55.

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And then, after reading for one hour,

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he would have 55 more pages than after

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reading for two hours.

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So 440 plus 55 is 495.

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And then before he started reading,

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or right when he started reading,

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he would have had to
read even 55 more pages,

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'cause after one hour, he
would have read those 55 pages.

00:03:21.323 --> 00:03:26.323
So 495 plus 55 is going to be, let's see,

00:03:26.384 --> 00:03:28.870
it's gonna be, add 5, you get to 500

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plus another 50 is 550 pages.

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So at time equals zero
had had 550 pages to read.

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So that's how long the book is.

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But how long does it take
Naoya to read the entire book?

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Well we could keep going.

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We could say, "Okay, at the fifth hour,

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"this thing's gonna go down by 55."

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So let's see, if this goes down by 50,

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if this goes down by 50,
we're going to get to 280,

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but then you go down five more, it's

00:03:56.488 --> 00:03:59.774
gonna go to 275, and we could keep going

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on and on and on.

00:04:01.155 --> 00:04:02.653
Or we could just say, "Look."

00:04:02.653 --> 00:04:06.124
"He's got 330 pages left
to read, and he's gonna,"

00:04:06.124 --> 00:04:08.086
Let's see, let me write this,
let me write the units down.

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"Pages, and he's reading at
a rate of 55 pages per hour,

00:04:13.659 --> 00:04:17.293
"pages per hour, this is the same thing,

00:04:17.293 --> 00:04:21.554
"this is going to be equal to 330 pages

00:04:21.554 --> 00:04:26.554
"times one over 55 hours per page."

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I'm dividing by something,
the same thing as

00:04:31.321 --> 00:04:34.874
multiplying by its reciprocal, so 55 pages

00:04:34.874 --> 00:04:36.720
per hours, if you divide by that,

00:04:36.720 --> 00:04:38.554
that's the same thing as multiplying by

00:04:38.554 --> 00:04:42.513
1/55th of an hour per page,
is one way to think about it.

00:04:42.513 --> 00:04:43.314
And so what do you get?

00:04:43.314 --> 00:04:45.961
The pages cancel out,
pages divided by pages,

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and you have 330 divided by 55 hours.

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330 divided by 55 hours.

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And what's that going to be?

00:04:55.690 --> 00:04:57.757
Let's see, 30 divided by 5 is 6,

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300 divided by 50 is 6,
so this is going to be

00:05:00.683 --> 00:05:02.053
equal to 6 hours.

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Now we have to be very
careful, you might want

00:05:03.353 --> 00:05:05.545
to write six hours here, but this is

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six hours after this point.

00:05:07.960 --> 00:05:10.293
Six hours after that point.

00:05:10.293 --> 00:05:13.161
So in total, it's going
to take him ten hours.

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At four hours, he had
330 pages left to read

00:05:15.053 --> 00:05:16.423
and then there was six more hours to

00:05:16.423 --> 00:05:18.490
finish those 330 pages.

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So it's gonna take a total of ten hours.

00:05:20.649 --> 00:05:22.623
And we could also see if this makes sense.

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If he reads 550 pages in ten hours,

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If he reads 550 pages in ten hours,

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in ten hours, what's his rate of reading?

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Well, 550 divided by 10, that's gonna be

00:05:38.958 --> 00:05:43.958
the same thing as 55 pages per hour.

00:05:44.113 --> 00:05:46.249
Which is consistent
with what we just read,

00:05:46.249 --> 00:05:49.616
that he reads at 55 pages per hour.

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Now another way that you
could have tackled this,

00:05:51.323 --> 00:05:52.351
you could say, "Hey, this in gonna be

00:05:52.351 --> 00:05:53.826
"a linear equation!"

00:05:53.826 --> 00:05:55.881
If you're well versed
with your linear equations

00:05:55.881 --> 00:05:58.279
you could say, "Look, he's
reading at a constant rate."

00:05:58.279 --> 00:05:59.754
And if we're talking about, let's say,

00:05:59.754 --> 00:06:00.891
let me just set up a function here,

00:06:00.891 --> 00:06:04.456
we could say left to read,
as a function of time,

00:06:04.456 --> 00:06:07.486
is going to be equal to,
he's reading at a rate of

00:06:07.486 --> 00:06:10.412
55 pages per hour, so the
amount he's left to read

00:06:10.412 --> 00:06:13.210
is gonna go down by 55 pages per hour.

00:06:13.210 --> 00:06:18.210
So it's gonna go down
55, 55 pages every hour.

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So you could write it like that, plus it's

00:06:21.557 --> 00:06:23.786
whatever the initial number of pages were.

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Plus whatever the initial
number of pages were.

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And then you could use this information

00:06:27.293 --> 00:06:29.417
right over here to solve for b.

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We know that when time
is equal to four hours,

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we know that when, let
me do it in that color.

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We know that when time
is equal to four hours,

00:06:38.090 --> 00:06:41.526
that the pages left to read are 330.

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Pages left to read are 330, and so you get

00:06:43.953 --> 00:06:48.953
330 is equal to, let's
see, negative 55 times 4 is

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negative 220.

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Negative 220 plus,

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negative 220 plus b,

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plus b, and then if you
add 220 to both sides,

00:07:01.914 --> 00:07:06.488
220, add 220, to both
sides, you're going to

00:07:06.488 --> 00:07:08.392
be left with, on this
side, you're gonna get

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550 is equal to, these cancel
out, they add up to zero,

00:07:13.291 --> 00:07:14.719
is equal to b.

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And so the function
that describes how much

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he has left to read,
the amount he has left

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to read is a function
of time, is, well he's

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reading 55 pages per hour, so the amount

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he has left to read's
gonna go down every hour,

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that's why we have that negative there.

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So the negative 55 pages per hour,

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times the number of hours, plus 550.

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So if someone said, "Well
how long is the book?"

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Well that's going to be L of zero.

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That's going to be how many pages

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he has left to read at time zero.

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And we see that L of
zero is going to be 550.

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When T is equal to zero,
let me write this down,

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L of zero is going to be equal to 550.

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When T is equal to zero,
this term goes away

00:07:55.789 --> 00:07:57.983
and you just have 550 pages.

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And, just like that,
we could have answered

00:08:01.025 --> 00:08:02.615
this question.

00:08:02.615 --> 00:08:04.520
And then you could say,
"Well look, if the book

00:08:04.520 --> 00:08:07.956
"is 550 pages, and if he's
reading at 55 pages per hour,

00:08:07.956 --> 00:08:09.581
"how many hours is it gonna take?"

00:08:09.581 --> 00:08:13.018
Well 550 pages divided
by 55 pages per hour

00:08:13.018 --> 00:08:14.794
is going to get you 10 hours.

00:08:14.794 --> 00:08:16.095
And we could have done that here actually,

00:08:16.095 --> 00:08:16.919
now that I think about it.

00:08:16.919 --> 00:08:19.112
Instead of saying 330
pages divided by 55 pages

00:08:19.112 --> 00:08:20.626
per hour, and say ,"Hey,
that's gonna take us

00:08:20.626 --> 00:08:22.620
"six more hours from this point."

00:08:22.620 --> 00:08:24.419
We already knew this,
and we could have said,

00:08:24.419 --> 00:08:26.994
"Look, if we have 550 pages to read,

00:08:26.994 --> 00:08:29.083
"and we're reading at 55 pages per hour,

00:08:29.083 --> 00:08:30.918
"it's gonna take us ten hours."

00:08:30.918 --> 00:08:32.624
So you see, there's a
ton of different ways

00:08:32.624 --> 00:08:33.659
to solve these problems.

00:08:33.659 --> 00:08:34.958
And they're all really,
at the end of the day,

00:08:34.958 --> 00:08:36.194
representing the same thing.

00:08:36.194 --> 00:08:38.389
That's what's neat about mathematics.

00:08:38.389 --> 00:08:42.255
They're just ways of representing how to

00:08:42.255 --> 00:08:43.822
think about things, and
at the end of the day

00:08:43.822 --> 00:08:46.690
we're really just thinking
about the same problem,

00:08:46.690 --> 00:08:49.058
just representing it with different ways

00:08:49.058 --> 00:08:51.087
and different symbols.


Constructing linear functions example 2

WEBVTT
Kind: captions
Language: en

00:00:00.423 --> 00:00:02.118
- [Voiceover] Hiro
painted his room at a rate

00:00:02.118 --> 00:00:05.114
of eight square meters per hour.

00:00:05.114 --> 00:00:06.855
After three hours of painting,

00:00:06.855 --> 00:00:10.327
he had 28 square meters left to paint.

00:00:10.327 --> 00:00:12.846
So after three hours of painting,

00:00:12.846 --> 00:00:15.701
he had 28 square meters left to paint.

00:00:15.701 --> 00:00:16.724
So they're talking in terms

00:00:16.724 --> 00:00:17.722
of how much we have left to paint,

00:00:17.722 --> 00:00:19.650
not how much we have painted.

00:00:19.650 --> 00:00:22.691
Let A of t denote the area to paint,

00:00:22.691 --> 00:00:26.194
A measured in square meters
as a function of time, t,

00:00:26.194 --> 00:00:27.517
measured in hours.

00:00:27.517 --> 00:00:30.141
So A of t, once again, this
is how much we have to paint,

00:00:30.141 --> 00:00:32.649
not how much we have painted.

00:00:32.649 --> 00:00:35.261
Write the function's formula.

00:00:35.261 --> 00:00:36.817
So what I like to do is,

00:00:36.817 --> 00:00:38.791
let's just think about
a couple points here.

00:00:38.791 --> 00:00:41.113
Let's just make it a
little bit tangible for us.

00:00:41.113 --> 00:00:43.481
So, let me, I'll do this
in a different color.

00:00:43.481 --> 00:00:44.990
So let's think about

00:00:44.990 --> 00:00:47.858
what A of t is at different times.

00:00:47.858 --> 00:00:50.087
So, this is time.

00:00:50.087 --> 00:00:53.326
This is A as a function of time.

00:00:53.326 --> 00:00:54.673
And they give us one of them.

00:00:54.673 --> 00:00:57.053
They say, "after three hours of painting,

00:00:57.053 --> 00:01:00.153
"after three hours of painting, he had 28,

00:01:00.153 --> 00:01:03.555
"28 square meters left to paint."

00:01:03.555 --> 00:01:06.086
And once again, A of t is how
much we have left to paint,

00:01:06.086 --> 00:01:08.025
not how much we have painted.

00:01:08.025 --> 00:01:09.139
So I'm gonna leave some space here

00:01:09.139 --> 00:01:10.289
for some other values,

00:01:10.289 --> 00:01:11.926
maybe zero, one, two.

00:01:11.926 --> 00:01:13.516
Let's write three over here.

00:01:13.516 --> 00:01:17.974
After three hours, he had 28
square meters left to paint.

00:01:17.974 --> 00:01:18.822
We're just assuming that

00:01:18.822 --> 00:01:20.621
this is going to be in square meters,

00:01:20.621 --> 00:01:22.421
and this is in hours.

00:01:22.421 --> 00:01:24.522
Now, they tell us that he painted his room

00:01:24.522 --> 00:01:27.286
at a rate of eight square meters per hour.

00:01:27.286 --> 00:01:29.317
So let's actually back up a little bit.

00:01:29.317 --> 00:01:30.153
Let's back up.

00:01:30.153 --> 00:01:33.717
Let's say after two hours,

00:01:33.717 --> 00:01:36.817
how much would he have had left to paint?

00:01:36.817 --> 00:01:38.861
What would A of t been?

00:01:38.861 --> 00:01:40.521
Would it have been more than 28

00:01:40.521 --> 00:01:43.319
or would it have been less than 28?

00:01:43.319 --> 00:01:46.384
Well, he's painting eight square meters

00:01:46.384 --> 00:01:48.253
per hour,

00:01:48.253 --> 00:01:49.751
so every hour that goes by,

00:01:49.751 --> 00:01:51.086
he's painting more,

00:01:51.086 --> 00:01:53.524
but A of t isn't how much he has painted,

00:01:53.524 --> 00:01:56.160
it's how much he has left to paint.

00:01:56.160 --> 00:01:59.724
So he should have less to paint

00:01:59.724 --> 00:02:01.721
as time goes up.

00:02:01.721 --> 00:02:03.951
So as time goes up,

00:02:03.951 --> 00:02:08.726
as time increases, A of t should go down.

00:02:08.726 --> 00:02:11.885
So at two hours, he
should have more to paint

00:02:11.885 --> 00:02:12.988
than at three hours,

00:02:12.988 --> 00:02:13.719
because remember,

00:02:13.719 --> 00:02:15.891
A of t is how much he has left to paint.

00:02:15.891 --> 00:02:17.992
So how much more would
he have had to paint

00:02:17.992 --> 00:02:20.627
at two hours than at three hours?

00:02:20.627 --> 00:02:23.251
Well it tells us that he
has eight square meters,

00:02:23.251 --> 00:02:26.386
he paints at a rate of eight
square meters per hour.

00:02:26.386 --> 00:02:27.860
So between two and three hours,

00:02:27.860 --> 00:02:30.217
he would've painted eight square meters.

00:02:30.217 --> 00:02:31.274
So at two hours, he would've had

00:02:31.274 --> 00:02:34.025
eight square meters more to paint.

00:02:34.025 --> 00:02:36.521
So, if you add eight to
this right over here,

00:02:36.521 --> 00:02:38.449
you would be at 36.

00:02:38.449 --> 00:02:41.421
So he would've had 36
square meters to paint

00:02:41.421 --> 00:02:43.186
at two hours.

00:02:43.186 --> 00:02:45.426
And what about one hour?

00:02:45.426 --> 00:02:48.085
So at one hour, he would've
had eight more square meters

00:02:48.085 --> 00:02:48.851
to paint.

00:02:48.851 --> 00:02:53.019
So 36 plus eight, that is 44.

00:02:53.019 --> 00:02:56.015
And at zero hours, what
would he have had to paint?

00:02:56.015 --> 00:02:58.186
So, let me do this another
color, at zero hours.

00:02:58.186 --> 00:03:01.483
Well he would've had to paint
eight more square meters.

00:03:01.483 --> 00:03:03.584
So 44 plus eight is 52.

00:03:03.584 --> 00:03:05.210
Let's think about
whether that makes sense.

00:03:05.210 --> 00:03:06.359
If right when he was starting,

00:03:06.359 --> 00:03:08.728
he had 52 square meters to paint.

00:03:08.728 --> 00:03:10.992
Then, an hour goes by,

00:03:10.992 --> 00:03:13.592
so your change in time is one hour,

00:03:13.592 --> 00:03:17.627
and then your change in how
much he has left to paint,

00:03:17.627 --> 00:03:19.925
it goes down by eight.

00:03:19.925 --> 00:03:23.548
Change in A is equal to negative eight.

00:03:23.548 --> 00:03:24.523
That makes sense.

00:03:24.523 --> 00:03:26.984
His rate of change should be negative

00:03:26.984 --> 00:03:30.421
because the amount he has
left to paint goes down

00:03:30.421 --> 00:03:32.394
as time goes forward.

00:03:32.394 --> 00:03:34.519
So this was pretty interesting.

00:03:34.519 --> 00:03:35.448
Now let's see if we can actually

00:03:35.448 --> 00:03:36.992
construct a formula,

00:03:36.992 --> 00:03:40.161
or the formula that
describes this function.

00:03:40.161 --> 00:03:42.251
Well this is happening at a constant rate.

00:03:42.251 --> 00:03:44.840
Every time t goes up by one,

00:03:44.840 --> 00:03:47.174
we see A of t goes down by eight.

00:03:47.174 --> 00:03:49.426
t goes up by one, A of
t goes down by eight.

00:03:49.426 --> 00:03:50.483
They tell us that, and that's because

00:03:50.483 --> 00:03:53.652
he paints it at a rate of
eight square meters per hour.

00:03:53.652 --> 00:03:55.591
So whenever you're describing something

00:03:55.591 --> 00:03:57.553
that's happening at a constant rate,

00:03:57.553 --> 00:04:00.711
that can be described
by a linear function.

00:04:00.711 --> 00:04:03.695
And a linear function will have the form

00:04:03.695 --> 00:04:08.420
A of t is equal to your
rate of change times time,

00:04:08.420 --> 00:04:10.417
plus wherever you started,

00:04:10.417 --> 00:04:12.646
and m and b are just the
letters that people tend to use

00:04:12.646 --> 00:04:14.086
for your rate of change,

00:04:14.086 --> 00:04:16.187
your slope, if you were graphing this,

00:04:16.187 --> 00:04:17.952
and b, where you started off,

00:04:17.952 --> 00:04:19.960
and this would be your vertical intercept,

00:04:19.960 --> 00:04:21.388
sometimes you call it your y-intercept,

00:04:21.388 --> 00:04:23.385
but in this case it would
be your A-intercept,

00:04:23.385 --> 00:04:26.125
if we're thinking about the actual,

00:04:26.125 --> 00:04:28.319
it would help us find the A-intercept

00:04:28.319 --> 00:04:30.618
if we were graphing this thing.

00:04:30.618 --> 00:04:33.126
But we actually already
know both of these things.

00:04:33.126 --> 00:04:35.424
We know what our rate of change is.

00:04:35.424 --> 00:04:36.620
It is negative eight.

00:04:36.620 --> 00:04:38.362
I mean, we could say,
"well what's our slope?"

00:04:38.362 --> 00:04:42.507
Our slope is change of A over change in t.

00:04:42.507 --> 00:04:44.515
Change in A,

00:04:44.515 --> 00:04:45.757
let me write it this way,

00:04:45.757 --> 00:04:47.789
let me do it in a different
color just for fun.

00:04:47.789 --> 00:04:49.623
So our,

00:04:49.623 --> 00:04:54.186
our slope is just our change

00:04:54.186 --> 00:04:57.358
in our dependent variable

00:04:57.358 --> 00:04:59.808
over our change in our
independent variable,

00:04:59.808 --> 00:05:01.688
which is equal to negative eight.

00:05:01.688 --> 00:05:02.478
They tell us that.

00:05:02.478 --> 00:05:03.952
It's equal to negative eight.

00:05:03.952 --> 00:05:06.390
So this thing is equal to negative eight,

00:05:06.390 --> 00:05:09.688
and b is going to be equal to A of zero.

00:05:09.688 --> 00:05:12.149
A of zero,

00:05:12.149 --> 00:05:13.890
well when t is equal to zero,

00:05:13.890 --> 00:05:15.551
this term right here goes away

00:05:15.551 --> 00:05:16.456
and you're just left with b.

00:05:16.456 --> 00:05:17.989
A of zero is equal to b.

00:05:17.989 --> 00:05:19.661
And we know what A of zero is.

00:05:19.661 --> 00:05:22.284
It is equal to 52.

00:05:22.284 --> 00:05:25.791
So we know this right over here is 52.

00:05:25.791 --> 00:05:26.824
And we're done!

00:05:26.824 --> 00:05:29.285
We know that, I'll just
rewrite it just for fun,

00:05:29.285 --> 00:05:30.957
A of t,

00:05:30.957 --> 00:05:35.322
the area that he has left to
paint as a function of time,

00:05:35.322 --> 00:05:38.619
is equal to negative eight

00:05:38.619 --> 00:05:42.141
times time,

00:05:42.141 --> 00:05:45.447
plus 52.

00:05:45.447 --> 00:05:47.387
And you can confirm that
the units make sense,

00:05:47.387 --> 00:05:49.553
because this negative eight,

00:05:49.553 --> 00:05:51.527
and actually let's let me write
it one time with the units,

00:05:51.527 --> 00:05:54.418
just 'cause it is an important
thing to think about.

00:05:54.418 --> 00:05:56.218
Area as a function of time,

00:05:56.218 --> 00:05:58.087
this is how much he has left to paint,

00:05:58.087 --> 00:06:01.830
is going to be equal to

00:06:01.830 --> 00:06:05.393
negative eight

00:06:05.393 --> 00:06:08.888
square meters per hour,

00:06:08.888 --> 00:06:12.580
so negative eight meters squared per hour,

00:06:12.580 --> 00:06:17.445
times t hours,

00:06:17.445 --> 00:06:19.046
maybe I'll write out "hours"

00:06:19.046 --> 00:06:20.521
so you don't think it's a variable,

00:06:20.521 --> 00:06:21.554
t hours,

00:06:21.554 --> 00:06:23.249
let me write the hours over here,

00:06:23.249 --> 00:06:28.044
t hours plus 52 square meters.

00:06:28.044 --> 00:06:32.214
Plus, let me do it right over here.

00:06:32.214 --> 00:06:35.940
Plus, I have trouble switching colors,

00:06:35.940 --> 00:06:40.886
plus 52 square meters.

00:06:40.886 --> 00:06:44.255
And you see hours divided
by hours cancels out

00:06:44.255 --> 00:06:45.718
and you'll just be left
with meters squared.

00:06:45.718 --> 00:06:48.226
You'd have negative eight t square meters

00:06:48.226 --> 00:06:50.374
plus 52 square meters,

00:06:50.374 --> 00:06:52.974
and so the A of t is
going to be given to you

00:06:52.974 --> 00:06:55.587
in square meters.

00:06:55.587 --> 00:06:57.792
Now there's other ways that you might have

00:06:57.792 --> 00:06:58.791
wanted to tackle this.

00:06:58.791 --> 00:06:59.488
You might've immediately said,

00:06:59.488 --> 00:07:00.823
"Hey look, my rate of change

00:07:00.823 --> 00:07:02.820
" is eight square meters per hour."

00:07:02.820 --> 00:07:03.981
But you have to be very careful there.

00:07:03.981 --> 00:07:06.024
You might've said, "Oh my rate of change,

00:07:06.024 --> 00:07:07.150
"maybe it's going to be positive

00:07:07.150 --> 00:07:07.928
"eight square meters per hour."

00:07:07.928 --> 00:07:08.717
But you have to be clear

00:07:08.717 --> 00:07:10.784
that A is not how much he's painting,

00:07:10.784 --> 00:07:12.775
it's how much he has left to paint.

00:07:12.775 --> 00:07:15.352
So his rate of change of how
much he has left to paint

00:07:15.352 --> 00:07:18.220
is decreasing at eight
square meters per hour.

00:07:18.220 --> 00:07:20.589
So you might've said, "Okay, immediately,

00:07:20.589 --> 00:07:23.561
"my formula would look like this:

00:07:23.561 --> 00:07:25.128
"A of t is going to to be equal to

00:07:25.128 --> 00:07:29.052
"negative eight times t plus some b."

00:07:29.052 --> 00:07:30.759
And then you could've
used this information

00:07:30.759 --> 00:07:32.106
right over here to solve for b.

00:07:32.106 --> 00:07:35.466
You say, "Hey, when t is equal to three,

00:07:35.466 --> 00:07:36.454
"A is equal to 28."

00:07:36.454 --> 00:07:37.592
You could've just used this information

00:07:37.592 --> 00:07:40.042
right over here and
substituted right over here.

00:07:40.042 --> 00:07:43.049
So when t is equal to
three, when this is three,

00:07:43.049 --> 00:07:44.686
A of t is 28.

00:07:44.686 --> 00:07:47.453
And you would've gotten 28
is equal to negative eight

00:07:47.453 --> 00:07:49.182
times three,

00:07:49.182 --> 00:07:51.845
so negative 24, plus b.

00:07:51.845 --> 00:07:54.585
And then you would've
added 24 to both sides.

00:07:54.585 --> 00:07:56.152
Whoops.

00:07:56.152 --> 00:08:00.320
You would've added 24 to both sides,

00:08:00.320 --> 00:08:02.784
and you would've gotten

00:08:02.784 --> 00:08:07.568
28 plus 24 is 52.

00:08:08.168 --> 00:08:10.550
And then on the right side,
you would just have b.

00:08:10.550 --> 00:08:13.813
You would get b is equal to 52,

00:08:13.813 --> 00:08:15.693
which is exactly what we got over there.

00:08:15.693 --> 00:08:17.017
I like to do it this way

00:08:17.017 --> 00:08:19.954
just to make sure that
we really conceptualize,

00:08:19.954 --> 00:08:22.468
we really got what was going on.


Constructing linear functions example 1

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:02.871
- [Voiceover] "A lake near
the Arctic Circle is covered

00:00:02.871 --> 00:00:06.006
"by a 2-meter-thick sheet of ice

00:00:06.006 --> 00:00:08.211
"during the cold winter months.

00:00:08.731 --> 00:00:12.043
"When spring arrives, the warm
air gradually melts the ice,

00:00:12.043 --> 00:00:16.686
"causing its thickness to
decrease at a constant rate."

00:00:16.686 --> 00:00:19.242
It's gonna decrease at a constant rate.

00:00:19.682 --> 00:00:24.677
"After 3 weeks, the sheet
is only 1.25 meters thick.

00:00:24.767 --> 00:00:29.767
"After 3 weeks, the sheet
is only 1.25 meters thick.

00:00:31.233 --> 00:00:34.612
"Let S(t) denote the
ice sheet's thickness S

00:00:34.612 --> 00:00:36.865
"(measured in meters) as a function

00:00:36.865 --> 00:00:39.139
"of time (measured in weeks).

00:00:39.509 --> 00:00:42.205
"Write the function's formula."

00:00:42.745 --> 00:00:45.316
Alright, so we have some
interesting things here.

00:00:45.726 --> 00:00:48.404
They've given us some
values for this function.

00:00:48.404 --> 00:00:50.378
We know when time is equal to zero.

00:00:50.378 --> 00:00:54.500
We know that S of zero,
when time equals zero,

00:00:54.740 --> 00:00:57.785
that's when the sheet is two meters thick.

00:00:58.465 --> 00:01:00.966
So S of zero is equal to two.

00:01:01.406 --> 00:01:04.031
And they also tell us
that after three weeks,

00:01:04.031 --> 00:01:07.815
the sheet is only 1.25 meters thick.

00:01:08.225 --> 00:01:10.718
And when we have the function S of t,

00:01:10.718 --> 00:01:14.411
S is measured in meters,
time is measured in weeks.

00:01:14.411 --> 00:01:16.987
So after zero weeks,
we're two meters thick,

00:01:16.987 --> 00:01:19.858
and then they tell us, after three weeks--

00:01:19.858 --> 00:01:21.553
So S of three.

00:01:21.743 --> 00:01:26.735
After three weeks,
we're 1.25 meters thick.

00:01:28.955 --> 00:01:30.353
Or another way to think about it,

00:01:32.473 --> 00:01:36.924
I could write t here in
weeks and S in meters,

00:01:37.234 --> 00:01:40.220
and when time is zero,
we're two meters thick,

00:01:40.480 --> 00:01:42.590
and when time is 1.--

00:01:42.590 --> 00:01:46.278
Sorry, when time is three weeks,

00:01:46.278 --> 00:01:49.184
we are 1.25 meters thick.

00:01:49.734 --> 00:01:54.727
So when our change in time
is equal to positive three,

00:01:55.407 --> 00:01:57.613
we increased our time by three,

00:01:57.613 --> 00:02:00.817
what's our change in thickness?

00:02:00.817 --> 00:02:04.484
Our change in thickness,
the triangle here,

00:02:04.484 --> 00:02:07.735
that's the Greek letter delta,
shorthand for "change in,"

00:02:07.735 --> 00:02:12.611
well, this was negative 0.75.

00:02:16.481 --> 00:02:19.415
So what was the rate of
change over this time?

00:02:19.415 --> 00:02:20.948
And they tell us that the rate

00:02:20.948 --> 00:02:23.270
of change is at a constant rate.

00:02:23.270 --> 00:02:25.615
So whatever it is between
these two periods of time,

00:02:25.615 --> 00:02:27.101
between zero weeks and three weeks,

00:02:27.101 --> 00:02:31.536
it would be that same rate
between any two periods of time,

00:02:31.536 --> 00:02:32.999
between zero week and one week,

00:02:32.999 --> 00:02:36.481
or one week and two weeks,
or 1 1/2 weeks and 1.6 weeks.

00:02:37.231 --> 00:02:40.518
So what is the rate of change

00:02:40.518 --> 00:02:43.911
of thickness relative to time?

00:02:44.721 --> 00:02:47.611
Well, it's going to be change in thickness

00:02:48.211 --> 00:02:49.901
over change in time.

00:02:49.901 --> 00:02:52.712
How much does our
thickness change per time?

00:02:53.412 --> 00:02:54.616
Well, we saw right over here.

00:02:55.116 --> 00:02:56.521
Our thickness went down,

00:02:56.751 --> 00:03:01.746
set 0.75 meters in three weeks.

00:03:06.726 --> 00:03:08.041
Or we could say that this right

00:03:08.041 --> 00:03:10.780
over here is equal to, let's see.

00:03:10.780 --> 00:03:15.076
75 divided by three is 25, so
0.75 divided by three is 0.25.

00:03:15.486 --> 00:03:16.818
We have the negative out there,

00:03:16.818 --> 00:03:21.817
negative 0.25 meters per week.

00:03:23.757 --> 00:03:25.721
So how can we take the information we have

00:03:25.721 --> 00:03:28.877
and express this as a function?

00:03:28.877 --> 00:03:30.340
It's going to be a linear function,

00:03:30.340 --> 00:03:34.102
because we see that we are
changing at a constant rate.

00:03:35.082 --> 00:03:36.169
Let's think about it a little bit.

00:03:36.479 --> 00:03:39.047
Linear functions, one way
we could write it is in--

00:03:39.927 --> 00:03:41.278
So we could write it--

00:03:41.278 --> 00:03:42.601
If we were dealing with x and y,

00:03:42.601 --> 00:03:46.060
you might recognize y
is equal to mx plus b,

00:03:46.060 --> 00:03:48.498
often written as slope-intercept form.

00:03:48.498 --> 00:03:50.773
This is when you're dealing with x as the,

00:03:50.773 --> 00:03:52.701
I guess you could say
the independent variable,

00:03:52.701 --> 00:03:54.256
y as the dependent variable,

00:03:54.256 --> 00:03:56.276
and b would be where you start.

00:03:56.276 --> 00:03:58.514
What happens when x equals zero

00:03:59.154 --> 00:04:01.525
and m is your rate of
change, it's your slope?

00:04:02.235 --> 00:04:05.675
So in this case, we don't have y and x,

00:04:05.675 --> 00:04:07.585
we're going to have S and t.

00:04:08.165 --> 00:04:09.721
We have S as a function of time,

00:04:10.401 --> 00:04:12.483
and it's going to be equal
to the rate of change...

00:04:13.483 --> 00:04:18.137
times time, plus where we started, plus b.

00:04:18.747 --> 00:04:22.513
Now, what is b going to be?

00:04:23.213 --> 00:04:24.535
Well, one way to think about it is,

00:04:24.535 --> 00:04:26.903
well, what's S of zero going to be?

00:04:26.903 --> 00:04:31.788
S of zero is going to
be m times zero, plus b.

00:04:31.788 --> 00:04:33.623
S of zero is going to be b.

00:04:33.623 --> 00:04:35.271
Well, we already know that this ice sheet,

00:04:35.271 --> 00:04:37.152
it starts off at two meters thick.

00:04:37.152 --> 00:04:40.031
So S of zero is equal
to b, is equal to two.

00:04:40.481 --> 00:04:42.191
So b is equal to two.

00:04:42.741 --> 00:04:44.120
And what is m?

00:04:44.120 --> 00:04:46.138
Well, we've already said,
that's our rate of change,

00:04:46.138 --> 00:04:47.732
that is our slope,

00:04:47.732 --> 00:04:52.307
that is how much our thickness changes

00:04:52.307 --> 00:04:54.373
with respect to time.

00:04:54.373 --> 00:04:57.809
And we already figured out
that that's negative 0.25.

00:04:58.479 --> 00:05:02.849
So m is negative 0.25.

00:05:02.849 --> 00:05:06.239
You could say that m is the
slope between this point,

00:05:06.759 --> 00:05:08.815
between the point zero comma two,

00:05:08.815 --> 00:05:11.045
and the point three comma 1.25,

00:05:11.045 --> 00:05:15.643
if we were plotting these points
on a t/S coordinate plane.

00:05:16.403 --> 00:05:18.451
So now we can write what
the function's going to be.

00:05:18.451 --> 00:05:20.657
Maybe I'll do this in a
new color just for fun.

00:05:21.427 --> 00:05:24.256
S of t, thickness as a function of time,

00:05:24.756 --> 00:05:29.756
is going to be equal to m, negative 0.25,

00:05:30.249 --> 00:05:34.636
times time, plus two.

00:05:35.086 --> 00:05:37.220
Or if you want, you
could write it like this,

00:05:37.220 --> 00:05:41.369
two minus 0.25t.

00:05:41.369 --> 00:05:43.273
I actually like this
form a little bit better.

00:05:43.273 --> 00:05:45.711
In my brain, it kind of describes

00:05:45.711 --> 00:05:47.267
what's happening a little bit more.

00:05:47.267 --> 00:05:48.776
When time is equal to
zero, you're starting

00:05:48.776 --> 00:05:52.700
at two meters thick, and then every week

00:05:52.700 --> 00:05:55.208
that goes by, as t increases by one,

00:05:55.208 --> 00:05:58.552
you're going to lose a quarter of a meter.

00:05:58.552 --> 00:06:01.315
You're going to lose, you
have a negative value right

00:06:01.315 --> 00:06:05.285
over here, you're gonna lose 0.25 meters.

00:06:05.725 --> 00:06:07.700
And if you really want
to kind of get this even

00:06:07.700 --> 00:06:09.465
in a deeper level, I
encourage you to graph it,

00:06:09.465 --> 00:06:11.740
and it'll become even
clearer what's going on here.

00:06:11.740 --> 00:06:13.379
That this right over here
is this right over here,

00:06:13.379 --> 00:06:15.069
is this right over here,

00:06:15.069 --> 00:06:20.065
this is the slope of the line

00:06:20.865 --> 00:06:24.673
that represents the solution
set to this equation,

00:06:24.673 --> 00:06:27.900
and this two, this would
be your vertical intercept.

00:06:27.900 --> 00:06:29.688
In this case, it would be your S-intercept

00:06:29.688 --> 00:06:32.637
as opposed to your y-intercept,
when y is the vertical axis.


Graphing linear functions example 2

WEBVTT
Kind: captions
Language: en

00:00:00.366 --> 00:00:04.142
- Omojobi is 220 centimeters tall.

00:00:04.142 --> 00:00:05.971
He wanted to fill up his pool

00:00:05.971 --> 00:00:07.823
so that the water level would be

00:00:07.823 --> 00:00:09.860
as high as he is tall.

00:00:09.860 --> 00:00:11.152
So that the water level, I guess,

00:00:11.152 --> 00:00:13.480
would be 220 centimeters tall.

00:00:13.581 --> 00:00:17.353
The water level rose by
six centimeters each minute

00:00:17.353 --> 00:00:19.338
and reached the desired height

00:00:19.338 --> 00:00:21.092
after 20 minutes.

00:00:21.231 --> 00:00:24.589
Graph the pool's water
level, in centimeters,

00:00:24.589 --> 00:00:28.011
as a function of time, in minutes.

00:00:28.011 --> 00:00:29.808
Well, they do tell us
one interesting thing.

00:00:29.808 --> 00:00:31.665
They say that the water level

00:00:31.665 --> 00:00:35.951
reached the desired
height after 20 minutes.

00:00:35.951 --> 00:00:37.583
And we know what the desired height is,

00:00:37.583 --> 00:00:40.004
the desired height is to be as tall as,

00:00:40.004 --> 00:00:41.721
or to be as deep as he is tall,

00:00:41.721 --> 00:00:43.198
or as high as he is tall.

00:00:43.198 --> 00:00:45.839
And that is 220 centimeters tall.

00:00:45.839 --> 00:00:46.872
So they're telling us essentially

00:00:46.872 --> 00:00:49.244
that the water level of the pool

00:00:49.244 --> 00:00:53.502
after 20 minutes is 220 centimeters,

00:00:53.502 --> 00:00:54.614
so we can plot that.

00:00:54.781 --> 00:00:57.019
So after 20 minutes, so we can plot

00:00:57.019 --> 00:00:58.241
that point right there.

00:00:58.241 --> 00:01:02.198
After 20 minutes, we
are at 220 centimeters.

00:01:02.238 --> 00:01:03.713
So we would be right there.

00:01:03.713 --> 00:01:04.560
Now the other question is

00:01:04.560 --> 00:01:05.683
where would we put this point?

00:01:05.683 --> 00:01:09.968
We need another data point
in order to define a line.

00:01:10.043 --> 00:01:12.425
And so they tell us that the water,

00:01:12.425 --> 00:01:13.381
you might be tempted to say,

00:01:13.381 --> 00:01:14.313
"Okay, maybe the water level was

00:01:14.313 --> 00:01:17.335
"at zero to begin with," but
they didn't tell us that.

00:01:17.335 --> 00:01:19.622
Maybe when he started filling the pool

00:01:19.622 --> 00:01:21.302
there was already some water in there.

00:01:21.302 --> 00:01:22.746
So we have to be a
little bit more careful.

00:01:22.746 --> 00:01:24.547
But they do give us some information.

00:01:24.827 --> 00:01:29.652
The water level rose by six
centimeters each minute,

00:01:29.652 --> 00:01:32.476
it rose by six centimeters each minute.

00:01:32.476 --> 00:01:36.745
So at 20 minutes, we know we're at 220.

00:01:37.585 --> 00:01:40.691
And if we rose six
centimeters each minute,

00:01:40.691 --> 00:01:44.885
where would we have been, let's say

00:01:44.885 --> 00:01:47.448
10 minutes ago?

00:01:47.448 --> 00:01:50.583
So, where would we have been at time 10?

00:01:50.583 --> 00:01:53.444
So, every 10 minutes, if the water level

00:01:53.444 --> 00:01:56.112
is rising six centimeters a minute,

00:01:56.112 --> 00:02:00.088
it would be rising 60
centimeters every 10 minutes.

00:02:00.088 --> 00:02:03.739
So 10 minutes ago, we would
be 60 centimeters less.

00:02:03.739 --> 00:02:06.311
So 60 centimeters less than 220.

00:02:06.311 --> 00:02:08.279
That's 20 centimeters less,

00:02:08.279 --> 00:02:10.935
that's 40 centimeters less,

00:02:10.935 --> 00:02:12.821
that's 60 centimeters less.

00:02:12.821 --> 00:02:15.197
So, if you go back in time 10 minutes,

00:02:15.197 --> 00:02:19.176
you would be 60 centimeters shallower,

00:02:19.176 --> 00:02:24.162
or less high, and 220 minus 60 is 160.

00:02:24.162 --> 00:02:27.230
Now, I think I'm done. I
think this describes it.

00:02:27.230 --> 00:02:28.571
Now let's see if it makes sense.

00:02:28.571 --> 00:02:30.760
This is telling us that at time zero

00:02:30.760 --> 00:02:32.438
there was, before he started filling it,

00:02:32.438 --> 00:02:35.543
there was already 100
centimeters in his pool.

00:02:35.543 --> 00:02:38.652
And then after 20 minutes, he's done.

00:02:38.652 --> 00:02:41.121
And is that consistent with
six centimeters a minute?

00:02:41.121 --> 00:02:44.839
Well, based on this,
it took him 20 minutes

00:02:46.859 --> 00:02:50.282
to get to 220, but that's
an incremental 120.

00:02:50.282 --> 00:02:55.138
To go from 100 to 220
is 120 more centimeters.

00:02:55.138 --> 00:03:00.138
So in 20 minutes, he got 120 centimeters.

00:03:00.345 --> 00:03:02.516
Well 120 centimeters divided by 20 minutes

00:03:02.516 --> 00:03:05.198
is six centimeters per minute.

00:03:05.198 --> 00:03:09.532
So this is looking good
and we got it right.


Graphing linear functions example 1

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:02.211
- [Voiceover] Karl filled
up the tank of his truck

00:00:02.211 --> 00:00:05.241
with 400 liters of fuel
and set out to deliver

00:00:05.241 --> 00:00:07.664
a shipment of bananas to Alaska.

00:00:07.664 --> 00:00:11.045
The truck consumed 0.8 liters of fuel

00:00:11.045 --> 00:00:13.227
or eight-tenths of a liter of fuel

00:00:13.227 --> 00:00:15.839
for each kilometer driven.

00:00:15.839 --> 00:00:18.846
Graph the amount of fuel
remaining in the truck's tank

00:00:18.846 --> 00:00:21.778
in liters as a function of distance

00:00:21.778 --> 00:00:23.928
driven in kilometers.

00:00:23.928 --> 00:00:27.076
And right over here we
have, we have a graph

00:00:27.076 --> 00:00:28.922
where we have a coordinate plane

00:00:28.922 --> 00:00:32.486
where our horizontal coordinate
is distance in kilometers,

00:00:32.486 --> 00:00:37.042
and our vertical, our vertical
axis is fuel in liters.

00:00:37.042 --> 00:00:41.088
So we can define the line by
moving these two points around,

00:00:41.088 --> 00:00:43.309
because two points define a line.

00:00:43.309 --> 00:00:45.497
And so, let's just think about two points

00:00:45.497 --> 00:00:46.525
that we could figure out.

00:00:46.525 --> 00:00:49.497
Can we figure out the fuel
at two different distances,

00:00:49.497 --> 00:00:52.162
and then that will help
us define the line.

00:00:52.162 --> 00:00:54.547
Well, the first thing that
we might want to think about

00:00:54.547 --> 00:00:57.107
is, well, what about before
we've traveled at all?

00:00:57.107 --> 00:00:59.411
That might be the easy
thing to figure out.

00:00:59.411 --> 00:01:01.657
What was the amount of fuel in the tank

00:01:01.657 --> 00:01:03.029
when we haven't traveled at all?

00:01:03.029 --> 00:01:06.083
And they tell us that in this passage.

00:01:06.083 --> 00:01:09.332
And I encourage you to pause
the video and think about that.

00:01:09.332 --> 00:01:12.251
Well, they tell us Karl filled
up the tank of his truck

00:01:12.251 --> 00:01:15.684
with 400 liters of fuel
and then set out to deliver

00:01:15.684 --> 00:01:17.102
a shipment of bananas.

00:01:17.102 --> 00:01:18.914
So before he had driven at all,

00:01:18.914 --> 00:01:20.814
right after he'd filled his tank,

00:01:20.814 --> 00:01:23.291
he had 400 liters of fuel.

00:01:23.291 --> 00:01:26.089
So we could say when
distance was zero kilometers,

00:01:26.089 --> 00:01:28.434
he had 400 liters of fuel.

00:01:28.434 --> 00:01:30.292
So we have one point on that line.

00:01:30.292 --> 00:01:32.572
Now we gotta think about
where we might want to put,

00:01:32.572 --> 00:01:34.791
where we want to put this other point.

00:01:34.791 --> 00:01:36.434
And the way I think about it is,

00:01:36.434 --> 00:01:38.822
well, let's just, we know he's consuming,

00:01:38.822 --> 00:01:42.283
he's consuming eight-tenths
of a liter of fuel

00:01:42.283 --> 00:01:44.409
for each kilometer driven.

00:01:44.409 --> 00:01:46.045
But they don't have, you
know, we're not going by

00:01:46.045 --> 00:01:47.508
one kilometer, two kilometers.

00:01:47.508 --> 00:01:48.505
They're going by, this is like

00:01:48.505 --> 00:01:50.623
50 kilometers, 100 kilometers.

00:01:50.623 --> 00:01:52.210
So let's think about how much fuel

00:01:52.210 --> 00:01:56.308
he would have consumed after
driving 100 kilometers,

00:01:56.308 --> 00:01:58.745
and if he consumed that
much, we would subtract that

00:01:58.745 --> 00:02:00.510
from the amount of fuel he started with,

00:02:00.510 --> 00:02:01.891
and then that would tell us,

00:02:01.891 --> 00:02:03.371
that would tell us where
this point would be.

00:02:03.371 --> 00:02:05.281
It's going to be some place over here,

00:02:05.281 --> 00:02:07.800
and it's going to be, it's
going to be below 400,

00:02:07.800 --> 00:02:09.332
'cause we're consuming fuel.

00:02:09.332 --> 00:02:12.711
Fuel should be going down
as distance increases.

00:02:12.711 --> 00:02:16.123
This should be a downward-sloping line.

00:02:16.123 --> 00:02:18.411
So I have my, I have my scratchpad here.

00:02:18.411 --> 00:02:20.636
Let me, let me get it out.

00:02:20.636 --> 00:02:21.975
And I have the same question there.

00:02:21.975 --> 00:02:24.025
It just gives us all the same information.

00:02:24.025 --> 00:02:27.664
But what I want to figure
out is, so we already know,

00:02:27.664 --> 00:02:29.509
we already know that.

00:02:29.509 --> 00:02:33.305
So we have distance, distance.

00:02:33.305 --> 00:02:35.840
Let me, I'll just write.

00:02:35.840 --> 00:02:37.345
Actually, let me just
write the whole thing.

00:02:37.345 --> 00:02:39.562
Give myself a little bit more space.

00:02:39.562 --> 00:02:43.510
Distance in kilometers.
Distance in kilometers.

00:02:43.510 --> 00:02:45.802
And then you have fuel,

00:02:45.802 --> 00:02:49.092
you have fuel in liters.

00:02:49.092 --> 00:02:51.197
You have fuel in liters.

00:02:51.197 --> 00:02:52.381
And we already figured out that

00:02:52.381 --> 00:02:54.738
before he got on, right
after he filled up his tank

00:02:54.738 --> 00:02:57.652
but before he set out on his trip,

00:02:57.652 --> 00:03:01.947
at distance zero kilometers
he had 400 liters of fuel.

00:03:01.947 --> 00:03:04.536
And we've already actually plotted that.

00:03:04.536 --> 00:03:07.055
But then we said, well what
happens at a hundred kilometers?

00:03:07.055 --> 00:03:10.290
At a hundred kilometers,
how much fuel will he have?

00:03:10.290 --> 00:03:15.019
Well, they tell us that he
consumes 0.8 liters of fuel

00:03:15.019 --> 00:03:16.517
for each kilometer.

00:03:16.517 --> 00:03:20.988
So, 0.8 liters

00:03:20.988 --> 00:03:23.634
per kilometer, and then
we just multiply that

00:03:23.634 --> 00:03:25.634
times the number of kilometers.

00:03:25.634 --> 00:03:29.148
So, times 100 kilometers.

00:03:29.148 --> 00:03:32.086
The units work out, kilometers
divided by kilometers.

00:03:32.086 --> 00:03:33.792
We're just going to be left with liters,

00:03:33.792 --> 00:03:35.452
and then we multiply the numbers.

00:03:35.452 --> 00:03:37.310
Eight-tenths of a hundred,

00:03:37.310 --> 00:03:39.492
well that's going to be equal to 80,

00:03:39.492 --> 00:03:43.636
and the units are liters, 80 liters.

00:03:43.636 --> 00:03:45.356
So at a distance of a hundred kilometers,

00:03:45.356 --> 00:03:47.359
he's going to have consumed, right,

00:03:47.359 --> 00:03:48.488
let me be careful here.

00:03:48.488 --> 00:03:51.263
He's going to have consumed 80 liters.

00:03:51.263 --> 00:03:53.748
He's going to have consumed 80 liters.

00:03:53.748 --> 00:03:57.404
So the fuel, the fuel
is actually going to be

00:03:57.404 --> 00:04:00.830
what he started with, what he started with

00:04:00.830 --> 00:04:03.111
minus how much he consumed.

00:04:03.111 --> 00:04:06.784
So it's going to be minus 0.8,

00:04:06.784 --> 00:04:07.915
and if we want to write the units there.

00:04:07.915 --> 00:04:10.210
I might as well, so,
you know, this is liters

00:04:10.210 --> 00:04:11.668
right over here.

00:04:11.668 --> 00:04:13.688
This is 400 liters.

00:04:13.688 --> 00:04:16.885
And I can write this
kilometers, kilometers.

00:04:16.885 --> 00:04:18.462
It's going to be 400 liters

00:04:18.462 --> 00:04:22.522
minus 0.8 liters

00:04:23.352 --> 00:04:27.835
per kilometer times, times,

00:04:27.835 --> 00:04:28.798
let me make that clear,

00:04:28.798 --> 00:04:33.360
times 100, times 100 kilometers.

00:04:34.370 --> 00:04:36.901
And same thing, kilometers
divided by kilometers,

00:04:36.901 --> 00:04:41.750
and we are left with 400 liters
minus 0.8 liters times 100.

00:04:41.750 --> 00:04:45.132
Well, that's just going
to be equal to 400 liters,

00:04:45.132 --> 00:04:46.908
which is how much he started with,

00:04:46.908 --> 00:04:49.718
minus eight-tenths of a hundred is 80,

00:04:49.718 --> 00:04:52.502
and the units left is 80 liters.

00:04:52.502 --> 00:04:54.139
So 400 liters minus 80 liters,

00:04:54.139 --> 00:04:56.857
that's going to be 320 liters.

00:04:56.857 --> 00:04:59.527
So when he has traveled
a hundred kilometers,

00:04:59.527 --> 00:05:03.033
he will have 320 liters left in his tank.

00:05:03.033 --> 00:05:04.368
So let's plot that.

00:05:04.368 --> 00:05:05.948
So when he has traveled
a hundred kilometers,

00:05:05.948 --> 00:05:08.095
actually, I just randomly
had put the point there,

00:05:08.095 --> 00:05:11.670
he is going to have 320
liters left in his tank.

00:05:11.670 --> 00:05:15.130
And just like that, we have
plotted the line that showed

00:05:15.130 --> 00:05:16.976
how much fuel he has in his tank

00:05:16.976 --> 00:05:21.596
as a function of, as a
function of distance traveled.

00:05:21.596 --> 00:05:22.943
And you can even see from this

00:05:22.943 --> 00:05:24.578
that he's going to run out of fuel

00:05:24.578 --> 00:05:27.070
at the 500-kilometer mark.


Interpreting linear formulas example 2

WEBVTT
Kind: captions
Language: en

00:00:00.418 --> 00:00:03.066
- [Voiceover] Agent Hunt is
transferring classified files

00:00:03.066 --> 00:00:06.107
from the CIA mainframe
into his flash drive.

00:00:06.107 --> 00:00:08.881
This is the formula for
the size of the files

00:00:08.881 --> 00:00:13.281
on Agent Hunt's drive S,
measured in megabytes,

00:00:13.281 --> 00:00:16.793
as a function of time t,

00:00:16.793 --> 00:00:18.603
measured in seconds.

00:00:18.603 --> 00:00:20.322
Complete the following sentences

00:00:20.322 --> 00:00:22.840
with the appropriate missing values.

00:00:22.840 --> 00:00:24.792
When the transfer began,

00:00:24.792 --> 00:00:28.784
the drive had blank
megabytes of files on it.

00:00:28.784 --> 00:00:30.920
So, when the transfer began,

00:00:30.920 --> 00:00:33.416
that would be t equals zero.

00:00:33.416 --> 00:00:36.655
T equals zero was right
when the transfer began.

00:00:36.655 --> 00:00:38.145
So t equals zero.

00:00:38.145 --> 00:00:42.688
S of zero is going to be five times zero,

00:00:42.688 --> 00:00:46.043
five times zero plus 45,

00:00:46.043 --> 00:00:49.537
which of course is just
going to be equal to 45.

00:00:49.537 --> 00:00:51.000
And S is measured,

00:00:51.000 --> 00:00:52.659
and they tell us, measured in megabytes.

00:00:52.659 --> 00:00:55.843
So there were 45 megabytes of files

00:00:55.843 --> 00:00:58.611
on his drive before,

00:00:58.611 --> 00:01:00.732
or right when the transfer began.

00:01:00.732 --> 00:01:03.000
Then they say, every 10 seconds,

00:01:03.000 --> 00:01:06.599
every 10 seconds, an
additional blank megabytes

00:01:06.599 --> 00:01:09.199
are transferred into the drive.

00:01:09.199 --> 00:01:11.834
So every 10 seconds, how many?

00:01:11.834 --> 00:01:12.930
Well, one way to think about,

00:01:12.930 --> 00:01:14.479
there's a bunch of ways
you could think about this

00:01:14.479 --> 00:01:17.160
one is, we know that
there were 45 megabytes

00:01:17.160 --> 00:01:19.006
at time equals zero.

00:01:19.006 --> 00:01:21.003
Let's think about what
happens 10 seconds later.

00:01:21.003 --> 00:01:23.313
How many megabytes are in his drive?

00:01:23.313 --> 00:01:26.703
So we can say S of 10 is going to be

00:01:26.703 --> 00:01:31.242
five times 10 plus 45,

00:01:31.242 --> 00:01:35.270
which is equal to 50 plus 45.

00:01:35.270 --> 00:01:38.504
So, time equals zero
there were 45 megabytes

00:01:38.504 --> 00:01:41.186
and 10 seconds later there's now

00:01:41.186 --> 00:01:43.474
50 plus 45 megabytes.

00:01:43.474 --> 00:01:46.641
So in 10 seconds an extra 50 megabytes

00:01:46.641 --> 00:01:48.534
are transferred into the drive.

00:01:48.534 --> 00:01:52.318
And that's expected,
because as t increases one,

00:01:52.318 --> 00:01:54.535
this function goes up by five,

00:01:54.535 --> 00:01:58.599
so if t increases by 10, this
is going to increase by 50.

00:01:58.599 --> 00:02:00.340
So for every 10 seconds you would have,

00:02:00.340 --> 00:02:03.402
50 megabytes are
transferred into the drive.

00:02:03.402 --> 00:02:04.830
And you can test that more if you want.

00:02:04.830 --> 00:02:07.416
You can say okay, let's
go another 10 seconds.

00:02:07.416 --> 00:02:09.691
S of 20 seconds,

00:02:09.691 --> 00:02:14.532
that's going to be 5 times 20 plus 45,

00:02:14.532 --> 00:02:18.236
which is equal to 100 plus 45.

00:02:18.236 --> 00:02:21.602
So notice, when 10 seconds
have passed in time,

00:02:21.602 --> 00:02:23.842
when we've increased time by 10 seconds,

00:02:23.842 --> 00:02:26.508
we've increased memory, our
memory that we're using,

00:02:26.508 --> 00:02:29.026
the megabytes we're
using has increased by,

00:02:29.026 --> 00:02:33.392
has increased by, has increased by 50.


Interpreting linear formulas example 1

WEBVTT
Kind: captions
Language: en

00:00:00.371 --> 00:00:02.798
- Andrei has a glass tank.

00:00:02.798 --> 00:00:04.845
First, he wants to put some marbles in it,

00:00:04.845 --> 00:00:06.389
all of the same volume.

00:00:06.389 --> 00:00:08.224
Then, he wants to fill the tank with water

00:00:08.224 --> 00:00:10.267
until it's completely full.

00:00:10.267 --> 00:00:12.322
This is the formula
for the volume of water

00:00:12.322 --> 00:00:16.002
Andrei should use W, measured
in liters, as a function

00:00:16.002 --> 00:00:18.683
of the number of marbles used n.

00:00:18.683 --> 00:00:21.142
And we see that the amount
of water he needs to put in

00:00:21.142 --> 00:00:24.047
as a function of the marbles used, n,

00:00:24.047 --> 00:00:28.996
is equal to 32 minus 0.05n.

00:00:29.406 --> 00:00:30.881
That's interesting.

00:00:30.881 --> 00:00:32.197
Complete the following sentences

00:00:32.197 --> 00:00:33.957
with the appropriate missing values.

00:00:33.957 --> 00:00:36.538
The glass tank's volume is blank liters.

00:00:36.538 --> 00:00:40.114
The volume of each marble is blank liters.

00:00:40.114 --> 00:00:41.716
So, one thing that we could do

00:00:41.716 --> 00:00:43.875
to think about the glass tank's volume is,

00:00:43.875 --> 00:00:45.886
is well, let's use this
function to figure out

00:00:45.886 --> 00:00:50.179
how much water would we need
if we had no marbles in it,

00:00:50.179 --> 00:00:52.141
because that amount of
water would be the volume

00:00:52.141 --> 00:00:53.207
of the entire tank.

00:00:53.207 --> 00:00:55.765
Remember, we're gonna fill
up the entire tank with water

00:00:55.765 --> 00:00:58.203
based on how many marbles are
in it, but if we figure out

00:00:58.203 --> 00:01:03.160
what W of zero is, this is
the amount of water we need

00:01:03.160 --> 00:01:04.171
if we had no marbles.

00:01:04.171 --> 00:01:07.479
This would be the volume
of the entire tank

00:01:07.479 --> 00:01:08.896
and what is this going to be?

00:01:08.896 --> 00:01:13.040
Well W of zero, when n is equal
to zero, this term goes away

00:01:13.040 --> 00:01:15.176
and we're left with 32.

00:01:15.176 --> 00:01:19.773
So, W of zero would be
volume of entire tank.

00:01:19.773 --> 00:01:24.741
Volume of tank, which
is equal to 32 liters

00:01:27.361 --> 00:01:30.426
and so, we could write
32 right over there.

00:01:30.426 --> 00:01:33.003
And then they ask us,
the volume of each marble

00:01:33.003 --> 00:01:34.617
is blank liters.

00:01:34.617 --> 00:01:37.558
Well, what's going on here
in this function definition?

00:01:37.558 --> 00:01:40.086
So, the highest that this could be is 32,

00:01:40.086 --> 00:01:42.245
'cause that's the most
water you would put in.

00:01:42.245 --> 00:01:44.822
If you had no marbles, you
would just fill up the tank

00:01:44.822 --> 00:01:46.691
and put in 32 liters.

00:01:46.691 --> 00:01:48.938
Then, every marble that you put in,

00:01:48.938 --> 00:01:53.922
you're going to have to
put in 0.05 less liters.

00:01:53.922 --> 00:01:58.001
You see that, you have minus 0.05 times n.

00:01:58.001 --> 00:02:00.854
So, every time n grows by one,

00:02:00.854 --> 00:02:04.831
you have to put in 0.05 less

00:02:04.831 --> 00:02:06.776
fewer liters of water,

00:02:06.776 --> 00:02:11.424
which implies that the
marble is 0.05 liters.

00:02:11.424 --> 00:02:16.424
That's it's volume, so let's
write that down, 0.05 liters.

00:02:17.946 --> 00:02:19.318
And, you could test this assumption.

00:02:19.318 --> 00:02:21.837
If you said, look, if you had one marble,

00:02:21.837 --> 00:02:23.995
if you had one marble, then you would have

00:02:23.995 --> 00:02:27.815
to put 0.05 less than 32 liters in

00:02:27.815 --> 00:02:29.138
and why would that be the case?

00:02:29.138 --> 00:02:32.725
Well, that would mean because
that one marble is taking up

00:02:32.725 --> 00:02:37.375
0.05 liters, that's how
much space it's taking up.


Converting from slope-intercept to standard form

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:02.160
- Rewrite Y is equal to

00:00:02.160 --> 00:00:06.322
2/3x plus 4/7 in standard form.

00:00:06.322 --> 00:00:08.490
The equation must be simplified,

00:00:08.490 --> 00:00:11.130
which means all numbers must be integers

00:00:11.130 --> 00:00:14.220
that do not share a common
factor other than one.

00:00:14.220 --> 00:00:17.050
Alright, we'll worry about this
second part in a little bit.

00:00:17.050 --> 00:00:19.210
Let's see if we can rewrite this.

00:00:19.210 --> 00:00:21.930
So it's Y is equal to 2/3x plus 4/7.

00:00:21.930 --> 00:00:24.140
Let me get my scratch pad out.

00:00:24.400 --> 00:00:27.010
So it's Y is equal to

00:00:27.010 --> 00:00:31.000
2/3x plus 4/7.

00:00:31.000 --> 00:00:32.300
So the way that it's written right now,

00:00:32.300 --> 00:00:33.690
this is slope intercept form.

00:00:33.690 --> 00:00:35.930
It's written in the form

00:00:35.930 --> 00:00:40.630
Y is equal to mx plus b,

00:00:40.630 --> 00:00:43.520
where m in this case is 2/3 and b is 4/7.

00:00:43.520 --> 00:00:45.100
It's very easy to figure
out what the slope

00:00:45.100 --> 00:00:48.460
and what the Y intercept
is from this equation.

00:00:48.460 --> 00:00:50.470
But we wanna write this in standard form.

00:00:50.470 --> 00:00:51.800
Which would be the form

00:00:51.800 --> 00:00:55.100
Ax plus By

00:00:55.100 --> 00:00:57.540
is equal to C.

00:00:57.790 --> 00:01:00.520
And that extra text they were saying,

00:01:00.520 --> 00:01:02.130
the equation must be simplified.

00:01:02.130 --> 00:01:04.260
Which means all numbers must be integers

00:01:04.260 --> 00:01:07.000
that do not share a common
factor other than one.

00:01:07.000 --> 00:01:11.190
That just means that A, B,
and C in the standard form

00:01:11.190 --> 00:01:13.160
they want need to be integers.

00:01:13.160 --> 00:01:15.750
And they want them to not
have any common factors.

00:01:15.750 --> 00:01:17.830
So if we got to the point of say

00:01:17.830 --> 00:01:20.290
4x plus 2y

00:01:20.290 --> 00:01:23.530
is equal to 10,

00:01:23.530 --> 00:01:26.590
well this number, and this
number, and this number

00:01:26.590 --> 00:01:27.920
are all divisible by two.

00:01:27.920 --> 00:01:29.530
They all have the common factor of two.

00:01:29.530 --> 00:01:31.220
So we would want to simplify it more.

00:01:31.220 --> 00:01:33.620
Divide them all by two
and then you would get

00:01:33.620 --> 00:01:34.970
two so you divide this by two.

00:01:34.970 --> 00:01:36.460
You get 2x, divide this by two

00:01:36.460 --> 00:01:38.960
you'll get plus y is equal to 5.

00:01:38.960 --> 00:01:40.960
So this is the form
that they're asking for

00:01:40.960 --> 00:01:42.420
and probably because it's just easier

00:01:42.420 --> 00:01:44.990
for the site to know that
this is the right answer.

00:01:44.990 --> 00:01:48.020
Because there's obviously a
bunch of forms in this way.

00:01:48.020 --> 00:01:50.790
So let's see if we can do that.

00:01:50.790 --> 00:01:55.500
Let's see if we can write
it in standard form.

00:01:55.750 --> 00:01:59.660
So the first thing I would wanna do is,

00:01:59.660 --> 00:02:01.160
well there's a bunch of ways
that you could approach it.

00:02:01.160 --> 00:02:02.770
The first thing you could try to do is

00:02:02.770 --> 00:02:05.850
well let's get rid of
all of these fractions.

00:02:05.850 --> 00:02:07.970
And the best way to get
rid of the fractions

00:02:07.970 --> 00:02:12.960
is to multiple by three
and to multiply by seven.

00:02:12.960 --> 00:02:14.730
If you multiply by three you
get rid of this fraction.

00:02:14.730 --> 00:02:16.730
If you multiply by seven you
get rid of this fraction.

00:02:16.730 --> 00:02:19.760
So if you multiply three
and you multiply by seven.

00:02:19.760 --> 00:02:21.580
Let me just rewrite it over here.

00:02:21.920 --> 00:02:23.890
There's actually also a couple
of ways that we can do this.

00:02:23.890 --> 00:02:25.260
So if you multiply.

00:02:25.260 --> 00:02:26.690
So one way to do it.

00:02:26.690 --> 00:02:28.400
So we start with Y is equal to

00:02:28.400 --> 00:02:32.280
2/3x plus 4/7.

00:02:32.530 --> 00:02:37.500
So if I multiply this side by
three and I multiply by seven,

00:02:37.500 --> 00:02:39.548
I have to do that to this side as well.

00:02:39.548 --> 00:02:42.778
So this is going to be multiplied by three

00:02:42.778 --> 00:02:44.690
and multiplied by seven.

00:02:44.690 --> 00:02:47.630
So the left hand side becomes 21y.

00:02:47.630 --> 00:02:49.870
21y.

00:02:50.291 --> 00:02:52.990
Three times seven of course is
21, we just figured that out.

00:02:52.990 --> 00:02:54.790
We would distribute the 21.

00:02:54.790 --> 00:02:57.460
21 times 2/3, well let's see.

00:02:57.460 --> 00:03:01.730
21 divided by three is
seven, times two is 14.

00:03:01.730 --> 00:03:03.830
So it's gonna be 14x.

00:03:04.050 --> 00:03:08.110
And then 21 divided by seven
is three times four is 12.

00:03:08.530 --> 00:03:11.510
So just like that I was able
to get rid of the fractions.

00:03:11.790 --> 00:03:15.330
And now I wanna get all the

00:03:15.330 --> 00:03:17.220
X's and Y's on one side.

00:03:17.220 --> 00:03:19.720
So I wanna get this
14x onto the left side.

00:03:19.720 --> 00:03:21.330
So let's see if I can do that.

00:03:21.330 --> 00:03:23.008
So I'm gonna do that by.

00:03:23.008 --> 00:03:25.730
To get rid of this I would
want to subtract 14x.

00:03:25.730 --> 00:03:27.060
I can't just do it on the right hand side

00:03:27.060 --> 00:03:29.290
I have to do it on the
left hand side as well.

00:03:29.290 --> 00:03:33.090
So I wanna subtract 14x, and
then what am I left with?

00:03:33.090 --> 00:03:34.990
Let me give myself a
little bit more space.

00:03:34.990 --> 00:03:38.520
So on the left hand side I have

00:03:38.520 --> 00:03:43.450
negative 14x plus 21y.

00:03:43.860 --> 00:03:47.620
Plus 21y

00:03:47.620 --> 00:03:49.550
is equal to.

00:03:49.550 --> 00:03:52.390
Let's see and I subtracted
14x to get rid of this.

00:03:52.390 --> 00:03:55.650
And then I have this is equal to 12.

00:03:55.650 --> 00:03:57.960
Now let's see, am I done?

00:03:57.960 --> 00:04:02.120
Do these share any, do 14, 21, and 12

00:04:02.120 --> 00:04:03.740
share any common factors?

00:04:03.996 --> 00:04:07.531
Let's see, 14 is divisible
by two and seven.

00:04:07.895 --> 00:04:10.260
21 is divisible by three and seven.

00:04:10.260 --> 00:04:13.770
12 is divisible by two, six, three, four.

00:04:13.770 --> 00:04:16.800
But all of these aren't
divisible by the same number.

00:04:17.290 --> 00:04:19.030
14, let's see.

00:04:19.030 --> 00:04:21.570
14 is divisible two,
so is 12, but 21 isn't.

00:04:21.750 --> 00:04:25.520
14 is divisible by seven,
so is 21, but 12 isn't.

00:04:27.950 --> 00:04:31.760
And 21 and 12 are divisible
by three but 14 isn't.

00:04:31.760 --> 00:04:34.960
So I think this is about as
simplified as I could get.

00:04:34.960 --> 00:04:38.030
If there was a common factor
for all three of these numbers

00:04:38.030 --> 00:04:39.860
then I would divide all
of them the way I did

00:04:39.860 --> 00:04:41.360
in that previous example.

00:04:41.360 --> 00:04:43.500
But that's not the case right over here.

00:04:43.500 --> 00:04:46.960
So it's negative 14x
plus 21y is equal to 12.

00:04:46.960 --> 00:04:49.660
So let me see if I can
remember that and type that in.

00:04:49.930 --> 00:04:53.070
So it is,

00:04:53.970 --> 00:04:58.970
negative 14x plus 21y

00:04:58.980 --> 00:05:03.480
is equal to 12.

00:05:03.730 --> 00:05:05.229
Now let's see if I got it right.

00:05:05.650 --> 00:05:06.780
It worked out.


Intuitive understanding of slope example

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.030
Graph a line that
has a slope that

00:00:03.030 --> 00:00:07.180
is negative and greater than
the slope of the blue line.

00:00:07.180 --> 00:00:09.640
So let's think for a second
about what slope means.

00:00:09.640 --> 00:00:12.680
So if you use the word
slope in your everyday life,

00:00:12.680 --> 00:00:15.220
you're really talking about
how inclined something is,

00:00:15.220 --> 00:00:17.030
like a ski slope.

00:00:17.030 --> 00:00:20.050
So for example, this orange
line isn't inclined at all.

00:00:20.050 --> 00:00:24.140
It's flat, so this one
actually has a slope of 0.

00:00:24.140 --> 00:00:27.860
Another way of thinking
about it is as x increases,

00:00:27.860 --> 00:00:29.370
what is happening to y?

00:00:29.370 --> 00:00:31.740
And you see here that y
isn't changing at all,

00:00:31.740 --> 00:00:34.570
so this orange line right
now has a slope of 0.

00:00:34.570 --> 00:00:36.540
If the orange line
looked like this,

00:00:36.540 --> 00:00:38.600
it now has a positive slope.

00:00:38.600 --> 00:00:41.550
Notice, when x is
negative, your y

00:00:41.550 --> 00:00:43.540
value-- or say when
x is negative 5,

00:00:43.540 --> 00:00:46.580
your y value is here, and
then when x is positive 5,

00:00:46.580 --> 00:00:48.260
your y value has increased.

00:00:48.260 --> 00:00:53.270
As x increases, y is increasing,
so this has a positive slope.

00:00:53.270 --> 00:00:56.050
This has an even more positive
slope, an even more positive

00:00:56.050 --> 00:00:56.860
slope.

00:00:56.860 --> 00:00:59.190
This has an even
more positive slope.

00:00:59.190 --> 00:01:03.120
As x is increasing, the y value
is increasing really fast.

00:01:03.120 --> 00:01:05.150
This line is going
up really fast

00:01:05.150 --> 00:01:08.670
as we move towards the right,
so it's a very positive slope.

00:01:08.670 --> 00:01:11.720
This is less positive,
less positive.

00:01:11.720 --> 00:01:16.510
This is a 0 slope, and then
this is a negative slope.

00:01:16.510 --> 00:01:20.740
Notice, as x is increasing,
the line is going down.

00:01:20.740 --> 00:01:22.770
Your y value is decreasing.

00:01:22.770 --> 00:01:27.310
When x is negative
5, your y is 7.

00:01:27.310 --> 00:01:30.040
Well, when x is 5, your y is 5.

00:01:30.040 --> 00:01:32.260
So x is increased,
but y has gone down,

00:01:32.260 --> 00:01:34.050
so this is a negative slope.

00:01:34.050 --> 00:01:36.135
So they say graph a line
that has a slope that

00:01:36.135 --> 00:01:39.430
is negative and greater than
the slope of the blue line.

00:01:39.430 --> 00:01:42.170
So the blue line also
has a negative slope.

00:01:42.170 --> 00:01:47.280
As x is increasing, your
blue line is going down.

00:01:47.280 --> 00:01:49.380
Over here, when x
has negative values,

00:01:49.380 --> 00:01:51.820
your y value is quite high.

00:01:51.820 --> 00:01:53.640
And here, when x
has positive values,

00:01:53.640 --> 00:01:56.070
your y value has gone
all the way down.

00:01:56.070 --> 00:01:57.840
So this has a
negative slope, but we

00:01:57.840 --> 00:01:59.790
want to have a slope
greater than this one.

00:01:59.790 --> 00:02:01.165
So we still want
to have a-- they

00:02:01.165 --> 00:02:03.340
say graph a line that has
a slope that is negative--

00:02:03.340 --> 00:02:05.900
so my orange line currently
has a negative slope--

00:02:05.900 --> 00:02:08.623
and greater than the
slope of the blue line.

00:02:08.623 --> 00:02:10.039
So it has a negative
slope, but it

00:02:10.039 --> 00:02:12.630
is less negative than this
blue line right over here.

00:02:12.630 --> 00:02:14.755
If I wanted to be more
negative than the blue line,

00:02:14.755 --> 00:02:16.490
I'd have to do
something like that.

00:02:16.490 --> 00:02:19.310
But I want to be negative
but less negative

00:02:19.310 --> 00:02:21.340
than the blue line, so
that would be like that.

00:02:21.340 --> 00:02:24.720
If we wanted a 0 slope, once
again, something like this.

00:02:24.720 --> 00:02:27.340
If we wanted a positive
slope, something like this.

00:02:27.340 --> 00:02:29.660
So once again, negative
slope, less negative

00:02:29.660 --> 00:02:31.440
than the blue line.

00:02:31.440 --> 00:02:33.390
Check our answer.


Positive and negative slope

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:03.250
- [Voiceover] Slope is
defined as your change

00:00:03.250 --> 00:00:05.880
in the vertical direction, and I could use

00:00:05.880 --> 00:00:08.080
the Greek letter delta,
this little triangle here

00:00:08.080 --> 00:00:10.650
is the Greek letter
delta, it means change in.

00:00:10.650 --> 00:00:14.020
Change in the vertical
direction divided by

00:00:14.020 --> 00:00:16.490
change in the horizontal direction.

00:00:16.490 --> 00:00:20.690
That is the standard definition of slope

00:00:20.690 --> 00:00:22.490
and it's a reasonable way for measuring

00:00:22.490 --> 00:00:24.250
how steep something is.

00:00:24.250 --> 00:00:27.290
So for example, if we're
looking at the xy plane here,

00:00:27.290 --> 00:00:28.550
our change in the vertical direction

00:00:28.550 --> 00:00:30.850
is gonna be a change in the y variable

00:00:30.850 --> 00:00:33.120
divided by change in horizontal direction,

00:00:33.120 --> 00:00:36.690
is gonna be a change in the x variable.

00:00:36.690 --> 00:00:40.180
So let's see why that is a
good definition for slope.

00:00:40.180 --> 00:00:43.210
Well I could draw something
with a slope of one.

00:00:43.210 --> 00:00:45.640
A slope of one might
look something like...

00:00:50.290 --> 00:00:53.620
so a slope of one, as x increases by one,

00:00:53.620 --> 00:00:58.040
y increases by one, so a slope of one...

00:01:05.380 --> 00:01:07.420
is going to look like this.

00:01:08.350 --> 00:01:10.550
Notice, however much my change in x is,

00:01:10.550 --> 00:01:14.810
so for example here, my
change in x is positive two,

00:01:14.810 --> 00:01:17.480
I'm gonna have the same change in y.

00:01:17.480 --> 00:01:20.750
My change in y is going to be plus two.

00:01:20.750 --> 00:01:23.150
So my change in y divided by change in x

00:01:23.150 --> 00:01:26.120
is two divided by two is one.

00:01:26.120 --> 00:01:30.680
So for this line I have
slope is equal to one.

00:01:30.680 --> 00:01:33.120
But what would a slope of two look like?

00:01:33.120 --> 00:01:36.650
Well, a slope of two should be
steeper and we can draw that.

00:01:41.280 --> 00:01:43.120
Let me start at a different point,

00:01:43.120 --> 00:01:47.670
so if I start over here a
slope of two would look like...

00:01:50.350 --> 00:01:52.610
for every one that I
increase in the x direction

00:01:52.610 --> 00:01:55.350
I'm gonna increase two in the y direction,

00:01:55.350 --> 00:01:57.440
so it's going to look like...

00:02:00.350 --> 00:02:01.550
that.

00:02:01.550 --> 00:02:03.450
This line right over here, you see it.

00:02:03.450 --> 00:02:07.870
If my change in x is equal to one,

00:02:12.020 --> 00:02:14.250
my change in y is two.

00:02:14.250 --> 00:02:16.110
So change in y over
change in x is gonna be

00:02:16.110 --> 00:02:19.620
two over one, the slope here is two.

00:02:19.620 --> 00:02:21.290
And now, hopefully, you're appreciating

00:02:21.290 --> 00:02:23.410
why this definition of
slope is a good one.

00:02:23.410 --> 00:02:26.420
The higher the slope, the
steeper it is, the faster

00:02:26.420 --> 00:02:29.420
it increases, the faster
we increase in the vertical

00:02:29.420 --> 00:02:33.920
direction as we increase in
the horizontal direction.

00:02:33.920 --> 00:02:36.090
Now what would a negative slope be?

00:02:36.090 --> 00:02:37.390
So let's just think about what a line

00:02:37.390 --> 00:02:39.290
with a negative slope would mean.

00:02:39.290 --> 00:02:42.150
A negative slope would mean,
well we could take an example.

00:02:42.150 --> 00:02:47.150
If we have our change
in y over change in x

00:02:47.690 --> 00:02:52.690
was equal to a negative one.

00:02:52.720 --> 00:02:57.110
That means that if we
have a change in x of one,

00:02:57.110 --> 00:02:58.890
then in order to get
negative one here, that means

00:02:58.890 --> 00:03:02.820
that our change in y would have
to be equal to negative one.

00:03:02.820 --> 00:03:06.500
So a line with a negative
one slope would look like...

00:03:14.820 --> 00:03:16.520
would look like this.

00:03:16.520 --> 00:03:21.520
Notice, as x increases
by a certain amount,

00:03:22.950 --> 00:03:25.150
so our delta x here is one,

00:03:25.150 --> 00:03:28.250
y decreases by that same
amount instead of increasing.

00:03:28.250 --> 00:03:32.279
So now this is what we consider
a downward sloping line.

00:03:32.279 --> 00:03:35.619
So change in y is equal to negative one.

00:03:35.619 --> 00:03:39.919
So our change in y over our change in x

00:03:39.919 --> 00:03:42.159
is equal to negative one over one

00:03:42.159 --> 00:03:43.789
which is equal to negative one.

00:03:43.789 --> 00:03:45.989
So the slope of this line is negative one.

00:03:45.989 --> 00:03:48.019
Now if you had a slope with negative two,

00:03:48.019 --> 00:03:50.009
it would decrease even faster.

00:03:50.009 --> 00:03:51.659
So a line with a slope of negative two

00:03:51.659 --> 00:03:54.059
could look something like this.

00:03:56.319 --> 00:04:00.089
So as x increases by one,
y would decrease by two.

00:04:02.319 --> 00:04:04.609
So it would look something like...

00:04:08.159 --> 00:04:11.019
it would look like that.

00:04:11.019 --> 00:04:15.119
Notice, as our x increases
by a certain amount,

00:04:15.119 --> 00:04:20.019
our y decreases by twice as much.

00:04:20.019 --> 00:04:24.259
So this right over here has a slope

00:04:24.259 --> 00:04:26.549
of negative two.

00:04:26.549 --> 00:04:27.759
So hopefully this gives you a little bit

00:04:27.759 --> 00:04:30.179
more intuition for what slope represents

00:04:30.179 --> 00:04:33.689
and how the number that
we use to represent slope,

00:04:33.689 --> 00:04:36.089
how you can use that to
visualize how steep a line is.

00:04:36.089 --> 00:04:38.479
A very high positive
slope, as x increases,

00:04:38.479 --> 00:04:41.789
y is going to increase
fairly dramatically.

00:04:41.789 --> 00:04:43.769
If you have a negative slope...

00:04:46.289 --> 00:04:49.259
as x increases, your y is
actually going to decrease.

00:04:49.259 --> 00:04:50.859
And then the higher
the slope, the steeper,

00:04:53.109 --> 00:04:55.589
the more you increase as x increases,

00:04:55.589 --> 00:04:57.179
and the more negative the slope,

00:04:57.179 --> 00:05:00.669
the more you decrease as x increases.


Graphing solutions to two-variable linear equations example 2

WEBVTT
Kind: captions
Language: en

00:00:00.085 --> 00:00:01.612
- [Voiceover] So this is an example from

00:00:01.612 --> 00:00:03.957
the Khan Academy exercise,
graphing solutions

00:00:03.957 --> 00:00:06.372
to two variable linear equations.

00:00:06.372 --> 00:00:07.904
And they tell us to complete the table

00:00:07.904 --> 00:00:10.319
so each row represents a solution of

00:00:10.319 --> 00:00:11.666
the following equation.

00:00:11.666 --> 00:00:13.059
And they give us the equation,

00:00:13.059 --> 00:00:13.918
and then they want us to figure out,

00:00:13.918 --> 00:00:15.125
what does y equal when x is

00:00:15.125 --> 00:00:16.658
equal to negative five?

00:00:16.658 --> 00:00:17.889
And what does x equal when

00:00:17.889 --> 00:00:19.955
y is equal to eight.

00:00:19.955 --> 00:00:21.116
And to figure this out,

00:00:21.116 --> 00:00:22.509
I've actually copied and pasted this

00:00:22.509 --> 00:00:24.808
part of the problem onto my scratchpad,

00:00:24.808 --> 00:00:27.455
so let me get that out.

00:00:27.455 --> 00:00:29.336
And so this is the exact same problem,

00:00:29.336 --> 00:00:30.613
there's a couple of ways that

00:00:30.613 --> 00:00:32.122
we could try to tackle it.

00:00:32.122 --> 00:00:34.583
One way, is you could try
to simplify this more,

00:00:34.583 --> 00:00:36.115
get all your xs on one side

00:00:36.115 --> 00:00:37.764
and all your ys on the other side.

00:00:37.764 --> 00:00:40.215
Or we could just literally substitute

00:00:40.215 --> 00:00:42.106
when x equals negative five,

00:00:42.106 --> 00:00:44.544
what must y equal?

00:00:44.544 --> 00:00:46.494
Actually, let me do it
the second way, first.

00:00:46.494 --> 00:00:47.934
So if we take this equation,

00:00:47.934 --> 00:00:49.931
and we substitute x with negative five,

00:00:49.931 --> 00:00:51.208
what do we get?

00:00:51.208 --> 00:00:54.993
We get negative three times,

00:00:54.993 --> 00:00:56.757
well, we're gonna say x is negative five,

00:00:56.757 --> 00:00:58.382
times negative five,

00:00:58.382 --> 00:01:00.612
plus seven y

00:01:00.612 --> 00:01:03.793
is equal to five times, x is once again,

00:01:03.793 --> 00:01:05.371
it's gonna be negative five,

00:01:05.371 --> 00:01:06.509
x is negative five,

00:01:06.509 --> 00:01:08.320
five times negative five,

00:01:08.320 --> 00:01:09.876
plus two y.

00:01:09.876 --> 00:01:11.850
See, negative three times negative five

00:01:11.850 --> 00:01:13.684
is positive 15,

00:01:13.684 --> 00:01:16.609
plus seven y, is equal to

00:01:16.609 --> 00:01:20.510
negative 25 plus two y.

00:01:20.510 --> 00:01:24.109
And now, to solve for y,

00:01:24.109 --> 00:01:27.197
let's see, I could subtract
two y from both sides,

00:01:27.197 --> 00:01:29.751
so that I get rid of the
two y here on the right.

00:01:29.751 --> 00:01:32.259
So let me subtract two y,

00:01:32.259 --> 00:01:34.720
subtract two y from both sides.

00:01:34.720 --> 00:01:37.555
And then if I want all my
constants on the right hand side,

00:01:37.555 --> 00:01:40.525
I can subtract 15 from both sides.

00:01:40.525 --> 00:01:43.427
So let me subtract 15 from both sides.

00:01:43.427 --> 00:01:47.403
And I'm going to be left with
15 minus 15, that's zero,

00:01:47.403 --> 00:01:48.628
that's the whole point of subtracting

00:01:48.628 --> 00:01:51.867
15 from both sides, so I
get rid of this 15 here.

00:01:51.867 --> 00:01:53.342
Seven y minus two y.

00:01:53.342 --> 00:01:56.069
Seven of something minus
two of that same something

00:01:56.069 --> 00:01:57.498
is gonna be five of that something.

00:01:57.498 --> 00:01:59.193
It's gonna be equal to five y,

00:01:59.193 --> 00:02:03.303
is equal to negative 25 minus 15.

00:02:03.303 --> 00:02:05.253
Well, that's gonna be negative 40.

00:02:05.253 --> 00:02:07.933
And then two y minus two y,
well, that's just gonna be zero.

00:02:07.933 --> 00:02:09.131
That was the whole point of subtracting

00:02:09.131 --> 00:02:11.058
two y from both sides.

00:02:11.058 --> 00:02:14.218
So you have five times y
is equal to negative 40.

00:02:14.218 --> 00:02:16.886
Or, if we divide both sides by five,

00:02:16.886 --> 00:02:18.906
we divide both sides by five,

00:02:18.906 --> 00:02:23.434
we would get y is equal to negative eight.

00:02:23.434 --> 00:02:26.197
So when x is equal to negative five,

00:02:26.197 --> 00:02:28.612
y is equal to negative eight.

00:02:28.612 --> 00:02:32.303
Y is equal to negative eight.

00:02:32.303 --> 00:02:34.486
And actually we can fill that in.

00:02:34.486 --> 00:02:38.084
So this y is going to be
equal to negative eight.

00:02:38.084 --> 00:02:39.733
And now we gotta figure this out.

00:02:39.733 --> 00:02:42.729
What does x equal when
y is positive eight?

00:02:42.729 --> 00:02:47.024
Well, we can go back
to our scratchpad here.

00:02:47.024 --> 00:02:48.789
And I'll take the same equation,

00:02:48.789 --> 00:02:51.807
but let's make y equal to positive eight.

00:02:51.807 --> 00:02:55.197
So you have negative three x

00:02:55.197 --> 00:02:58.262
plus seven, now y is going to be eight,

00:02:58.262 --> 00:02:59.748
y is eight,

00:02:59.748 --> 00:03:03.231
seven times eight is
equal to five times x,

00:03:03.231 --> 00:03:05.367
plus two times, once again,

00:03:05.367 --> 00:03:08.595
y is eight, two times eight.

00:03:08.595 --> 00:03:11.149
So we get negative three x

00:03:11.149 --> 00:03:14.446
plus 56, that's 56,

00:03:14.446 --> 00:03:18.441
is equal to five x plus 16.

00:03:18.441 --> 00:03:20.669
Now, if we wanna get all of our

00:03:20.669 --> 00:03:22.410
constants on one side,

00:03:22.410 --> 00:03:24.582
and of all of our x
terms on the other side,

00:03:24.582 --> 00:03:26.288
well, what could we do?

00:03:26.288 --> 00:03:29.011
Let's see, we could add
three x to both sides.

00:03:29.011 --> 00:03:31.790
That would get rid of
all the xs on this side,

00:03:31.790 --> 00:03:33.114
and put 'em all on this side.

00:03:33.114 --> 00:03:35.598
So we're gonna add three x to both sides.

00:03:35.598 --> 00:03:36.806
And, let's see,

00:03:36.806 --> 00:03:39.058
if we want to get all the
constants on the left hand side,

00:03:39.058 --> 00:03:40.637
we'd wanna get rid of the 16,

00:03:40.637 --> 00:03:43.516
so we could subtract 16
from the right hand side,

00:03:43.516 --> 00:03:44.305
if we do it from the right,

00:03:44.305 --> 00:03:46.860
we're gonna have to do
it from the left as well.

00:03:46.860 --> 00:03:49.977
And we're gonna be left
with, these cancel out,

00:03:49.977 --> 00:03:54.708
56 minus 16 is positive 40.

00:03:54.708 --> 00:03:56.008
And then, let's see,

00:03:56.008 --> 00:03:57.564
16 minus 16 is zero.

00:03:57.564 --> 00:04:00.593
Five x plus three x is equal to eight x.

00:04:00.593 --> 00:04:03.531
We get eight x is equal to 40.

00:04:03.531 --> 00:04:06.062
We could divide both sides by eight,

00:04:06.062 --> 00:04:10.380
and we get, five is equal to x.

00:04:10.380 --> 00:04:12.609
So this right over here is going

00:04:12.609 --> 00:04:15.187
to be equal to five.

00:04:15.187 --> 00:04:18.441
So let's go back, let's go back, now.

00:04:18.441 --> 00:04:20.690
So when y is positive eight,

00:04:20.690 --> 00:04:23.174
x is positive five.

00:04:23.174 --> 00:04:25.270
Now they ask us, "Use your two solutions

00:04:25.270 --> 00:04:27.516
"to graph the equation."

00:04:27.516 --> 00:04:28.631
So let's see if we can do,

00:04:28.631 --> 00:04:32.308
oh, whoops, let me, let
me use my mouse now.

00:04:32.308 --> 00:04:33.785
So to graph the equations.

00:04:33.785 --> 00:04:36.382
So when x is negative
five, y is negative eight.

00:04:36.382 --> 00:04:40.890
So the point negative
five comma negative eight.

00:04:40.890 --> 00:04:43.110
So that's right over there.

00:04:43.110 --> 00:04:46.563
So let me move my browser
up so you can see that.

00:04:46.563 --> 00:04:49.429
Negative five, when x is negative five,

00:04:49.429 --> 00:04:51.403
y is negative eight.

00:04:51.403 --> 00:04:54.328
And when x is positive five,

00:04:54.328 --> 00:04:55.397
and we see that up here,

00:04:55.397 --> 00:04:58.391
when x is positive five,
y is positive eight.

00:04:58.391 --> 00:05:01.441
When x is positive five,
y is positive eight.

00:05:01.441 --> 00:05:02.246
And we're done.

00:05:02.246 --> 00:05:04.243
We can check our answer, if we like.

00:05:04.243 --> 00:05:05.520
We got it right.

00:05:05.520 --> 00:05:07.076
Now, I said there was
two ways to tackle it,

00:05:07.076 --> 00:05:09.824
I kind of just did it, I guess
you could say, the naive way.

00:05:09.824 --> 00:05:11.162
I just substituted negative five

00:05:11.162 --> 00:05:12.880
directly into this and solved for y.

00:05:12.880 --> 00:05:15.760
And then I substituted
y equals positive eight

00:05:15.760 --> 00:05:18.291
directly into this, and then solved for x.

00:05:18.291 --> 00:05:19.660
Another way that I could have done it,

00:05:19.660 --> 00:05:20.868
that actually probably would have been,

00:05:20.868 --> 00:05:22.957
or, it would for sure, would have been

00:05:22.957 --> 00:05:24.490
the easier way to do it,

00:05:24.490 --> 00:05:27.978
is ahead of time to try to
simplify this expression.

00:05:27.978 --> 00:05:30.165
So what I could have done,
right from the get-go,

00:05:30.165 --> 00:05:31.802
is said, "Hey, let's put
all my xs on one side,

00:05:31.802 --> 00:05:33.936
"and all my ys on the other side."

00:05:33.936 --> 00:05:37.863
So this is negative three x plus seven y

00:05:37.863 --> 00:05:41.670
is equal to five x plus two y.

00:05:41.670 --> 00:05:43.807
Now let's say I wanna
get all my ys on the left

00:05:43.807 --> 00:05:45.711
and all my xs on the right.

00:05:45.711 --> 00:05:48.218
So I don't want this
negative three x on the left,

00:05:48.218 --> 00:05:50.308
so I'd wanna add three x.

00:05:50.308 --> 00:05:51.840
Adding three x would cancel this out,

00:05:51.840 --> 00:05:53.187
but I can't just do it
on the left hand side,

00:05:53.187 --> 00:05:55.811
I have to do it on the
right hand side as well.

00:05:55.811 --> 00:05:58.667
And then, if I wanna get rid
of this two y on the right,

00:05:58.667 --> 00:06:00.640
I could subtract two y from the right,

00:06:00.640 --> 00:06:03.176
but, of course, I'd also
wanna do it from the left.

00:06:03.176 --> 00:06:04.657
And then what am I left with?

00:06:04.657 --> 00:06:07.188
So negative three x plus three x is zero,

00:06:07.188 --> 00:06:10.485
seven y minus two y is five y.

00:06:10.485 --> 00:06:13.109
And then I have five x plus three x

00:06:13.109 --> 00:06:14.665
is eight x.

00:06:14.665 --> 00:06:17.730
Two y minus two y is zero.

00:06:17.730 --> 00:06:19.425
And then if I wanted
to, I could solve for y,

00:06:19.425 --> 00:06:21.003
I could divide both sides by five

00:06:21.003 --> 00:06:25.787
and I'd get y is equal to 8/5 x.

00:06:25.787 --> 00:06:27.992
So, this right over here represents

00:06:27.992 --> 00:06:30.384
the same exact equation as this over here,

00:06:30.384 --> 00:06:32.009
it's just written in a different way.

00:06:32.009 --> 00:06:34.169
All of the xy pairs that satisfy this,

00:06:34.169 --> 00:06:36.398
would satisfy this, and vice versa.

00:06:36.398 --> 00:06:37.350
And this is much easier.

00:06:37.350 --> 00:06:39.671
Because if x is now negative five,

00:06:39.671 --> 00:06:41.018
if x is negative five,

00:06:41.018 --> 00:06:43.712
y would be 8/5 times negative five,

00:06:43.712 --> 00:06:45.848
well, that's going to be negative eight.

00:06:45.848 --> 00:06:48.193
And when y is equal to eight,

00:06:48.193 --> 00:06:50.190
well, you actually could
even do this up here,

00:06:50.190 --> 00:06:53.951
you could say five times
eight is equal to eight x,

00:06:53.951 --> 00:06:54.918
and then you could see, well

00:06:54.918 --> 00:06:56.936
five times eight the same
thing as eight times five,

00:06:56.936 --> 00:06:58.868
so x would be equal to five.

00:06:58.868 --> 00:07:00.135
So I think this would actually have been

00:07:00.135 --> 00:07:00.824
a simpler way to do it.

00:07:00.824 --> 00:07:02.658
You see it all, I was able
to do the entire problem

00:07:02.658 --> 00:07:04.051
in this little white space here,

00:07:04.051 --> 00:07:05.537
instead of having to do all of this,

00:07:05.537 --> 00:07:08.295
slightly, slightly hairier, algebra.


Checking ordered pair solutions to equations example 2

WEBVTT
Kind: captions
Language: en

00:00:00.411 --> 00:00:01.919
- [Voiceover] "Which of
the ordered pairs is a

00:00:01.919 --> 00:00:04.544
"solution of the following equation?"

00:00:04.544 --> 00:00:09.544
4x minus one is equal to 3y plus five.

00:00:09.721 --> 00:00:11.730
Now, when we look at an
ordered pair we wanna

00:00:11.730 --> 00:00:13.563
figure out whether it's a
solution, we just have to

00:00:13.563 --> 00:00:15.781
remind ourselves that
in these ordered pairs

00:00:15.781 --> 00:00:18.289
the convention, the standard,
is is that the first

00:00:18.289 --> 00:00:21.980
coordinate is the x coordinate,
and the second coordinate

00:00:21.980 --> 00:00:23.745
is the y coordinate.

00:00:23.745 --> 00:00:26.682
So they're gonna, if this is
a solution, if this ordered

00:00:26.682 --> 00:00:30.107
pair is a solution, that means
that if x is equal to three

00:00:30.107 --> 00:00:32.928
and y is equal to two,
that that would satisfy

00:00:32.928 --> 00:00:34.077
this equation up here.

00:00:34.077 --> 00:00:35.540
So let's try that out.

00:00:35.540 --> 00:00:38.105
So, we have four times x.

00:00:38.105 --> 00:00:40.764
Well we're saying x needs
to be equal to three,

00:00:40.764 --> 00:00:44.258
minus one, is going to be
equal to three times y.

00:00:44.258 --> 00:00:47.357
Well, if this ordered
pair is a solution then

00:00:47.357 --> 00:00:50.503
y is going to be equal
to two, so three times y,

00:00:50.503 --> 00:00:52.756
y is two, plus five.

00:00:52.756 --> 00:00:54.579
Notice all I did is wherever I saw the x,

00:00:54.579 --> 00:00:56.505
I substituted it with three,

00:00:56.505 --> 00:00:59.559
wherever I saw the y, I
substituted it with two.

00:00:59.559 --> 00:01:01.416
Now let's see if this is true.

00:01:01.416 --> 00:01:04.237
Four times three is twelve, minus one.

00:01:04.237 --> 00:01:06.903
Is this really the same
thing as three times two

00:01:06.903 --> 00:01:09.132
which is six, plus five?

00:01:09.132 --> 00:01:11.094
See, 12 minus one is 11,

00:01:11.094 --> 00:01:13.253
six plus five is also 11.

00:01:13.253 --> 00:01:16.202
This is true, 11 equals 11.

00:01:16.202 --> 00:01:21.159
This pair three, two does
satisfy this equation.

00:01:21.159 --> 00:01:23.911
Now let's see whether
this one does, two, three.

00:01:23.911 --> 00:01:25.977
So this is saying when x is equal to two,

00:01:25.977 --> 00:01:29.378
y would be equal to
three for this equation.

00:01:29.378 --> 00:01:30.516
Let's see if that's true.

00:01:30.516 --> 00:01:33.952
So four times x, we're
now gonna see if when

00:01:33.952 --> 00:01:35.914
x is two, y can be three.

00:01:35.914 --> 00:01:38.259
So four times x,

00:01:38.259 --> 00:01:39.977
four times two,

00:01:39.977 --> 00:01:43.370
minus one is equal to three times y,

00:01:43.370 --> 00:01:45.837
now y we're testing to
see if it can be three.

00:01:45.837 --> 00:01:48.028
Three times three plus five,

00:01:48.028 --> 00:01:49.386
let's see if this is true.

00:01:49.386 --> 00:01:51.637
Four times two is eight, minus one,

00:01:51.637 --> 00:01:56.637
is this equal to three times three?

00:01:57.992 --> 00:02:00.620
So that's nine plus five.

00:02:00.620 --> 00:02:03.697
So is seven equal to 14?

00:02:03.697 --> 00:02:06.903
No, clearly seven is not equal to 14.

00:02:06.903 --> 00:02:11.546
So these things are not
equal to each other.

00:02:11.546 --> 00:02:15.366
So this is not a solution,
when x equals two

00:02:15.366 --> 00:02:19.718
y cannot be to three and
satisfy this equation.

00:02:19.718 --> 00:02:24.368
So only three, two is a solution.


Checking ordered pair solutions to equations example 1

WEBVTT
Kind: captions
Language: en

00:00:00.504 --> 00:00:02.164
- [Voiceover] Which of the ordered pairs

00:00:02.164 --> 00:00:04.567
is a solution of the following equation?

00:00:05.347 --> 00:00:09.693
Negative three x minus y is equal to six.

00:00:09.930 --> 00:00:11.323
What we have to remind ourselves

00:00:11.323 --> 00:00:12.647
is when we're give an ordered pair,

00:00:12.647 --> 00:00:15.978
the first number is the x
coordinate and the second number

00:00:15.978 --> 00:00:18.765
is the y coordinate, or the y value.

00:00:18.765 --> 00:00:20.262
So when they tell us the ordered pair,

00:00:20.262 --> 00:00:22.363
negative four comma four,

00:00:22.363 --> 00:00:24.848
they're saying "hey look, if x is equal to

00:00:24.848 --> 00:00:28.458
"negative four, and y is
equal to positive four,

00:00:28.458 --> 00:00:31.372
"does that satisfy this equation?"

00:00:31.372 --> 00:00:33.090
And what we can do, is
we can just try that out.

00:00:33.090 --> 00:00:36.991
So we have negative three
and everywhere we see an x,

00:00:36.991 --> 00:00:39.394
everywhere we see an x,

00:00:39.394 --> 00:00:41.890
we can replace it with negative four.

00:00:41.890 --> 00:00:44.722
So it's negative three
times negative four,

00:00:45.372 --> 00:00:48.333
minus, minus

00:00:53.224 --> 00:00:55.287
and everywhere we see a y,

00:00:55.287 --> 00:00:57.945
we can replace it with positive four.

00:00:57.945 --> 00:00:59.988
We replace it with positive four.

00:00:59.988 --> 00:01:04.716
So negative three times x minus y,

00:01:04.716 --> 00:01:08.123
which is four, needs to be equal to six.

00:01:08.123 --> 00:01:09.783
Needs to be equal to six.

00:01:09.783 --> 00:01:11.582
Now is this indeed the case?

00:01:11.582 --> 00:01:13.579
Negative three times negative four

00:01:13.579 --> 00:01:15.448
is positive 12.

00:01:15.448 --> 00:01:19.639
Positive 12 minus four,
positive 12 minus four

00:01:19.639 --> 00:01:22.286
is equal to eight, it's not equal to six.

00:01:22.286 --> 00:01:25.466
Is not equal, is not equal to six.

00:01:25.466 --> 00:01:28.032
So this one does not work out.

00:01:28.032 --> 00:01:30.365
So let's see, negative three comma three.

00:01:30.365 --> 00:01:32.467
We can do the same thing here.

00:01:32.467 --> 00:01:35.973
Let's see what happens when
x is equal to negative three

00:01:35.973 --> 00:01:40.135
and y is equal to positive three.

00:01:40.148 --> 00:01:43.611
So we substitute back in,
we get negative three.

00:01:43.611 --> 00:01:47.930
Negative three times x,
which now we're going to

00:01:47.930 --> 00:01:50.728
try out x being equal to negative three.

00:01:50.728 --> 00:01:54.678
Minus y, minus y.

00:01:54.678 --> 00:01:56.957
Y is positive three here.

00:01:56.957 --> 00:02:00.719
Minus y, gonna do that y color blue.

00:02:00.719 --> 00:02:04.143
Minus y now needs to be equal to,

00:02:04.143 --> 00:02:06.976
now needs to be equal, just like before

00:02:06.976 --> 00:02:09.135
needs to be equal to six.

00:02:09.135 --> 00:02:11.736
So negative three times negative three.

00:02:11.736 --> 00:02:13.814
That's going to be positive nine.

00:02:13.814 --> 00:02:17.552
Nine minus three is indeed equal to six.

00:02:17.552 --> 00:02:21.075
Nine minus three is
indeed equal to six.

00:02:21.075 --> 00:02:22.575
Nine minus three is six.

00:02:22.575 --> 00:02:23.904
That is equal to six.

00:02:23.904 --> 00:02:25.425
This works out.

00:02:25.425 --> 00:02:29.685
So negative three comma
three is an ordered pair

00:02:29.685 --> 00:02:33.174
that is a solution to this equation.