Introduction to square roots

WEBVTT
Kind: captions
Language: en

00:00:00.117 --> 00:00:02.120
- [Voiceover] If you're
watching a movie and someone is

00:00:02.120 --> 00:00:05.217
attempting to do fancy
mathematics on a chalkboard,

00:00:05.217 --> 00:00:08.117
you'll almost always see a
symbol that looks like this.

00:00:08.117 --> 00:00:10.083
This radical symbol.

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And this is used to show the square root

00:00:13.117 --> 00:00:15.251
and we'll see other
types of roots as well,

00:00:15.251 --> 00:00:16.284
but your question is, well,

00:00:16.284 --> 00:00:18.183
what does this thing actually mean?

00:00:18.183 --> 00:00:19.950
And now that we know a
little bit about exponents,

00:00:19.950 --> 00:00:22.317
we'll see that the square
root symbol or the root symbol

00:00:22.317 --> 00:00:27.283
or the radical is not
so hard to understand.

00:00:27.283 --> 00:00:29.183
So, let's start with an example.

00:00:29.183 --> 00:00:33.818
So, we know that three to
the second power is what?

00:00:33.818 --> 00:00:35.483
Three squared is what?

00:00:35.483 --> 00:00:38.283
Well, that's the same
thing as three times three

00:00:38.283 --> 00:00:41.183
and that's going to be equal to nine.

00:00:41.183 --> 00:00:42.817
But what if we went the other way around?

00:00:42.817 --> 00:00:44.850
What if we started with the nine,

00:00:44.850 --> 00:00:47.350
and we said, well, what times
itself is equal to nine?

00:00:47.350 --> 00:00:48.917
We already know that answer is three,

00:00:48.917 --> 00:00:52.383
but how could we use a
symbol that tells us that?

00:00:52.383 --> 00:00:57.250
So, as you can imagine,
that symbol is going to be

00:00:57.250 --> 00:00:58.751
the radical here.

00:00:58.751 --> 00:01:03.317
So, we could write the
square root of nine,

00:01:03.317 --> 00:01:04.917
and when you look at
this way, you say, okay,

00:01:04.917 --> 00:01:07.683
what squared is equal to nine?

00:01:07.683 --> 00:01:11.051
And you would say, well,
this is going to be equal to,

00:01:11.051 --> 00:01:14.651
this is going to be equal to, three.

00:01:14.651 --> 00:01:15.783
And I want you to really look at

00:01:15.783 --> 00:01:20.184
these two equations right over here,

00:01:20.184 --> 00:01:22.650
because this is the essence
of the square root symbol.

00:01:22.650 --> 00:01:24.483
If you say the square root of nine,

00:01:24.483 --> 00:01:28.783
you're saying what times
itself is equal to nine?

00:01:28.783 --> 00:01:31.317
And, well, that's going to be three.

00:01:31.317 --> 00:01:34.950
And three squared is equal
to nine, I can do that again.

00:01:34.950 --> 00:01:36.150
I can do that many times.

00:01:36.150 --> 00:01:41.150
I can write four, four
squared, is equal to 16.

00:01:42.251 --> 00:01:45.517
Well, what's the square
root of 16 going to be?

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Well, it's going to be equal to four.

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Let me do it again.

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Actually, let me start
with the square root.

00:01:52.417 --> 00:01:57.383
What is the square root of 25 going to be?

00:01:57.383 --> 00:02:00.053
Well, this is the number that
times itself is going to be

00:02:00.053 --> 00:02:03.151
equal to 25 or the number,
where if I were to square it,

00:02:03.151 --> 00:02:04.484
I'd get to 25.

00:02:04.484 --> 00:02:05.650
Well, what number is that, well,

00:02:05.650 --> 00:02:07.817
that's going to be equal to five.

00:02:07.817 --> 00:02:12.534
Why, because we know that
five squared is equal to,

00:02:13.007 --> 00:02:16.585
five squared is equal to 25.

00:02:16.585 --> 00:02:18.852
Now, I know that there's a nagging feeling

00:02:18.852 --> 00:02:20.917
that some of you might be having,

00:02:20.917 --> 00:02:25.520
because if I were to take negative three,

00:02:25.520 --> 00:02:30.520
and square it, and square
it I would also get

00:02:30.770 --> 00:02:33.517
positive nine, and the same
thing if I were to take

00:02:33.517 --> 00:02:35.450
negative four and I were
to square the whole thing,

00:02:35.450 --> 00:02:38.450
I would also get positive
16, or negative five,

00:02:38.450 --> 00:02:41.517
and if I square that I
would also get positive 25.

00:02:41.517 --> 00:02:44.183
So, why couldn't this thing
right over here, why can't this

00:02:44.183 --> 00:02:48.184
square root be positive
three or negative three?

00:02:48.184 --> 00:02:49.917
Well, depending on who you talk to,

00:02:49.917 --> 00:02:52.083
that's actually a reasonable
thing to think about.

00:02:52.083 --> 00:02:55.500
But when you see a
radical symbol like this,

00:02:55.500 --> 00:02:58.883
people usually call
this the principal root.

00:02:58.883 --> 00:03:00.784
Principal root.

00:03:00.784 --> 00:03:04.634
Principal, principal square root.

00:03:05.550 --> 00:03:07.417
Square root.

00:03:07.417 --> 00:03:10.150
And another way to think
about it, it's the positive,

00:03:10.150 --> 00:03:12.117
this is going to be the
positive square root.

00:03:12.117 --> 00:03:15.383
If someone wants the
negative square root of nine,

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they might say something like this.

00:03:17.083 --> 00:03:20.617
They might say the negative,
let me scroll up a little bit,

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they might say something like

00:03:22.151 --> 00:03:24.784
the negative square root of nine.

00:03:24.784 --> 00:03:27.537
Well, that's going to be
equal to negative three.

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And what's interesting
about this is, well,

00:03:29.317 --> 00:03:32.917
if you square both sides
of this, of this equation,

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if you were to square both
sides of this equation,

00:03:35.283 --> 00:03:36.817
what do you get?

00:03:36.817 --> 00:03:40.817
Well negative, anything negative
squared becomes a positive.

00:03:40.817 --> 00:03:43.250
And then the square root
of nine squared, well,

00:03:43.250 --> 00:03:45.162
that's just going to be nine.

00:03:45.162 --> 00:03:48.983
And on the right-hand side,
negative three squared,

00:03:48.983 --> 00:03:52.157
well, negative three times
negative three is positive nine.

00:03:52.157 --> 00:03:53.317
So, it all works out.

00:03:53.317 --> 00:03:58.250
Nine is equal, nine is equal to nine.

00:03:58.250 --> 00:03:59.818
And so this is an
interesting thing, actually.

00:03:59.818 --> 00:04:03.183
Let me write this a little
bit more algebraically now.

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If we were to write, if we were to write

00:04:05.858 --> 00:04:10.417
the principal root of nine is equal to x.

00:04:10.417 --> 00:04:14.650
This is, there's only one
possible x here that satisfies it,

00:04:14.650 --> 00:04:16.653
because the standard convention,

00:04:16.653 --> 00:04:19.583
what most mathematicians
have agreed to view

00:04:19.583 --> 00:04:21.517
this radical symbol as, is that this is

00:04:21.517 --> 00:04:24.584
a principal square root, this
is the positive square root,

00:04:24.584 --> 00:04:26.083
so there's only one x here.

00:04:26.083 --> 00:04:28.385
There's only one x that
would satisfy this,

00:04:28.385 --> 00:04:31.517
and that is x is equal to three.

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Now, if I were to write x
squared is equal to nine,

00:04:36.350 --> 00:04:37.750
now, this is slightly different.

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X equals three definitely satisfies this.

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This could be x equals
three, but the other thing,

00:04:43.617 --> 00:04:45.183
the other x that satisfies this

00:04:45.183 --> 00:04:49.917
is x could also be
equal to negative three,

00:04:49.917 --> 00:04:53.983
'cause negative three squared
is also equal to nine.

00:04:53.983 --> 00:04:56.684
So, these two things,
these two statements,

00:04:56.684 --> 00:04:59.614
are almost equivalent,
although when you're looking

00:04:59.614 --> 00:05:02.617
at this one, there's two
x's that satisfy this one,

00:05:02.617 --> 00:05:04.417
while there's only one x
that satisfies this one,

00:05:04.417 --> 00:05:07.116
because this is a positive square root.

00:05:07.116 --> 00:05:08.750
If people wanted to write
something equivalent

00:05:08.750 --> 00:05:11.152
where you would have two
x's that could satisfy it,

00:05:11.152 --> 00:05:12.718
you might see something like this.

00:05:12.718 --> 00:05:17.317
Plus or minus square root
of nine is equal to x,

00:05:17.317 --> 00:05:21.567
and now x could take on positive
three or negative three.


Standard form for linear equations

WEBVTT
Kind: captions
Language: en

00:00:00.119 --> 00:00:01.606
- [Voiceover] We've already
looked at several ways

00:00:01.606 --> 00:00:02.998
of writing linear equations.

00:00:02.998 --> 00:00:05.410
You could write it in
slope-intercept form,

00:00:05.410 --> 00:00:10.400
where it would be of the form
of Y is equal to MX plus B,

00:00:10.400 --> 00:00:12.766
where M and B are constants.

00:00:12.766 --> 00:00:16.181
M is the coefficient on
this MX term right over here

00:00:16.181 --> 00:00:19.617
and M would represent the slope.

00:00:19.617 --> 00:00:22.171
And then from B you're able to figure out

00:00:22.171 --> 00:00:23.888
the y-intercept.

00:00:23.888 --> 00:00:25.190
The Y,

00:00:25.190 --> 00:00:28.277
you're able to figure out
the y-intercept from this.

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Literally the graph that
represents the XY pairs

00:00:32.271 --> 00:00:34.987
that satisfy this equation,

00:00:34.987 --> 00:00:37.496
it would intersect the y-axis at the point

00:00:37.496 --> 00:00:40.793
X equals zero, Y is equal to B.

00:00:40.793 --> 00:00:42.674
And it's slope would be M.

00:00:42.674 --> 00:00:44.804
We've already seen that multiple times.

00:00:44.804 --> 00:00:46.222
We've also seen that you can also

00:00:46.222 --> 00:00:48.892
express things in point-slope form.

00:00:48.892 --> 00:00:50.401
So let me make it clear.

00:00:50.401 --> 00:00:51.872
This is slope-intercept.

00:00:52.852 --> 00:00:54.803
Slope- intercept.

00:00:54.803 --> 00:00:56.660
And these are just different ways of

00:00:56.660 --> 00:00:58.378
writing the same equations.

00:00:58.378 --> 00:01:00.979
You can algebraically manipulate
from one to the other.

00:01:00.979 --> 00:01:02.825
Another way is point-slope.

00:01:03.991 --> 00:01:07.079
Point-slope form.

00:01:07.079 --> 00:01:09.285
And in point-slope form,
if you know that some,

00:01:09.285 --> 00:01:12.768
if you know that there's
an equation where the line

00:01:12.768 --> 00:01:15.156
that represents the
solutions of that equation

00:01:15.156 --> 00:01:16.968
has a slope M.

00:01:16.968 --> 00:01:19.243
Slope is equal to M.

00:01:19.243 --> 00:01:21.774
And if you know that X equals,

00:01:21.774 --> 00:01:25.280
X equals A, Y equals B,

00:01:25.280 --> 00:01:27.323
satisfies that equation,

00:01:27.323 --> 00:01:30.226
then in point-slope form you
can express the equation as

00:01:30.226 --> 00:01:32.961
Y minus B is equal to

00:01:34.807 --> 00:01:38.303
M times X minus A.

00:01:38.306 --> 00:01:41.652
This is point-slope form
and we do videos on that.

00:01:41.652 --> 00:01:43.788
But what I really want
to get into in this video

00:01:43.788 --> 00:01:44.740
is another form.

00:01:44.740 --> 00:01:46.876
And it's a form that you
might have already seen.

00:01:46.876 --> 00:01:48.556
And that is standard form.

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Standard.

00:01:51.070 --> 00:01:52.509
Standard form.

00:01:52.509 --> 00:01:56.063
And standard form takes the shape of

00:01:56.063 --> 00:02:01.054
AX plus BY is equal to C,

00:02:01.054 --> 00:02:04.956
where A, B, and C are integers.

00:02:04.956 --> 00:02:06.325
And what I want to do in this video,

00:02:06.325 --> 00:02:07.974
like we've done in the ones on point-slope

00:02:07.974 --> 00:02:10.064
and slope-intercept is get an appreciation

00:02:10.064 --> 00:02:12.014
for what is standard form good at

00:02:12.014 --> 00:02:14.637
and what is standard form less good at?

00:02:14.637 --> 00:02:16.657
So let's give a tangible example here.

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So let's say I have the linear equation,

00:02:18.980 --> 00:02:20.652
it's in standard form,

00:02:20.652 --> 00:02:22.977
9X plus

00:02:22.977 --> 00:02:27.118
16Y is equal to 72.

00:02:27.936 --> 00:02:30.165
And we wanted to graph this.

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So the thing that standard
form is really good for

00:02:33.276 --> 00:02:35.830
is figuring out, not just the y-intercept,

00:02:35.830 --> 00:02:37.475
y-intercept is pretty good if you're

00:02:37.475 --> 00:02:39.285
using slope-intercept form,

00:02:39.285 --> 00:02:41.375
but we can find out the
y-intercept pretty clearly

00:02:41.375 --> 00:02:44.463
from standard form and the x-intercept.

00:02:44.463 --> 00:02:47.111
The x-intercept isn't
so easy to figure out

00:02:47.111 --> 00:02:49.687
from these other forms right over here.

00:02:49.687 --> 00:02:51.128
So how do we do that?

00:02:51.128 --> 00:02:53.147
Well to figure out the x and y-intercepts,

00:02:53.147 --> 00:02:55.213
let's just set up a little table here,

00:02:55.213 --> 00:02:57.188
X comma Y,

00:02:57.188 --> 00:03:00.322
and so the x-intercept is going to happen

00:03:00.322 --> 00:03:02.784
when Y is equal to zero.

00:03:02.784 --> 00:03:04.920
And the y-intercept is going to happen

00:03:04.920 --> 00:03:06.985
when X is equal to zero.

00:03:06.985 --> 00:03:09.471
So when Y is zero, what is X?

00:03:09.471 --> 00:03:12.187
So when Y is zero, 16 times zero is zero,

00:03:12.187 --> 00:03:13.627
that term disappears,

00:03:13.627 --> 00:03:17.783
and you're left with 9X is equal to 72.

00:03:17.783 --> 00:03:20.035
So if nine times X is 72,

00:03:20.035 --> 00:03:22.682
72 divided by nine is eight.

00:03:22.682 --> 00:03:25.282
So X would be equal to eight.

00:03:25.282 --> 00:03:27.232
So once again, that was
pretty easy to figure out.

00:03:27.232 --> 00:03:28.905
This term goes away and
you just have to say hey,

00:03:28.905 --> 00:03:32.249
nine times X is 72, X would be eight.

00:03:32.249 --> 00:03:34.733
When Y is equal to zero, X is eight.

00:03:34.733 --> 00:03:36.614
So the point, let's see,

00:03:36.614 --> 00:03:39.423
Y is zero, X is one, two, three,

00:03:39.423 --> 00:03:41.560
four, five, six, seven, eight.

00:03:41.560 --> 00:03:42.628
That's this point,

00:03:42.628 --> 00:03:43.486
that right over here.

00:03:43.486 --> 00:03:46.064
This point right over
here is the x-intercept.

00:03:46.064 --> 00:03:48.851
When we talk about x-intercepts
we're referring to the point

00:03:48.851 --> 00:03:52.297
where the line actually
intersects the x-axis.

00:03:53.045 --> 00:03:54.646
Now what about the y-intercept?

00:03:54.646 --> 00:03:57.526
Well, we said X equals
zero, this disappears.

00:03:57.526 --> 00:04:00.823
And we're left with 16Y is equal to 72.

00:04:00.823 --> 00:04:02.204
And so we could solve,

00:04:02.204 --> 00:04:03.295
we could solve that.

00:04:03.295 --> 00:04:07.752
So we could say, alright
16Y is equal to 72.

00:04:07.752 --> 00:04:09.425
And then divide both sides by 16.

00:04:09.425 --> 00:04:13.674
We get Y is equal to 72 over 16.

00:04:13.674 --> 00:04:15.207
And let's see, what is that equal to?

00:04:15.207 --> 00:04:17.691
That is equal to, let's see,

00:04:17.691 --> 00:04:20.245
they're both divisible by eight,

00:04:20.245 --> 00:04:23.496
so that's nine over two.

00:04:23.496 --> 00:04:26.120
Or we could say it's 4.5.

00:04:26.120 --> 00:04:29.500
So when X is zero, Y is 4.5.

00:04:30.457 --> 00:04:32.593
And so, we could plot that point as well.

00:04:32.593 --> 00:04:34.869
X is zero, Y is one, two, three,

00:04:34.869 --> 00:04:37.237
4.5.

00:04:37.237 --> 00:04:39.165
And just with these two
points, two points are enough

00:04:39.165 --> 00:04:41.254
to graph a line, we can now graph it.

00:04:41.254 --> 00:04:42.671
So let's do that.

00:04:42.671 --> 00:04:44.319
So let me,

00:04:44.319 --> 00:04:45.643
oops, though I was using the tool

00:04:45.643 --> 00:04:47.499
that would draw a straight line.

00:04:47.499 --> 00:04:50.182
Let me see if I can...

00:04:51.279 --> 00:04:55.087
So the line will look something like that.

00:04:55.087 --> 00:04:55.853
There you have it.

00:04:55.853 --> 00:04:57.409
I've just graphed,

00:04:57.409 --> 00:04:59.011
I've just graphed, this is

00:04:59.011 --> 00:05:01.611
the line that represents
all the X and Y pairs

00:05:01.611 --> 00:05:03.492
that satisfy the equation

00:05:03.492 --> 00:05:08.285
9X plus 16Y is equal to 72.

00:05:09.056 --> 00:05:11.308
Now, I mentioned standard
form's good at certain things

00:05:11.308 --> 00:05:13.212
and the good thing that standard form is,

00:05:13.212 --> 00:05:15.070
where it's maybe somewhat unique relative

00:05:15.070 --> 00:05:16.672
to the other forms we looked at,

00:05:16.672 --> 00:05:19.431
is it's very easy to
figure out the x-intercept.

00:05:19.431 --> 00:05:21.265
It was very easy to
figure out the x-intercept

00:05:21.265 --> 00:05:22.263
from standard form.

00:05:22.263 --> 00:05:25.444
And it wasn't too hard to figure
out the y-intercept either.

00:05:25.444 --> 00:05:27.186
If we looked at slope-intercept form,

00:05:27.186 --> 00:05:29.299
the y-intercept just
kinda jumps out at you.

00:05:29.299 --> 00:05:33.083
At point-slope form, neither
the x nor the y-intercept

00:05:33.083 --> 00:05:34.941
kind of jump out at you.

00:05:34.941 --> 00:05:37.263
The place where slope-intercept
or point-slope form

00:05:37.263 --> 00:05:39.375
are frankly better is
that it's pretty easy

00:05:39.375 --> 00:05:41.534
to pick out the slope here,

00:05:41.534 --> 00:05:44.089
while in standard form you would have to

00:05:44.089 --> 00:05:45.506
do a little bit of work.

00:05:45.506 --> 00:05:47.317
You could use these two points,

00:05:47.317 --> 00:05:49.105
you could use the x and y-intercepts

00:05:49.105 --> 00:05:51.357
as two points and figure
out the slope from there.

00:05:51.357 --> 00:05:53.303
So you can literally say,
"Okay, if I'm going from

00:05:53.303 --> 00:05:55.870
"this point to this point, my change in X

00:05:57.129 --> 00:05:59.637
"to go from eight to
zero is negative eight.

00:05:59.637 --> 00:06:01.660
"And to go from zero to 4.5,"

00:06:02.464 --> 00:06:04.011
I wrote that little delta
there unnecessarily.

00:06:04.011 --> 00:06:04.926
Let me.

00:06:05.604 --> 00:06:07.880
So when you go from eight to zero,

00:06:07.880 --> 00:06:11.177
your change in X is
equal to negative eight.

00:06:11.177 --> 00:06:13.963
And to go from zero to 4.5,

00:06:13.963 --> 00:06:17.678
your change in Y is going to be 4.5.

00:06:17.678 --> 00:06:18.899
So your slope,

00:06:19.647 --> 00:06:20.576
once you've figured this out,

00:06:20.576 --> 00:06:22.689
you could say, "Okay, this is going to be

00:06:22.689 --> 00:06:26.033
"change in Y, 4.5, over change in X,

00:06:26.033 --> 00:06:27.774
"over negative 8."

00:06:27.774 --> 00:06:29.793
And since I, at least I
don't like a decimal up here,

00:06:29.793 --> 00:06:32.395
let's multiply the numerator
and the denominator by two.

00:06:32.395 --> 00:06:36.760
You get negative nine over 16.

00:06:36.760 --> 00:06:38.356
Now once again, we had to do
a little bit of work here.

00:06:38.356 --> 00:06:39.657
We either use these two points,

00:06:39.657 --> 00:06:42.791
it didn't just jump
immediately out of this,

00:06:42.791 --> 00:06:45.299
although you might see a
little bit of a pattern

00:06:45.299 --> 00:06:46.855
of what's going on here.

00:06:46.855 --> 00:06:47.621
But you still have to think about

00:06:47.621 --> 00:06:48.861
is it negative? Is it positive?

00:06:48.861 --> 00:06:50.929
You have to do a little bit
of algebraic manipulation.

00:06:50.929 --> 00:06:52.856
Or, what I typically do if
I'm looking for the slope,

00:06:52.856 --> 00:06:55.131
I actually might put this into,

00:06:55.131 --> 00:06:56.757
into one of the other forms.

00:06:56.757 --> 00:06:58.963
Especially slope-intercept form.

00:06:58.963 --> 00:07:00.681
But standard form by itself,

00:07:00.681 --> 00:07:03.931
great for figuring out
both the x and y-intercepts

00:07:03.931 --> 00:07:06.551
and it's frankly not that
hard to convert it to

00:07:06.551 --> 00:07:08.153
slope-intercept form.

00:07:08.153 --> 00:07:09.444
Let's do that

00:07:09.444 --> 00:07:10.813
just to make it clear.

00:07:10.813 --> 00:07:12.717
So if you start with 9X,

00:07:12.717 --> 00:07:14.993
let me do that in yellow.

00:07:14.993 --> 00:07:19.283
If we start with 9X plus 16Y

00:07:19.283 --> 00:07:21.210
is equal to 72

00:07:21.210 --> 00:07:23.765
and we want to put it
in slope-intercept form,

00:07:23.765 --> 00:07:25.715
we can subtract 9X from both sides.

00:07:25.715 --> 00:07:30.373
You get 16Y is equal to
negative 9X, plus 72.

00:07:31.065 --> 00:07:33.540
And then divide both sides by 16.

00:07:34.706 --> 00:07:36.587
So divide everthing

00:07:36.587 --> 00:07:37.957
by 16.

00:07:37.957 --> 00:07:40.464
And you'll be left with Y is equal to

00:07:40.464 --> 00:07:43.900
negative 9/16X, that's the slope,

00:07:43.900 --> 00:07:45.595
you see it right there,

00:07:45.595 --> 00:07:49.891
plus 72 over 16, we already figured out

00:07:49.891 --> 00:07:52.723
that's 9/2 or 4.5.

00:07:52.723 --> 00:07:55.789
So I could write, oh I'll
just write that as 4.5.

00:07:55.789 --> 00:07:59.225
And this form over here, much
easier to figure out the slope

00:07:59.225 --> 00:08:01.477
and, actually, the
y-intercept jumps out at you.

00:08:01.477 --> 00:08:04.671
But the x-intercept isn't as obvious.


Point-slope and slope-intercept form from two points

WEBVTT
Kind: captions
Language: en

00:00:00.306 --> 00:00:01.861
- [Voiceover] Let's say
we have a linear equation

00:00:01.861 --> 00:00:05.065
and we know when X is equal to four

00:00:05.065 --> 00:00:08.456
that Y is equal to nine

00:00:08.456 --> 00:00:11.869
and we've plotted that
point here on our XY plane.

00:00:11.869 --> 00:00:15.189
Actually, I forgot to label
the x-axis right over there.

00:00:15.189 --> 00:00:16.768
Now let's say we also know,

00:00:16.768 --> 00:00:19.903
we also know that when X is equal to six

00:00:19.903 --> 00:00:22.271
Y is equal to one.

00:00:22.271 --> 00:00:23.966
And we've plotted that point there.

00:00:23.966 --> 00:00:27.054
And so this green line
represents all of the solutions

00:00:27.054 --> 00:00:28.912
to this linear equation.

00:00:28.912 --> 00:00:30.584
Now what I want to do in this video is

00:00:30.584 --> 00:00:33.648
I want to say, well can we
find that linear equation

00:00:33.648 --> 00:00:37.735
and can we express it
in both point-slope form

00:00:37.735 --> 00:00:40.079
and in slope-intercept form.

00:00:40.079 --> 00:00:41.613
And I encourage you, like always,

00:00:41.613 --> 00:00:44.446
pause the video and see if you can do it.

00:00:44.446 --> 00:00:47.069
So let's first think
about point-slope form.

00:00:47.069 --> 00:00:48.509
Point,

00:00:48.509 --> 00:00:52.921
point-slope form.

00:00:52.921 --> 00:00:55.752
And point-slope form is
very easy to generate

00:00:55.752 --> 00:00:58.655
if you know a point on the line,

00:00:58.655 --> 00:01:01.046
or if you know a point that satisfies,

00:01:01.046 --> 00:01:04.089
where the X and Y coordinates
satisfy the linear equation,

00:01:04.089 --> 00:01:07.456
and if you were to know
the slope of the line

00:01:07.456 --> 00:01:10.614
that represents the solution
set of that linear equation.

00:01:10.614 --> 00:01:12.889
Now for sure we actually
were given two points

00:01:12.889 --> 00:01:15.420
that are solutions, that represent

00:01:15.420 --> 00:01:17.369
solutions to the linear equation.

00:01:17.369 --> 00:01:21.109
To fully apply point-slope, or
to apply point-slope easily,

00:01:21.109 --> 00:01:22.850
we just have to figure out the slope.

00:01:22.850 --> 00:01:24.614
And what we could do is,
we could just evaluate

00:01:24.614 --> 00:01:27.680
well what's the slope between
the two points that we know?

00:01:27.680 --> 00:01:30.511
And we just have to remind
ourselves that slope,

00:01:30.511 --> 00:01:33.136
slope, is equal to,

00:01:33.136 --> 00:01:38.127
slope is equal to change
in y over change in x.

00:01:38.127 --> 00:01:41.208
Sometimes people say rise over run.

00:01:41.208 --> 00:01:42.717
And what's that going to be?

00:01:42.717 --> 00:01:47.533
Well, if we say that this
second point right over here,

00:01:48.319 --> 00:01:50.315
if we say this is kind of our,

00:01:50.315 --> 00:01:52.638
if we're starting at this
point and we go to that point,

00:01:52.638 --> 00:01:56.213
then our change in Y, going
from this point to that point

00:01:56.213 --> 00:01:57.421
is going to be,

00:01:57.421 --> 00:02:01.537
it's going to be equal to one minus,

00:02:02.727 --> 00:02:05.700
one minus nine.

00:02:05.700 --> 00:02:08.045
One minus nine.

00:02:08.045 --> 00:02:12.828
This point right over here
is the point six comma one.

00:02:12.828 --> 00:02:15.684
So we started at Y equals nine,

00:02:15.684 --> 00:02:17.587
we finish at Y equals one,

00:02:17.587 --> 00:02:20.699
our change in Y is going
to be one minus nine.

00:02:20.699 --> 00:02:22.835
We have a negative eight change in Y,

00:02:22.835 --> 00:02:23.574
which makes sense.

00:02:23.574 --> 00:02:25.617
We've gone down eight.

00:02:25.617 --> 00:02:27.544
So this is going to be equal to,

00:02:27.544 --> 00:02:29.773
this is going to be
equal to negative eight.

00:02:29.773 --> 00:02:31.352
That's our change in Y.

00:02:31.352 --> 00:02:32.769
And what's our change in X?

00:02:32.769 --> 00:02:36.391
Well we go from X equals
four to X equals six.

00:02:36.391 --> 00:02:39.061
So we end up at X equals six

00:02:39.061 --> 00:02:42.428
and we started at X equals four.

00:02:42.428 --> 00:02:44.424
We started at X equals four,

00:02:44.424 --> 00:02:46.699
so our change in X is six minus four,

00:02:46.699 --> 00:02:48.371
which is equal to two.

00:02:48.371 --> 00:02:49.928
Which is equal to two.

00:02:49.928 --> 00:02:51.344
And you could have even done it visually.

00:02:51.344 --> 00:02:54.896
To go from this point to
this point your change in Y,

00:02:54.896 --> 00:02:56.940
your change in Y is

00:02:56.940 --> 00:02:58.217
you went down eight.

00:02:58.217 --> 00:03:00.144
So your change in, let me write this.

00:03:00.144 --> 00:03:04.277
So your change in Y is
equal to negative eight.

00:03:04.277 --> 00:03:05.508
And what was your change in X?

00:03:05.508 --> 00:03:07.215
Again to get to this point?

00:03:08.010 --> 00:03:10.462
Well, your change in X is positive two.

00:03:11.418 --> 00:03:15.389
So your change in X is equal to two.

00:03:15.389 --> 00:03:16.225
And so what's your slope?

00:03:16.225 --> 00:03:17.432
Change in Y over change in X.

00:03:17.432 --> 00:03:21.891
Negative eight over two
is equal to negative four.

00:03:21.891 --> 00:03:23.377
So now that we have a,

00:03:23.377 --> 00:03:24.840
now that we know the slope

00:03:24.840 --> 00:03:28.820
and we know a point, we know a,

00:03:28.820 --> 00:03:30.957
we actually know two points on the line,

00:03:30.957 --> 00:03:33.856
we can express this in point-slope form.

00:03:33.856 --> 00:03:35.481
And so let's do that.

00:03:35.481 --> 00:03:37.200
And the way I like to
it is I always like to

00:03:37.200 --> 00:03:41.164
just take it straight from the
definition of what slope is.

00:03:41.164 --> 00:03:44.090
We know that the slope between
any two points on this line

00:03:44.090 --> 00:03:46.157
is going to be negative four.

00:03:46.157 --> 00:03:50.940
So if we take an arbitrary
Y that sits on this line

00:03:50.940 --> 00:03:52.351
and if we find the difference
between that Y and,

00:03:52.351 --> 00:03:54.928
let's focus on this point up here.

00:03:54.928 --> 00:03:56.437
So if we find the difference between

00:03:56.437 --> 00:03:58.981
that Y and this Y, and nine,

00:04:00.356 --> 00:04:04.165
and it's over the difference between

00:04:04.165 --> 00:04:08.610
some X on the line and this X, and four.

00:04:11.052 --> 00:04:15.233
This is going to be the slope
between any XY on this line

00:04:15.233 --> 00:04:17.392
and this point right over here.

00:04:17.392 --> 00:04:19.852
And the slope between
any two points on a line

00:04:19.852 --> 00:04:21.525
are going to have to be constant.

00:04:21.525 --> 00:04:24.265
So this is going to be equal
to the slope of the line.

00:04:24.265 --> 00:04:27.167
It's going to be equal to negative four.

00:04:27.167 --> 00:04:28.723
And we're not in point-slope form

00:04:28.723 --> 00:04:30.929
or classic point-slope form just yet.

00:04:30.929 --> 00:04:34.666
To do that, we just multiply
both sides times X minus four.

00:04:34.666 --> 00:04:37.819
So we get Y minus 9,

00:04:37.819 --> 00:04:40.756
we get Y minus nine is equal to

00:04:40.756 --> 00:04:43.679
our slope, negative four

00:04:43.679 --> 00:04:45.931
times X minus four.

00:04:45.931 --> 00:04:47.950
Time X minus four.

00:04:47.950 --> 00:04:50.203
And this right over here is our classic,

00:04:50.203 --> 00:04:53.547
this right over here is our
classic point-slope form.

00:04:53.547 --> 00:04:54.562
We have the point,

00:04:54.562 --> 00:04:56.815
sometimes they even put
parenthesis like this,

00:04:56.815 --> 00:04:58.394
but we could figure out the point

00:04:58.394 --> 00:04:59.949
from this point-slope form.

00:04:59.949 --> 00:05:01.389
The point that sits on
this line with things

00:05:01.389 --> 00:05:04.198
that make both sides of
this equation equal to zero.

00:05:04.198 --> 00:05:07.078
So it would be X equals
four, Y equals nine,

00:05:07.078 --> 00:05:08.446
which we have right up there,

00:05:08.446 --> 00:05:10.584
and then the slope is right over here,

00:05:10.584 --> 00:05:12.232
it's negative four.

00:05:12.232 --> 00:05:16.494
Now from this can we now express this

00:05:16.494 --> 00:05:20.024
linear equation in y-intercept form?

00:05:20.024 --> 00:05:23.159
And y-intercept form, just
as a bit of a reminder,

00:05:23.159 --> 00:05:26.966
it's Y is equal to MX plus B.

00:05:26.966 --> 00:05:29.382
Where this coefficient is our slope

00:05:29.382 --> 00:05:31.332
and this constant right over here

00:05:31.332 --> 00:05:34.652
allows us to figure out our y-intercept.

00:05:34.652 --> 00:05:37.509
And to get this in this
form we just have to

00:05:37.509 --> 00:05:39.761
simplify a little bit of this algebra.

00:05:39.761 --> 00:05:41.665
So you have Y minus nine.

00:05:41.665 --> 00:05:44.590
Y minus nine is equal to,

00:05:44.590 --> 00:05:46.448
well let's distribute this negative four.

00:05:46.448 --> 00:05:47.655
And I'll just switch some colors.

00:05:47.655 --> 00:05:49.559
Let's distribute this negative four.

00:05:49.559 --> 00:05:52.555
Negative four times X is negative four X.

00:05:52.555 --> 00:05:56.386
Negative four time
negative four is plus 16.

00:05:56.386 --> 00:05:57.454
And now, if we just want to isolate

00:05:57.454 --> 00:05:59.172
the Y on the left hand side,

00:05:59.172 --> 00:06:01.680
we can add nine to both sides.

00:06:01.680 --> 00:06:02.539
So let's do that.

00:06:02.539 --> 00:06:04.257
Let's add nine,

00:06:04.257 --> 00:06:06.127
let's add nine to both sides.

00:06:07.084 --> 00:06:08.050
Let's add nine to both sides.

00:06:08.050 --> 00:06:10.441
On the left-hand side
we're just left with Y.

00:06:10.441 --> 00:06:13.009
And on the right-hand side we're left with

00:06:13.009 --> 00:06:17.982
negative four X and then
16 plus nine is plus 25.

00:06:18.598 --> 00:06:19.511
And there you have it.

00:06:19.511 --> 00:06:22.088
We have the same linear
equation, but it's now

00:06:22.088 --> 00:06:24.480
represented in slope-intercept form.

00:06:24.480 --> 00:06:26.756
Once again, we see the
slope right over here

00:06:26.756 --> 00:06:29.147
and now we can figure out
what the y-intercept is.

00:06:29.147 --> 00:06:31.562
The y-intercept when X is equal to zero,

00:06:31.562 --> 00:06:34.302
Y is going to be equal to 25.

00:06:34.302 --> 00:06:36.810
My axis right here, I
haven't drawn it high enough,

00:06:36.810 --> 00:06:39.085
but if I made it even taller
and taller and you see

00:06:39.085 --> 00:06:42.622
this line is gonna intersect the y-axis

00:06:42.622 --> 00:06:45.502
when Y is equal to,

00:06:45.502 --> 00:06:47.614
Y is equal to 25.

00:06:47.614 --> 00:06:50.169
So there you go, we wrote
it in point-slope form,

00:06:50.169 --> 00:06:52.839
that is that right over there,

00:06:52.839 --> 00:06:54.613
and we wrote it in Y,

00:06:55.718 --> 00:06:57.335
sorry, we wrote it in slope,

00:06:57.335 --> 00:07:00.284
we wrote it in slope-intercept form.

00:07:00.284 --> 00:07:02.884
Point-slope and slope-intercept.

00:07:02.884 --> 00:07:05.061
Hopefully you enjoyed that.


Slope-intercept form

WEBVTT
Kind: captions
Language: en

00:00:00.364 --> 00:00:01.769
- [Voiceover] There's
a lot of different ways

00:00:01.769 --> 00:00:04.301
that you could represent
a linear equation.

00:00:04.301 --> 00:00:06.970
So for example, if you
had the linear equation

00:00:06.970 --> 00:00:10.221
y is equal to 2x plus three,

00:00:10.221 --> 00:00:11.788
that's one way to represent it,

00:00:11.788 --> 00:00:14.749
but I could represent this in
an infinite number of ways.

00:00:14.749 --> 00:00:17.280
I could, let's see, I could

00:00:17.280 --> 00:00:19.589
subtract 2x from both sides,

00:00:19.589 --> 00:00:23.420
I could write this as negative 2x plus y

00:00:23.420 --> 00:00:25.987
is equal to three.

00:00:25.987 --> 00:00:29.615
I could manipulate it in
ways where I get it to,

00:00:29.615 --> 00:00:31.403
and I'm gonna do it right now,

00:00:31.403 --> 00:00:33.759
but this is another way of
writing that same thing.

00:00:33.759 --> 00:00:37.683
y minus five is equal to
two times x minus one.

00:00:37.683 --> 00:00:39.518
You could actually simplify this

00:00:39.518 --> 00:00:41.514
and you could get either
this equation here

00:00:41.514 --> 00:00:43.337
or that equation up on top.

00:00:43.337 --> 00:00:45.077
These are all equivalent,
you can get from one

00:00:45.077 --> 00:00:48.893
to the other with logical
algebraic operations.

00:00:48.893 --> 00:00:51.691
So there's an infinite number of ways

00:00:51.691 --> 00:00:54.175
to represent a given linear equation,

00:00:54.175 --> 00:00:56.033
but I what I wanna focus on in this video

00:00:56.033 --> 00:00:58.633
is this representation in particular,

00:00:58.633 --> 00:01:01.129
because this one is a
very useful representation

00:01:01.129 --> 00:01:03.509
of a linear equation and
we'll see in future videos,

00:01:03.509 --> 00:01:05.471
this one and this one can also be useful,

00:01:05.471 --> 00:01:07.746
depending on what you are looking for,

00:01:07.746 --> 00:01:09.255
but we're gonna focus on this one,

00:01:09.255 --> 00:01:10.917
and this one right over
here is often called

00:01:10.917 --> 00:01:13.377
slope-intercept form.

00:01:13.377 --> 00:01:17.173
Slope-intercept form.

00:01:17.173 --> 00:01:18.508
And hopefully in a few minutes,

00:01:18.508 --> 00:01:21.909
it will be obvious why it
called slop-intercept form.

00:01:21.909 --> 00:01:24.115
And before I explain that
to you, let's just try

00:01:24.115 --> 00:01:25.869
to graph this thing.

00:01:25.869 --> 00:01:26.890
I'm gonna try to graph it,

00:01:26.890 --> 00:01:28.666
I'm just gonna plot some points here,

00:01:28.666 --> 00:01:32.009
so x comma y, and I'm
gonna pick some x values

00:01:32.009 --> 00:01:34.633
where it's easy to calculate the y values.

00:01:34.633 --> 00:01:37.198
So maybe the easiest is
if x is equal to zero.

00:01:37.198 --> 00:01:40.159
If x is equal to zero, then
two times zero is zero,

00:01:40.159 --> 00:01:43.200
that term goes away, and
you're only left with this term

00:01:43.200 --> 00:01:46.336
right over here, y is equal to three.

00:01:46.336 --> 00:01:48.854
Y is equal to three.

00:01:48.854 --> 00:01:50.618
And so if we were to plot this.

00:01:50.618 --> 00:01:54.658
Actually let me start plotting it, so

00:01:54.658 --> 00:01:58.502
that is my y axis,

00:01:58.502 --> 00:02:03.098
and let me do the x axis, so

00:02:03.098 --> 00:02:06.256
that can be my x, oh that's not

00:02:06.256 --> 00:02:09.100
as straight as I would like it.

00:02:09.100 --> 00:02:12.491
So that looks pretty good, alright.

00:02:12.491 --> 00:02:15.810
That is my x axis and let me mark off

00:02:15.810 --> 00:02:19.304
some hash marks here,
so this is x equals one,

00:02:19.304 --> 00:02:22.938
x equals two, x equals three,

00:02:22.938 --> 00:02:27.938
this is y equals one,

00:02:28.758 --> 00:02:32.242
y equals two, y equals three,

00:02:32.242 --> 00:02:34.285
and obviously I could keep going

00:02:34.285 --> 00:02:38.267
and keep going, this would be
y is equal to negative one,

00:02:38.267 --> 00:02:41.401
this would be x is equal to negative one,

00:02:41.401 --> 00:02:43.828
negative two, negative three,

00:02:43.828 --> 00:02:45.707
so on and so forth.

00:02:45.707 --> 00:02:48.900
So this point right over
here, zero comma three,

00:02:48.900 --> 00:02:52.464
this is x is zero, y is three.

00:02:52.464 --> 00:02:55.518
Well, the point that represents
when x is equal to zero

00:02:55.518 --> 00:02:57.885
and y equals three, this is,

00:02:57.885 --> 00:03:00.022
we're right on the y axis.

00:03:00.022 --> 00:03:01.868
If they have a line going
through it and this line

00:03:01.868 --> 00:03:05.083
contains this point, this is
going to be the y- intercept.

00:03:05.083 --> 00:03:07.556
So one way to think about it,
the reason why this is called

00:03:07.556 --> 00:03:10.156
slope-intercept form is it's very easy

00:03:10.156 --> 00:03:12.246
to calculate the y-intercept.

00:03:12.246 --> 00:03:13.849
The y-intercept here is going to happen

00:03:13.849 --> 00:03:15.287
when it's written in this form,

00:03:15.287 --> 00:03:17.284
it's going to happen
when x is equal to zero

00:03:17.284 --> 00:03:18.573
and y is equal to three,

00:03:18.573 --> 00:03:20.836
it's gonna be this point right over here.

00:03:20.836 --> 00:03:23.054
So it's very easy to
figure out the intercept,

00:03:23.054 --> 00:03:25.724
the y-intercept from this form.

00:03:25.724 --> 00:03:27.083
Now you might be saying, well it says

00:03:27.083 --> 00:03:28.882
slope-intercept form, it must also be easy

00:03:28.882 --> 00:03:31.019
to figure out the slope from this form.

00:03:31.019 --> 00:03:32.666
And if you made that conclusion,

00:03:32.666 --> 00:03:33.781
you would be correct!

00:03:33.781 --> 00:03:35.917
And we're about to see
that in a few seconds.

00:03:35.917 --> 00:03:37.124
So let's plot some more points here

00:03:37.124 --> 00:03:39.585
and I'm just gonna keep
increasing x by one.

00:03:39.585 --> 00:03:42.499
So if you increase x by
one, so we could write

00:03:42.499 --> 00:03:45.901
that our delta x, our change
in x, delta Greek letter,

00:03:45.901 --> 00:03:47.955
this triangle is a Greek letter, delta,

00:03:47.955 --> 00:03:49.256
represents change in.

00:03:49.256 --> 00:03:51.403
Change in x here is one.

00:03:51.403 --> 00:03:53.539
We just increased x by
one, what's gonna be

00:03:53.539 --> 00:03:55.966
our corresponding change in y?

00:03:55.966 --> 00:03:58.161
What's going to be our change in y?

00:03:58.161 --> 00:04:00.261
So let's see, when x is equal to one,

00:04:00.261 --> 00:04:03.442
we have two times one, plus three

00:04:03.442 --> 00:04:06.054
is going to be five.

00:04:06.054 --> 00:04:08.840
So our change in y is going to be two.

00:04:08.840 --> 00:04:09.979
Let's do that again.

00:04:09.979 --> 00:04:12.219
Let's increase our x by one.

00:04:12.219 --> 00:04:14.795
Change in x is equal to one.

00:04:14.795 --> 00:04:16.956
So then if we're gonna increase by one,

00:04:16.956 --> 00:04:18.092
we're gonna go from x equals one

00:04:18.092 --> 00:04:19.497
to x equals two.

00:04:19.497 --> 00:04:21.761
Well what's our corresponding change in y?

00:04:21.761 --> 00:04:22.933
Well when x is equal to two,

00:04:22.933 --> 00:04:27.577
two times two is four,
plus three is seven.

00:04:27.577 --> 00:04:31.931
Well our change in y, our
change in y is equal to two.

00:04:31.931 --> 00:04:32.930
Went from five-

00:04:32.930 --> 00:04:34.705
when x went from one to two,

00:04:34.705 --> 00:04:37.247
y went from five to seven.

00:04:37.247 --> 00:04:39.244
So for every one that we increase x,

00:04:39.244 --> 00:04:41.729
y is increasing by two.

00:04:41.729 --> 00:04:46.535
So for this linear
equation, our change in y

00:04:46.535 --> 00:04:49.971
over change in x is always going to be,

00:04:49.971 --> 00:04:53.070
our change in y is two when
our change in x is one,

00:04:53.070 --> 00:04:54.487
or it's equal to two,

00:04:54.487 --> 00:04:57.251
or we could say that our
slope is equal to two.

00:04:57.251 --> 00:04:59.096
Well let's just graph this to make sure

00:04:59.096 --> 00:05:01.232
that we understand this.

00:05:01.232 --> 00:05:03.855
So when x equals one, y is equal to five.

00:05:03.855 --> 00:05:06.653
And actually we're gonna
have to graph five up here.

00:05:06.653 --> 00:05:09.033
So when x is equal to one, y is equal to,

00:05:09.033 --> 00:05:11.146
and actually this is a little bit higher,

00:05:11.146 --> 00:05:13.410
this, let me clean this up a little bit.

00:05:13.410 --> 00:05:15.291
So this one would be,

00:05:15.291 --> 00:05:17.567
erase that a little bit.

00:05:17.567 --> 00:05:18.774
Just like that.

00:05:18.774 --> 00:05:21.595
So that's y is equal to four,

00:05:21.595 --> 00:05:24.229
and this is y is equal to five.

00:05:24.229 --> 00:05:27.933
So when x is one, y is equal to five,

00:05:27.933 --> 00:05:30.871
so it's that point right over there.

00:05:30.871 --> 00:05:32.576
So our line is going to look-

00:05:32.576 --> 00:05:34.666
you only need two points to define a line,

00:05:34.666 --> 00:05:36.732
our line is going to like,

00:05:36.732 --> 00:05:40.586
let me do this in this
color right over here.

00:05:40.586 --> 00:05:44.034
Our line is going to look like,

00:05:44.034 --> 00:05:46.636
is going to look,

00:05:46.636 --> 00:05:49.409
is going to look something like,

00:05:49.409 --> 00:05:52.288
is going to look, let me see if I can,

00:05:52.288 --> 00:05:53.751
I didn't draw it completely at scale,

00:05:53.751 --> 00:05:56.212
but it's going to look
something like this.

00:05:56.212 --> 00:05:58.650
This is the line, this is the line,

00:05:58.650 --> 00:06:01.808
y is equal to 2x plus three.

00:06:01.808 --> 00:06:03.967
But we already figured out
that its slope is equal to two,

00:06:03.967 --> 00:06:07.067
when our change in x is one,

00:06:07.067 --> 00:06:11.444
when our change in x is
one, our change in y is two.

00:06:11.444 --> 00:06:13.626
If our change in x was negative one,

00:06:13.626 --> 00:06:15.844
if our change in x was negative one,

00:06:15.844 --> 00:06:19.593
our change in y is negative two.

00:06:19.593 --> 00:06:22.309
And you can see that,
if from zero we went,

00:06:22.309 --> 00:06:25.467
we went down one, if we
went to negative one,

00:06:25.467 --> 00:06:26.976
then what's our y going to be?

00:06:26.976 --> 00:06:29.948
Two times negative one is

00:06:29.948 --> 00:06:32.596
negative two plus three is one.

00:06:32.596 --> 00:06:36.554
So we see that, the point
negative one comma one

00:06:36.554 --> 00:06:38.957
is on the line as well.

00:06:38.957 --> 00:06:41.708
So the slope here, our
change in y over change in x,

00:06:41.708 --> 00:06:44.134
if we're going from between any two points

00:06:44.134 --> 00:06:47.606
on this line, is always going to be two.

00:06:47.606 --> 00:06:50.671
But where do you see two
in this original equation?

00:06:50.671 --> 00:06:53.086
Well you see the two right over here.

00:06:53.086 --> 00:06:55.186
And when you write something
in slop-intercept form,

00:06:55.186 --> 00:06:56.870
where you explicitly solve for y,

00:06:56.870 --> 00:07:01.549
y is equal to some constant
times x to the first power

00:07:01.549 --> 00:07:04.566
plus some other constant,
the second one is going

00:07:04.566 --> 00:07:07.573
to be your intercept, your y-intercept,

00:07:07.573 --> 00:07:09.286
or it's going to be a way to
figure out the y-intercept,

00:07:09.286 --> 00:07:11.601
the intercept itself is this
point, the point at which

00:07:11.601 --> 00:07:14.261
the line intercepts the y axis,

00:07:14.261 --> 00:07:17.545
and then this two is going
to represent your slope.

00:07:17.545 --> 00:07:19.205
And that makes sense because
every time you increase

00:07:19.205 --> 00:07:22.061
x by one, you're gonna
multiply that by two,

00:07:22.061 --> 00:07:25.544
so you're gonna increase y by two.

00:07:25.544 --> 00:07:28.378
So this is just a, kinda
of a get your feet wet

00:07:28.378 --> 00:07:29.945
with the idea of slope-intercept form,

00:07:29.945 --> 00:07:33.403
but you'll see, at least for
me, this is the easiest form

00:07:33.403 --> 00:07:35.377
for me to think about what the graph

00:07:35.377 --> 00:07:36.782
of something looks like,

00:07:36.782 --> 00:07:38.395
because if you were given another,

00:07:38.395 --> 00:07:40.740
if you were given another linear equation,

00:07:40.740 --> 00:07:44.456
let's say y is equal to negative x,

00:07:44.456 --> 00:07:46.778
negative x plus two.

00:07:46.778 --> 00:07:48.623
Well immediately you say, okay look,

00:07:48.623 --> 00:07:51.107
my yintercept is going to be the point

00:07:51.107 --> 00:07:55.089
zero comma two, so I'm
gonna intersect the y axis

00:07:55.089 --> 00:07:57.724
right at that point, and
then I have a slope of,

00:07:57.724 --> 00:08:00.302
the coefficient here is
really just negative one,

00:08:00.302 --> 00:08:02.333
so I have a slope of negative one.

00:08:02.333 --> 00:08:04.203
So as we increase x by one, we're gonna

00:08:04.203 --> 00:08:06.641
decrease y by one.

00:08:06.641 --> 00:08:10.076
Increase x by one, you're
gonna decrease y by one.

00:08:10.076 --> 00:08:13.874
If you increase x by two,
you're gonna decrease y by two.

00:08:13.874 --> 00:08:16.438
And so our line is gonna
look something like this.

00:08:16.438 --> 00:08:19.433
Let me see if I can draw
it relatively neatly.

00:08:19.433 --> 00:08:21.616
It's going to look something,

00:08:21.616 --> 00:08:25.017
I think I can do a little
bit better than that.

00:08:25.017 --> 00:08:27.677
It's 'cus my graph paper is hand drawn.

00:08:27.677 --> 00:08:30.891
It's not ideal, but I think you get,

00:08:30.891 --> 00:08:32.819
you get the point.

00:08:32.819 --> 00:08:34.409
It's gonna look something like that.

00:08:34.409 --> 00:08:36.208
So from slope-intercept form,

00:08:36.208 --> 00:08:38.740
very easy to figure out
what the y-intercept is,

00:08:38.740 --> 00:08:41.038
and very easy to figure out the slope.

00:08:41.038 --> 00:08:45.044
The slope here, slope
here is negative one.

00:08:45.044 --> 00:08:46.925
That's this negative one right over here,

00:08:46.925 --> 00:08:50.662
and the y-intercept, y-intercept

00:08:50.662 --> 00:08:54.040
is the point zero comma two,

00:08:54.040 --> 00:08:56.408
very easy to figure out 'cus essentially

00:08:56.408 --> 00:08:58.771
that gave you the information right there.


Introduction to intercepts

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:02.698
- [Voiceover] Let's say that
we have the linear equation,

00:00:02.698 --> 00:00:07.698
y = 1/2x - 3.

00:00:09.989 --> 00:00:13.077
So if we wanted to draw
the line that represents

00:00:13.077 --> 00:00:16.560
the set of all points, all the
coordinates where the x value

00:00:16.560 --> 00:00:18.743
and the y value satisfy this equation,

00:00:18.743 --> 00:00:20.438
we could start off by just trying to draw,

00:00:20.438 --> 00:00:23.990
by trying to draw a few of those points,

00:00:23.990 --> 00:00:26.591
and then connecting them with a line.

00:00:26.591 --> 00:00:30.584
Let's set up a little table here x, y.

00:00:30.584 --> 00:00:32.906
And we can just try a
couple of x values here,

00:00:32.906 --> 00:00:35.878
then figure out what the
corresponding y values are.

00:00:35.878 --> 00:00:37.304
I'm going to pick x
values where it's going

00:00:37.304 --> 00:00:39.175
to be fairly easy to
calculate the y values.

00:00:39.175 --> 00:00:41.846
Let's say when x is equal to zero,

00:00:41.846 --> 00:00:44.144
then you're gonna have 1/2 x 0 - 3,

00:00:44.144 --> 00:00:46.930
well then y is going to be -3.

00:00:46.930 --> 00:00:50.785
When x is, let me try x
= 2, because then 1/2 x 2

00:00:50.785 --> 00:00:52.387
is just gonna be 1.

00:00:52.387 --> 00:00:55.184
So when x = 2, you're going to have

00:00:55.184 --> 00:01:00.184
1/2 x 2 = 1, -3 is -2.

00:01:00.421 --> 00:01:04.925
When x is equal to, let's try 4.

00:01:04.925 --> 00:01:09.925
So 1/2 x 4 is 2, and then -3 is -1,

00:01:12.704 --> 00:01:14.375
and we could keep going but actually

00:01:14.375 --> 00:01:16.674
all we need is two points for a line.

00:01:16.674 --> 00:01:18.996
So we're ready to plot
this line if we'd like.

00:01:18.996 --> 00:01:22.293
The point 0, -3 is on this line.

00:01:22.293 --> 00:01:25.822
0, -3 and actually let me do
this in a slightly darker color

00:01:25.822 --> 00:01:27.762
so that we can see it on
this white background.

00:01:27.762 --> 00:01:32.091
0, -3 is on the line,
2, -2 is on the line.

00:01:32.091 --> 00:01:37.091
So 2, -2 and then we have 4, -1.

00:01:38.175 --> 00:01:42.006
So when x is 4, y is -1,

00:01:42.006 --> 00:01:47.006
and I could draw a line
that connects all of these

00:01:47.439 --> 00:01:49.831
so it would look something like...

00:01:49.831 --> 00:01:51.688
If I, let's see if I could do this.

00:01:51.688 --> 00:01:54.219
It would look something like,

00:01:54.219 --> 00:01:59.219
it would something like, like that.

00:01:59.861 --> 00:02:01.765
So this right over
here, this is literally,

00:02:01.765 --> 00:02:06.765
this is the graph of y = 1/2x - 3.

00:02:09.195 --> 00:02:11.006
Now when we look at a graph like this

00:02:11.006 --> 00:02:13.839
an interesting thing that
we might want to ask ourself

00:02:13.839 --> 00:02:18.228
is where does the graph
intersect our axes?

00:02:18.228 --> 00:02:19.621
So first we can say, well where does

00:02:19.621 --> 00:02:22.237
in intersect our x-axis?

00:02:22.237 --> 00:02:23.472
When you look at this, it looks like

00:02:23.472 --> 00:02:26.212
it happens at this point right over here.

00:02:26.212 --> 00:02:30.159
This point where a
graph intersects an axes

00:02:30.159 --> 00:02:31.924
this is called an intercept.

00:02:31.924 --> 00:02:35.917
This one in particular is
called the x-intercept.

00:02:35.917 --> 00:02:37.612
Why is it called the x-intercept?

00:02:37.612 --> 00:02:38.820
Because that's where the graph

00:02:38.820 --> 00:02:41.768
is intersecting the x-axis

00:02:41.768 --> 00:02:44.067
and the x-intercept, it looks like

00:02:44.067 --> 00:02:47.898
this is at the .6, 0.

00:02:47.898 --> 00:02:49.520
Now it's very interesting, the x-intercept

00:02:49.520 --> 00:02:51.846
happens when y = 0.

00:02:51.846 --> 00:02:53.587
Remember, you're on the x-axis when

00:02:53.587 --> 00:02:56.884
you haven't moved up
or down from that axis

00:02:56.884 --> 00:02:58.927
which means y = 0.

00:02:58.927 --> 00:03:02.828
So your x-intercept
happens at x = 6, y = 0.

00:03:02.828 --> 00:03:04.082
It's this coordinate.

00:03:04.082 --> 00:03:06.009
Now what about the y-intercept?

00:03:06.009 --> 00:03:08.285
Well the y-intercept is
this point right over here.

00:03:08.285 --> 00:03:10.467
This is where you intersect or I guess

00:03:10.467 --> 00:03:13.462
you could say intercept the y-axis.

00:03:13.462 --> 00:03:16.365
So this right over here, that over there

00:03:16.365 --> 00:03:19.128
is the y-intercept.

00:03:19.128 --> 00:03:21.705
The y-intercept is at the coordinate

00:03:21.705 --> 00:03:24.329
that has a 0 for the x-coordinate.

00:03:24.329 --> 00:03:27.696
X is 0 here and y is -3.

00:03:27.696 --> 00:03:30.157
X is 0 and y is -3.

00:03:30.157 --> 00:03:31.736
This was actually one of the points,

00:03:31.736 --> 00:03:36.426
or one of the pairs
that we first tried out.

00:03:36.426 --> 00:03:40.327
You can validate that 6,
0 satisfies this equation

00:03:40.327 --> 00:03:41.093
right over here.

00:03:41.093 --> 00:03:46.093
If x is 6, 1/2 x 6 is 3,
-3 is indeed equal to 0.

00:03:46.898 --> 00:03:49.080
So now that we know
what an x-intercept is,

00:03:49.080 --> 00:03:52.006
it's the point where a
graph intersects the x-axis

00:03:52.006 --> 00:03:54.514
or intercepts the x-axis
and the y-intercept

00:03:54.514 --> 00:03:58.282
is the point where a graph
intercepts the y-axis

00:03:58.282 --> 00:04:00.152
or intersects the y-axis.

00:04:00.152 --> 00:04:01.777
Let's try to see if we can
find the x and y-intercepts

00:04:01.777 --> 00:04:05.562
for a few other linear equations.

00:04:05.562 --> 00:04:09.172
So let's say that I had
the linear equation.

00:04:09.172 --> 00:04:14.172
Let's say that I have 5x + 6y = 30.

00:04:19.958 --> 00:04:21.722
I encourage you to pause this video,

00:04:21.722 --> 00:04:24.578
and figure out what are
the x and y-intercepts

00:04:24.578 --> 00:04:27.620
for the graph that
represents the solutions,

00:04:27.620 --> 00:04:31.312
all the xy pairs that
satisfy this equation.

00:04:31.312 --> 00:04:33.680
Well the easiest thing to do here,

00:04:33.680 --> 00:04:38.680
let's see what the y value is when x = 0

00:04:38.765 --> 00:04:42.201
and what x value is when y = 0.

00:04:42.201 --> 00:04:46.590
When x = 0 this becomes 6y = 30.

00:04:46.590 --> 00:04:48.354
So 6 times what is 30?

00:04:48.354 --> 00:04:50.955
Well y would be equal to 5 here.

00:04:50.955 --> 00:04:54.090
So when x is 0, y is 5.

00:04:54.090 --> 00:04:55.808
What about when y is 0?

00:04:55.808 --> 00:04:57.990
Well when y is 0, that's going to be 0,

00:04:57.990 --> 00:04:59.964
and you have 5x = 30.

00:04:59.964 --> 00:05:02.332
Well then x would be equal to 6.

00:05:02.332 --> 00:05:05.072
Then x would be equal to 6.

00:05:05.072 --> 00:05:08.462
So we could plot those points, 0, 5.

00:05:08.462 --> 00:05:12.084
When x is 0, y is 5.

00:05:12.084 --> 00:05:16.496
When x is 6, y is 0.

00:05:16.496 --> 00:05:20.490
So those are both points on this graph

00:05:20.490 --> 00:05:22.958
and then the actual graph is going to,

00:05:22.958 --> 00:05:26.662
or the actual line that
represents the x and y pairs

00:05:26.662 --> 00:05:30.288
that satisfy this equation
is going to look like,

00:05:30.288 --> 00:05:32.749
it's going to look like this.

00:05:32.749 --> 00:05:34.630
I'll just try.

00:05:34.630 --> 00:05:38.577
So I can make it go, it's
going to look like...

00:05:38.577 --> 00:05:41.195
It's going to go through those two points.

00:05:41.195 --> 00:05:44.126
So it going to...I can make
it go the other way too.

00:05:44.126 --> 00:05:45.404
Let me see.

00:05:45.404 --> 00:05:48.004
It's going to go through those two points

00:05:48.004 --> 00:05:51.549
and so it's going to
look something like that.

00:05:51.549 --> 00:05:54.451
Now what are its' x and y-intercepts?

00:05:54.451 --> 00:05:55.774
Well, we already kind of figured it out

00:05:55.774 --> 00:05:56.889
but the intercepts themselves,

00:05:56.889 --> 00:05:58.979
these are the points on the graph where

00:05:58.979 --> 00:06:01.231
they intersect the axes.

00:06:01.231 --> 00:06:03.669
So this right over here,
this is the y-intercept.

00:06:03.669 --> 00:06:06.293
That point is the y-intercept

00:06:06.293 --> 00:06:08.684
and it happens, it's
always going to happen

00:06:08.684 --> 00:06:13.491
when x = 0, and when x
= 0 we know that y = 5.

00:06:13.491 --> 00:06:15.859
It's that point, the point 0, 5.

00:06:15.859 --> 00:06:19.272
And what is the y inter...what
is the x-intercept?

00:06:19.272 --> 00:06:21.246
The x-intercept is the
point, it's actually

00:06:21.246 --> 00:06:25.077
the same x-intercept for this
equation right over here.

00:06:25.077 --> 00:06:30.077
It's the point 6, 0. That
point right over there.


Introduction to slope

WEBVTT
Kind: captions
Language: en

00:00:00.516 --> 00:00:02.118
- [Voiceover] As we start to graph lines,

00:00:02.118 --> 00:00:04.300
we might notice that they're
differences between lines.

00:00:04.300 --> 00:00:07.760
For example, this pink or
this magenta line here,

00:00:07.760 --> 00:00:10.361
it looks steeper than this blue line.

00:00:11.091 --> 00:00:13.681
And what we'll see is
this notion of steepness,

00:00:13.681 --> 00:00:17.404
how steep a line is, how
quickly does it increase

00:00:17.424 --> 00:00:18.719
or how quickly does it decrease,

00:00:19.039 --> 00:00:21.716
is a really useful idea in mathematics.

00:00:22.156 --> 00:00:25.160
So ideally, we'd be
able to assign a number

00:00:25.380 --> 00:00:27.357
to each of these lines
or to any lines that

00:00:27.357 --> 00:00:29.563
describes how steep it is,

00:00:29.563 --> 00:00:33.278
how quickly does it increase or decrease?

00:00:33.968 --> 00:00:35.577
So what's a reasonable way to do that?

00:00:35.577 --> 00:00:37.497
What's a reasonable way to assign a number

00:00:37.497 --> 00:00:40.058
to these lines that
describe their steepness?

00:00:41.328 --> 00:00:43.061
Well one way to think
about it, could say well,

00:00:43.061 --> 00:00:45.461
how much does a line increase

00:00:47.758 --> 00:00:48.781
in the vertical direction

00:00:48.781 --> 00:00:51.892
for a given increase in
the horizontal direction?

00:00:52.482 --> 00:00:53.388
So let's write this down.

00:00:54.038 --> 00:00:56.606
So let's say if we an increase

00:00:57.366 --> 00:00:59.358
increase, in vertical,

00:01:00.493 --> 00:01:01.560
in vertical,

00:01:02.480 --> 00:01:05.382
for a given increase in horizontal

00:01:05.382 --> 00:01:07.924
for a given increase

00:01:08.707 --> 00:01:10.292
a given increase

00:01:11.243 --> 00:01:12.773
in horizontal.

00:01:13.623 --> 00:01:17.340
So, how can this give us a value?

00:01:18.094 --> 00:01:20.927
Well let's look at that
magenta line again.

00:01:21.430 --> 00:01:23.299
Now let's just start at an arbituary point

00:01:23.299 --> 00:01:24.244
in that magenta line.

00:01:24.244 --> 00:01:25.176
But I'll start at a point

00:01:25.176 --> 00:01:26.466
where it's going to be easy for me

00:01:26.466 --> 00:01:28.532
to figure out what point we're at.

00:01:29.212 --> 00:01:30.731
So if we were to start right here,

00:01:31.941 --> 00:01:34.423
and if I were to increase
in the horizontal

00:01:34.563 --> 00:01:36.280
direction by one.

00:01:37.090 --> 00:01:38.360
So I move one to the right.

00:01:38.825 --> 00:01:40.066
To get back on the line, how much

00:01:40.066 --> 00:01:42.342
do I have to increase in
the vertical direction?

00:01:42.862 --> 00:01:45.714
Well I have to increase in
the vertical direction by two.

00:01:47.214 --> 00:01:47.951
By two.

00:01:47.951 --> 00:01:49.444
So at least for this magenta line,

00:01:49.444 --> 00:01:52.062
it looks like our increase in vertical is

00:01:52.236 --> 00:01:54.761
two, whenever we have an increase in

00:01:54.831 --> 00:01:56.084
one in the horizontal direction.

00:01:57.947 --> 00:02:00.585
Let's see, does that
still work if I were to

00:02:01.608 --> 00:02:03.386
start here, instead of
increasing the horizontal

00:02:03.386 --> 00:02:05.917
direction by one, if I were increase in

00:02:05.917 --> 00:02:07.519
the horizontal direction...

00:02:07.519 --> 00:02:08.657
So let's increase by three.

00:02:08.657 --> 00:02:11.722
So now, I've gone plus three

00:02:11.722 --> 00:02:12.790
in the horizontal direction,

00:02:12.790 --> 00:02:14.206
then to get back on the line,

00:02:14.206 --> 00:02:15.929
how much do I have to increase
in the vertical direction?

00:02:15.929 --> 00:02:20.336
I have to increase by one,
two, three, four, five, six

00:02:20.876 --> 00:02:22.426
I have to increase by six.

00:02:23.016 --> 00:02:24.028
So plus six.

00:02:24.688 --> 00:02:27.489
So when I increase by three

00:02:27.689 --> 00:02:30.225
in the horizontal direction,

00:02:30.865 --> 00:02:32.983
I increase by six in the vertical.

00:02:32.983 --> 00:02:34.701
We were just saying,
hey, let's just measure

00:02:34.701 --> 00:02:35.653
how much to we increase in vertical

00:02:35.653 --> 00:02:37.325
for a given increase in the horizontal?

00:02:38.115 --> 00:02:39.415
Well two over one is just two

00:02:39.415 --> 00:02:41.831
and that's the same
thing as six over three.

00:02:42.487 --> 00:02:44.479
So no matter where I start on this line,

00:02:44.834 --> 00:02:46.887
no matter where I start on this line,

00:02:47.684 --> 00:02:51.053
if I take and if I increase
in the horizontal direction

00:02:51.463 --> 00:02:52.991
by a given amount,

00:02:52.991 --> 00:02:54.631
I'm going to increase twice as much

00:02:55.171 --> 00:02:56.210
twice as much

00:02:56.680 --> 00:02:59.239
in the vertical direction.

00:02:59.476 --> 00:03:01.493
Twice as much in the vertical direction.

00:03:01.563 --> 00:03:03.919
So this notion of this
increase in vertical

00:03:03.919 --> 00:03:05.617
divided by increase in horizontal,

00:03:05.877 --> 00:03:09.027
this is what mathematicians
use to describe

00:03:09.027 --> 00:03:10.211
the steepness of lines.

00:03:10.901 --> 00:03:12.543
And this is called the slope.

00:03:13.242 --> 00:03:14.669
So this is called the slope of a line.

00:03:15.609 --> 00:03:17.525
And you're probably
familiar with the notion

00:03:18.275 --> 00:03:21.637
of the word slope being
used for a ski slope,

00:03:21.987 --> 00:03:24.201
and that's because a ski slope
has a certain inclination.

00:03:24.201 --> 00:03:25.582
It could have a steep
slope or a shallow slope.

00:03:26.512 --> 00:03:29.149
So slope is a measure for
how steep something is.

00:03:29.649 --> 00:03:32.867
And the convention is, is
we measure the increase

00:03:32.867 --> 00:03:35.172
in vertical for a given
in increase in horizontal.

00:03:36.052 --> 00:03:38.628
So six two over one is
equal to six over three

00:03:38.628 --> 00:03:40.744
is equal to two, this
is equal to the slope

00:03:40.744 --> 00:03:42.416
of this magenta line.

00:03:44.720 --> 00:03:46.252
So let me write this down.

00:03:46.880 --> 00:03:50.108
So this slope right over
here, the slope of that line,

00:03:50.548 --> 00:03:52.708
is going to be equal to two.

00:03:53.168 --> 00:03:54.598
And one way to interpret that,

00:03:54.598 --> 00:03:56.778
for whatever amount you increase in the

00:03:56.778 --> 00:03:58.339
horizontal direction,
you're going to increase

00:03:58.339 --> 00:04:00.745
twice as much in the vertical direction.

00:04:01.655 --> 00:04:03.309
Now what about this blue line here?

00:04:03.309 --> 00:04:04.983
What would be the slope of the blue line?

00:04:05.412 --> 00:04:08.268
Well, let me rewrite another
way that you'll typically

00:04:08.268 --> 00:04:09.758
see the definition of slope.

00:04:09.758 --> 00:04:12.223
And this is just the
convention that mathematicians

00:04:12.223 --> 00:04:13.515
have defined for slope

00:04:13.515 --> 00:04:14.607
but it's a valuable one.

00:04:15.017 --> 00:04:16.580
What is are is our change in vertical

00:04:16.580 --> 00:04:19.042
for a given change in horizontal?

00:04:19.822 --> 00:04:22.655
And I'll introduce a new notation for you.

00:04:23.275 --> 00:04:25.680
So, change in vertical,

00:04:26.726 --> 00:04:27.908
and in this coordinate,

00:04:28.357 --> 00:04:30.125
the vertical is our Y coordinate.

00:04:30.975 --> 00:04:33.213
divided by our change in horizontal.

00:04:34.103 --> 00:04:35.466
And X is our horizontal coordinate

00:04:35.876 --> 00:04:39.536
in this coordinate plane right over here.

00:04:40.098 --> 00:04:42.346
So wait, you said change in but then you

00:04:42.346 --> 00:04:43.465
drew this triangle.

00:04:43.465 --> 00:04:45.880
Well this is the Greek letter delta.

00:04:46.060 --> 00:04:48.248
This is the Greek letter delta.

00:04:48.248 --> 00:04:52.474
And it's a math symbol used
to represent change in.

00:04:53.054 --> 00:04:55.237
So that's delta, delta.

00:04:57.197 --> 00:04:59.319
And it literally means, change in Y,

00:04:59.909 --> 00:05:01.649
change in Y,

00:05:02.349 --> 00:05:04.214
divided by change in X,

00:05:04.657 --> 00:05:06.602
change in X.

00:05:07.141 --> 00:05:08.720
So if we want to find the
slope of the blue line,

00:05:09.050 --> 00:05:11.994
we just have to say, well
how much does Y change

00:05:11.994 --> 00:05:13.201
for a given change in X?

00:05:14.225 --> 00:05:15.618
So, the slope of the blue line.

00:05:17.228 --> 00:05:19.997
So let's see, let me do it this way.

00:05:23.876 --> 00:05:25.502
Let's just start at some point here.

00:05:25.502 --> 00:05:28.242
And let's say my X changes by two

00:05:28.242 --> 00:05:31.515
so my delta X is equal to positive two.

00:05:31.515 --> 00:05:33.257
What's my delta Y going to be?

00:05:33.257 --> 00:05:34.511
What's going to be my change in Y?

00:05:34.511 --> 00:05:37.367
Well, if I go by the right by two,

00:05:37.367 --> 00:05:38.411
to get back on the line,

00:05:38.411 --> 00:05:40.014
I'll have to increase my Y by two.

00:05:40.014 --> 00:05:44.345
So my change in Y is also
going to be plus two.

00:05:44.955 --> 00:05:46.190
So the slope of this blue line,

00:05:47.020 --> 00:05:48.442
the slope of the blue line,

00:05:48.952 --> 00:05:51.809
which is change in Y over change in X.

00:05:52.629 --> 00:05:55.311
We just saw that when our
change in X is positive two,

00:05:55.741 --> 00:05:58.449
our change in Y is also positive two.

00:05:58.919 --> 00:06:00.771
So our slope is two divided by two,

00:06:00.991 --> 00:06:01.863
which is equal to one.

00:06:02.483 --> 00:06:04.440
Which tells us however much we increase in

00:06:04.440 --> 00:06:06.971
X, we're going to increase
the same amount in Y.

00:06:06.971 --> 00:06:09.630
We see that, we increase
one in X, we increase

00:06:09.630 --> 00:06:10.709
one in Y.

00:06:10.709 --> 00:06:12.404
Increase one in X, increase one in Y.

00:06:12.404 --> 00:06:14.703
>From any point on the line,
that's going to be true.

00:06:14.703 --> 00:06:17.291
You increase three in X,

00:06:17.740 --> 00:06:20.391
you're going to increase three in Y.

00:06:20.851 --> 00:06:22.063
It's actually true the other way.

00:06:22.283 --> 00:06:24.130
If you decrease one in X,

00:06:24.130 --> 00:06:25.941
you're going to decrease one in Y.

00:06:26.681 --> 00:06:28.541
If you decrease two in X,

00:06:28.541 --> 00:06:30.621
you're going to decrease two in Y.

00:06:31.491 --> 00:06:34.694
And that makes sense from
the math of it as well

00:06:35.234 --> 00:06:37.806
Because if you're change
in X is negative two,

00:06:37.806 --> 00:06:39.547
that's what we did right over here,

00:06:39.727 --> 00:06:41.660
our change is X is negative two,

00:06:41.790 --> 00:06:42.542
we went two back,

00:06:42.542 --> 00:06:45.445
then your change in Y is going
to be negative two as well.

00:06:46.295 --> 00:06:47.906
Your change in Y is
going to be negative two,

00:06:47.906 --> 00:06:50.088
and negative two divided by negative two,

00:06:50.088 --> 00:06:53.246
is positive one, which
is your slope again.


Subtracting in scientific notation

WEBVTT
Kind: captions
Language: en

00:00:00.237 --> 00:00:01.166
- [Voiceover] What I
want to do in this video

00:00:01.166 --> 00:00:03.441
is get a little bit of
practice subtracting

00:00:03.441 --> 00:00:05.670
in scientific notation.

00:00:05.670 --> 00:00:10.670
So let's say that I have
4.1 x 10 to the -2 power.

00:00:13.077 --> 00:00:15.956
4.1 x 10 to the -2 power

00:00:15.956 --> 00:00:19.137
and from that I want to subtract,

00:00:19.137 --> 00:00:24.137
I want to subtract 2.6,
2.6 x 10 to the -3 power.

00:00:27.148 --> 00:00:29.191
Like always, I encourage
you to pause this video

00:00:29.191 --> 00:00:30.700
and see if you can solve this on your own

00:00:30.700 --> 00:00:32.999
and then we could work
through it together.

00:00:32.999 --> 00:00:35.344
All right, I'm assuming
you've had a go it.

00:00:35.344 --> 00:00:38.130
So the easiest thing
that I can think of doing

00:00:38.130 --> 00:00:40.592
is try to convert one of these numbers

00:00:40.592 --> 00:00:43.749
so that it has the same,
it's being multiplied

00:00:43.749 --> 00:00:46.280
by the same power of ten as the other one.

00:00:46.280 --> 00:00:47.952
What I could think about doing,

00:00:47.952 --> 00:00:52.294
well can we express
4.1 times 10 to the -2?

00:00:52.294 --> 00:00:56.566
Can we express it as
something times 10 to the -3?

00:00:56.566 --> 00:01:01.566
So we have 4.1 times 10 to the -2.

00:01:02.023 --> 00:01:06.156
Well if we want 10 to the
-2 to go to 10 to the -3

00:01:06.156 --> 00:01:09.871
we would divide by 10, but
we can't just divide by 10.

00:01:09.871 --> 00:01:12.146
That would literally change
the value of the number.

00:01:12.146 --> 00:01:13.261
In order to not change it, we want

00:01:13.261 --> 00:01:15.351
to multiply by 10 as well.

00:01:15.351 --> 00:01:18.578
So we're multiplying by
10 and dividing by 10.

00:01:18.578 --> 00:01:21.086
I could have written it like this.

00:01:21.086 --> 00:01:24.963
I could have written 10/10 times,

00:01:24.963 --> 00:01:26.565
let me write this a little bit neater.

00:01:26.565 --> 00:01:30.536
I could have written 10/10 x this

00:01:30.536 --> 00:01:34.437
and then you take 10 x 4.1, you get 41,

00:01:34.437 --> 00:01:37.246
and then 10 to the -2 divided by 10

00:01:37.246 --> 00:01:39.095
is going to be 10 to the -3.

00:01:39.095 --> 00:01:42.554
So this right over here,
this is equal to 10 x 4.1

00:01:42.554 --> 00:01:47.554
is 41 times 10 to the -3.

00:01:47.988 --> 00:01:49.218
And that makes sense.

00:01:49.218 --> 00:01:54.218
41 thousandths is the same
thing as 4.1 hundredths

00:01:54.303 --> 00:01:56.672
and all we did is we
multiplied this times 10

00:01:56.672 --> 00:01:58.831
and we divided this times 10.

00:01:58.831 --> 00:01:59.945
So let's rewrite this.

00:01:59.945 --> 00:02:04.945
We can rewrite it now as 41 X 10 to the -3

00:02:06.075 --> 00:02:11.075
minus 2.6, -2.6 x 10 to the -3.

00:02:15.107 --> 00:02:16.895
So now we have two things.

00:02:16.895 --> 00:02:21.702
We have 41 X 10 to the
-3 - 2.6 x 10 to the -3.

00:02:21.702 --> 00:02:24.674
Well this is going to be the same thing as

00:02:24.674 --> 00:02:28.598
41 - 2.6.

00:02:28.598 --> 00:02:33.265
- 2.6, let me do it in that same color.

00:02:33.265 --> 00:02:35.331
That was purple.

00:02:35.331 --> 00:02:39.902
- 2.6, 10 to the -3.

00:02:39.902 --> 00:02:43.186
10, whoops, 10 to the -3.

00:02:43.186 --> 00:02:45.995
There's 10 to the -3
there, 10 to the -3 there.

00:02:45.995 --> 00:02:48.550
One way to think about it,
I have just factored out

00:02:48.550 --> 00:02:50.895
a 10 to the -3.

00:02:50.895 --> 00:02:53.751
Now what's 41 - 2.6?

00:02:53.751 --> 00:02:58.751
Well 41 - 2 is 39, and then
-.6 is going to be 38.4.

00:03:02.527 --> 00:03:05.198
38.4 and then you're going to have

00:03:05.198 --> 00:03:08.263
times 10 to the -3 power.

00:03:08.263 --> 00:03:10.979
x 10 to the -3 power.

00:03:10.979 --> 00:03:13.394
Now, this is what the product,

00:03:13.394 --> 00:03:16.041
or this is what the difference
of these two numbers is

00:03:16.041 --> 00:03:18.363
but this is no longer
in scientific notation.

00:03:18.363 --> 00:03:20.429
In order to be scientific notation

00:03:20.429 --> 00:03:25.429
this number right over
here has to be between 1,

00:03:25.723 --> 00:03:27.859
has to be greater than or equal to 1

00:03:27.859 --> 00:03:31.876
and less than 10.

00:03:31.876 --> 00:03:34.593
So what we could do is we
could divide this number

00:03:34.593 --> 00:03:36.543
right over here by 10.

00:03:36.543 --> 00:03:40.374
We can divide this by 10
and then we can multiply,

00:03:40.374 --> 00:03:42.998
then we can multi...so we could do this.

00:03:42.998 --> 00:03:46.405
We could, kind of the opposite
of what we did up here.

00:03:46.405 --> 00:03:49.407
We could divide this by 10
and then multiply by 10.

00:03:49.407 --> 00:03:50.939
So if you divide by 10 and multiply by 10

00:03:50.939 --> 00:03:52.541
you're not changing the value.

00:03:52.541 --> 00:03:56.790
So 38.4 divided by 10 is 3.84

00:03:56.790 --> 00:03:58.508
and then all of this business,

00:03:58.508 --> 00:04:03.508
10 to the -3, 10 to the -3
x 10 is 10 to the -2 power.

00:04:05.103 --> 00:04:10.103
So this is going to be
3.84 x 10 to the -2 power.


Cheryl's birthday

WEBVTT
Kind: captions
Language: en

00:00:00.279 --> 00:00:02.494
- [Voiceover] So, this is a
fun deductive reasoning problem

00:00:02.494 --> 00:00:04.769
that's being forwarded
around the Internet.

00:00:04.769 --> 00:00:06.679
Several people have
actually forwarded it to me,

00:00:06.679 --> 00:00:08.934
and so, I thought it would be
a fun thing to work through.

00:00:08.934 --> 00:00:11.819
The grammar isn't perfect,
but the problem is a fun one.

00:00:11.819 --> 00:00:13.465
So, it says Albert and Bernard just,

00:00:13.465 --> 00:00:16.045
it should say became friends with Cheryl

00:00:16.045 --> 00:00:18.466
and they wanted to know
when her birthday is.

00:00:18.466 --> 00:00:20.965
Cheryl gives them a list
of 10 possible dates

00:00:20.965 --> 00:00:22.590
and we have our 10 dates here.

00:00:22.590 --> 00:00:25.028
Cheryl then tells Albert and
Bernard separately the month

00:00:25.028 --> 00:00:28.260
and the day of her birthday, respectively.

00:00:28.260 --> 00:00:30.636
So, just to make sure we're
parsing that sentence correctly,

00:00:30.636 --> 00:00:32.464
she gives Albert,

00:00:32.464 --> 00:00:36.734
she tells Albert the
month of her birthday.

00:00:36.734 --> 00:00:39.146
And she gives Bernard, she gives Bernard,

00:00:39.146 --> 00:00:41.526
the day of her birthday.

00:00:41.526 --> 00:00:44.870
And, I think we, it's fair
to assume that Albert knows,

00:00:44.870 --> 00:00:46.455
he's been given the month and he knows

00:00:46.455 --> 00:00:48.203
that Bernard's been given the day.

00:00:48.203 --> 00:00:50.487
And Bernard's been given the
day and he knows that Albert

00:00:50.487 --> 00:00:51.866
has been given the month.

00:00:51.866 --> 00:00:54.226
And we should also assume
that they are good logicians.

00:00:54.226 --> 00:00:57.002
That they are going to
do very logical things.

00:00:57.002 --> 00:00:58.927
And then, we have this dialogue.

00:00:58.927 --> 00:01:01.629
Albert says, I don't know
when Cheryl's birthday is,

00:01:01.629 --> 00:01:04.352
but I know that Bernard does not know too.

00:01:04.352 --> 00:01:05.449
That's fascinating.

00:01:05.449 --> 00:01:07.279
And then, Bernard says,

00:01:07.279 --> 00:01:10.439
at first I didn't know
when Cheryl's birthday is,

00:01:10.439 --> 00:01:14.002
but now I know, but now I know.

00:01:14.002 --> 00:01:15.867
Fascinating, and then Albert says,

00:01:15.867 --> 00:01:18.726
then I also know when
Cheryl's birthday is.

00:01:18.726 --> 00:01:21.682
So, when is Cheryl's birthday?

00:01:21.682 --> 00:01:23.795
So like always, pause the video.

00:01:23.795 --> 00:01:26.186
See if you can work
through this on your own.

00:01:26.186 --> 00:01:28.013
It's a fun problem to tackle,

00:01:28.013 --> 00:01:29.475
and I'll give you a little bit of a hint.

00:01:29.475 --> 00:01:32.178
Diagram out, write the
information that each,

00:01:32.178 --> 00:01:34.006
that Albert knows and
how does the world look

00:01:34.006 --> 00:01:35.327
from Albert's point of view.

00:01:35.327 --> 00:01:36.485
And then, write out how the world looks

00:01:36.485 --> 00:01:38.781
from Bernard's point of
view and then tell us,

00:01:38.781 --> 00:01:40.540
and then think about what these,

00:01:40.540 --> 00:01:43.779
what you can deduce from these sentences.

00:01:43.779 --> 00:01:45.837
So, I'm assuming you've had a go at it,

00:01:45.837 --> 00:01:46.917
so let's just do what I just said.

00:01:46.917 --> 00:01:48.644
Let's diagram out the world from each

00:01:48.644 --> 00:01:50.289
of their points of view.

00:01:50.289 --> 00:01:53.862
First, from Albert's point
of view, A for Albert.

00:01:53.862 --> 00:01:55.997
So, he's been given the month.

00:01:55.997 --> 00:02:00.017
so he's been given either
May, and if it's May,

00:02:00.017 --> 00:02:02.248
he says, well then if
it's May it's gonna be,

00:02:02.248 --> 00:02:04.554
the actual day is gonna be 15, 16, or 7,

00:02:04.554 --> 00:02:09.554
the 15th, 16th or 19th,
15th, 16th or 19th.

00:02:09.633 --> 00:02:13.270
He was told, or he was
told the month June.

00:02:13.270 --> 00:02:16.155
And if he was told the
month of June, then he says,

00:02:16.155 --> 00:02:21.000
well I don't know if the
date is 17 or 18, 17 or 18.

00:02:21.000 --> 00:02:24.871
If he was told the month of
July, if he was told July,

00:02:24.871 --> 00:02:26.738
he'd say, well I don't
know whether the date is

00:02:26.738 --> 00:02:28.995
the 14th or the 16th.

00:02:28.995 --> 00:02:31.475
And if he was told the month August,

00:02:31.475 --> 00:02:32.835
if he was told the month August,

00:02:32.835 --> 00:02:34.050
he'd be thinking to himself,

00:02:34.050 --> 00:02:35.514
well, the date's either gonna be

00:02:35.514 --> 00:02:39.803
the 14th, the 15th, or the 17th.

00:02:39.803 --> 00:02:40.859
Now, let's think about the world

00:02:40.859 --> 00:02:43.135
from Bernard's point of view.

00:02:43.135 --> 00:02:46.752
>From Bernard's point of view,
he's been given the date.

00:02:46.752 --> 00:02:49.271
So, if he's been given the date the 14th,

00:02:49.271 --> 00:02:51.912
if he was told the
14th, then he would say,

00:02:51.912 --> 00:02:54.167
all right, let's see, May's not an option.

00:02:54.167 --> 00:02:56.942
June wouldn't be an option,
July would be an option

00:02:56.942 --> 00:02:59.421
and August would be an option,
'cause there's a July 14.

00:02:59.421 --> 00:03:00.416
and an August 14.

00:03:00.416 --> 00:03:02.427
There's no May 14th or June 14th.

00:03:02.427 --> 00:03:04.256
So he would say, well, I'm not sure.

00:03:04.256 --> 00:03:05.477
If he was told it was the 14th,

00:03:05.477 --> 00:03:09.781
then it's either gonna be July or August.

00:03:09.781 --> 00:03:13.662
If he was told the 15th,
then it would be, let's see,

00:03:13.662 --> 00:03:16.140
May would be an option,
there's a May 15th here,

00:03:16.140 --> 00:03:18.619
or August would be an
option, there's no June 15th

00:03:18.619 --> 00:03:20.205
or July 15th here.

00:03:20.205 --> 00:03:24.145
So, May, May or August.

00:03:24.145 --> 00:03:27.599
If he was told the 16th, let's
see what are the options?

00:03:27.599 --> 00:03:29.651
It could be May,

00:03:29.651 --> 00:03:31.928
there's a May 16th as one of the options.

00:03:31.928 --> 00:03:35.585
There's no June 16th,
there is a July 16th.

00:03:35.585 --> 00:03:37.311
So he'd say maybe it's July.

00:03:37.311 --> 00:03:39.831
And there's no August 16th either.

00:03:39.831 --> 00:03:42.695
If he was told, it's a 17th,

00:03:42.695 --> 00:03:44.361
if he was told it's the 17th,

00:03:44.361 --> 00:03:45.620
he could say, well it's
not gonna be in May,

00:03:45.620 --> 00:03:46.657
there's no May 17th.

00:03:46.657 --> 00:03:49.216
There is a June 17th.

00:03:49.216 --> 00:03:51.309
There is a June 17th.

00:03:51.309 --> 00:03:55.190
There's no July 17th, but
there is an August 17th.

00:03:55.190 --> 00:03:58.909
So, there is an August, there is,

00:03:58.909 --> 00:04:00.371
I'll just write Aug for short.

00:04:00.371 --> 00:04:02.727
There is an August 17th.

00:04:02.727 --> 00:04:05.287
And then, if he was told an
18th, he would say, let's see,

00:04:05.287 --> 00:04:08.233
there's no May 18th, there is a June 18th,

00:04:08.233 --> 00:04:10.529
and I think that's the only
18th that we have here.

00:04:10.529 --> 00:04:13.069
So, if he was told that
the day is the 18th,

00:04:13.069 --> 00:04:17.620
then he would know, he would
know that the month is June.

00:04:17.620 --> 00:04:19.796
And, if he was told that
the day is the 19th,

00:04:19.796 --> 00:04:24.722
then up here, the only
19th here is May 19th,

00:04:24.722 --> 00:04:26.945
May, May 19th.

00:04:26.945 --> 00:04:28.735
So, if Bernard, Bernard's interesting.

00:04:28.735 --> 00:04:31.181
If he was told the 18th or the 19th,

00:04:31.181 --> 00:04:34.269
then he would know what the month is.

00:04:34.269 --> 00:04:35.529
So, with this out of the way,

00:04:35.529 --> 00:04:37.398
and if you're getting
inspired, I encourage you

00:04:37.398 --> 00:04:38.861
to pause the video again and see

00:04:38.861 --> 00:04:40.649
if you could work
through these statements,

00:04:40.649 --> 00:04:43.615
but now let's work through the statements.

00:04:43.615 --> 00:04:46.216
So, Albert says, I don't know
when Cheryl's birthday is,

00:04:46.216 --> 00:04:47.333
which makes sense 'cause no matter

00:04:47.333 --> 00:04:48.613
what month he's been given here,

00:04:48.613 --> 00:04:51.437
there's multiple options
for each of those months,

00:04:51.437 --> 00:04:52.656
so it makes sense that he doesn't know

00:04:52.656 --> 00:04:54.345
when Cheryl's birthday is.

00:04:54.345 --> 00:04:59.345
But, I know, I know that
Bernard does not know too.

00:04:59.503 --> 00:05:01.352
So how could he be so confident

00:05:01.352 --> 00:05:03.932
that Bernard does not know too?

00:05:03.932 --> 00:05:06.228
Well, Albert has the same information

00:05:06.228 --> 00:05:10.555
that we have right over here
and Albert says well look,

00:05:10.555 --> 00:05:15.555
if the date, if Bernard
was given an 18 or a 19,

00:05:16.145 --> 00:05:20.389
then he would know for sure
that Bernard does know the date.

00:05:20.389 --> 00:05:23.111
So, the only way that
he can feel confident,

00:05:23.111 --> 00:05:25.185
the only way that he can
feel confident that Bernard,

00:05:25.185 --> 00:05:26.301
the only way he could say,

00:05:26.301 --> 00:05:30.904
I know that Bernard does not know too,

00:05:30.904 --> 00:05:33.708
means that Albert knows that Bernard

00:05:33.708 --> 00:05:36.267
did not get an 18th or 19th.

00:05:36.267 --> 00:05:39.193
So, another way to
translate this statement is,

00:05:39.193 --> 00:05:43.744
Bernard did not get an 18th or 19th.

00:05:43.744 --> 00:05:47.828
Bernard not get

00:05:47.828 --> 00:05:52.562
18th or 19th.

00:05:52.562 --> 00:05:54.370
Now how can Albert be so confident

00:05:54.370 --> 00:05:57.499
that Bernard did not get an 18th or 19th?

00:05:57.499 --> 00:06:00.851
Well, Albert knows,
Albert knows the month,

00:06:00.851 --> 00:06:02.172
Albert knows the month.

00:06:02.172 --> 00:06:04.549
And if he's so confident that Bernard

00:06:04.549 --> 00:06:06.438
did not get an 18th or 19th,

00:06:06.438 --> 00:06:09.059
that's because he knows
it's not May or June.

00:06:09.059 --> 00:06:12.130
If the month that Albert got
was May, then he would say,

00:06:12.130 --> 00:06:13.796
oh maybe he did get an 19th here,

00:06:13.796 --> 00:06:15.564
in which I can't, couldn't
make the statement.

00:06:15.564 --> 00:06:17.433
Or, if the month was June, well then 18th

00:06:17.433 --> 00:06:20.074
would've been a possibility
for what Bernard got.

00:06:20.074 --> 00:06:21.620
And so, he wouldn't have been able

00:06:21.620 --> 00:06:23.286
to make this statement here.

00:06:23.286 --> 00:06:24.485
But because he says,

00:06:24.485 --> 00:06:28.020
I know that Bernard does not know too,

00:06:28.020 --> 00:06:30.174
the reason why he can say that is because

00:06:30.174 --> 00:06:32.795
he knows it's not May or June.

00:06:32.795 --> 00:06:34.016
So we can rule out,

00:06:34.016 --> 00:06:37.733
we can rule out May or June.

00:06:37.733 --> 00:06:39.377
So we can rule out,

00:06:39.377 --> 00:06:42.811
we can rule out the 18th, the 19th here,

00:06:42.811 --> 00:06:45.838
and we can also rule out May
or June now from Bernard.

00:06:45.838 --> 00:06:47.466
Now from Bernard's point
of view, we can say look,

00:06:47.466 --> 00:06:49.170
Albert was, the only
reason why he was able

00:06:49.170 --> 00:06:51.568
to make that statement
is because Albert knows

00:06:51.568 --> 00:06:54.148
that the month is not May or June.

00:06:54.148 --> 00:06:56.850
So, let me rule out all
of the Mays and Junes.

00:06:56.850 --> 00:06:58.435
So, let me rule out June here.

00:06:58.435 --> 00:07:02.742
Let me rule out, let me
rule out May over here.

00:07:02.742 --> 00:07:05.485
Now I think you see where this is going.

00:07:05.485 --> 00:07:07.252
Now Bernard, and once again, at any point,

00:07:07.252 --> 00:07:11.448
if you get so inspired,
please pause the video

00:07:11.448 --> 00:07:13.520
and continue on with
the deductive reasoning

00:07:13.520 --> 00:07:15.287
that we have started.

00:07:15.287 --> 00:07:17.969
All right, now Bernard says,

00:07:17.969 --> 00:07:22.175
At first, I didn't know
when Cheryl's birthday is.

00:07:22.175 --> 00:07:24.958
Which is fair because it
wasn't the 18th or 19th

00:07:24.958 --> 00:07:26.523
which Albert already knew.

00:07:26.523 --> 00:07:29.024
But now I know.

00:07:29.024 --> 00:07:31.033
So, now I know.

00:07:31.033 --> 00:07:34.304
So before he didn't
know, but now he knows.

00:07:34.304 --> 00:07:37.717
So, if now he knows, what
are the possibilities?

00:07:37.717 --> 00:07:42.187
What are the possibilities going on here?

00:07:42.187 --> 00:07:46.352
So he knows if the date
he was given is the 17th,

00:07:46.352 --> 00:07:47.835
we've been able to, then
he could deduce that,

00:07:47.835 --> 00:07:49.420
hey, the month is now August.

00:07:49.420 --> 00:07:51.136
If the date he was given was the 16th,

00:07:51.136 --> 00:07:53.483
the month that he could
now deduce is July.

00:07:53.483 --> 00:07:55.475
If the date he was given was the 15th,

00:07:55.475 --> 00:07:57.547
he could deduce August.

00:07:57.547 --> 00:07:59.498
But if the date he was given was 14th,

00:07:59.498 --> 00:08:01.652
he still wouldn't know what the month is.

00:08:01.652 --> 00:08:03.540
It could still be July or August.

00:08:03.540 --> 00:08:05.511
So we could rule out the 14th,

00:08:05.511 --> 00:08:08.449
'cause if the date he
was given was the 14th

00:08:08.449 --> 00:08:09.707
he still wouldn't know.

00:08:09.707 --> 00:08:11.454
But he says, now I do know.

00:08:11.454 --> 00:08:13.404
So we could rule out the 14th.

00:08:13.404 --> 00:08:15.924
And once again, we're assuming
Albert is hearing everything.

00:08:15.924 --> 00:08:18.504
Albert and Bernard,
they both could kind of

00:08:18.504 --> 00:08:20.961
draw these diagrams out.

00:08:20.961 --> 00:08:23.682
And so Albert could now say, okay, look,

00:08:23.682 --> 00:08:28.051
if he now knows, then the date
couldn't have been the 14th.

00:08:28.051 --> 00:08:30.407
So then he rules out the 14th.

00:08:30.407 --> 00:08:32.398
He rules out the 14th.

00:08:32.398 --> 00:08:34.816
So the stuff in orange that I just did

00:08:34.816 --> 00:08:36.992
I got from this statement there.

00:08:36.992 --> 00:08:38.026
And then Albert says,

00:08:38.026 --> 00:08:41.419
then I also know when
Cheryl's birthday is.

00:08:41.419 --> 00:08:43.654
So, from Albert's point of view,

00:08:43.654 --> 00:08:45.949
the only way that he now
knows what the birthday is

00:08:45.949 --> 00:08:48.489
if the month he was given was August

00:08:48.489 --> 00:08:50.602
he still has two options there.

00:08:50.602 --> 00:08:53.304
But he says he now knows
when Cheryl's birthday is.

00:08:53.304 --> 00:08:55.925
So it can't be August because
all the deductive reasoning

00:08:55.925 --> 00:08:57.972
we've just now done,
if he was given August

00:08:57.972 --> 00:08:59.191
then he would still not know

00:08:59.191 --> 00:09:01.345
whether it was the 15th or 16th.

00:09:01.345 --> 00:09:03.295
But now that he knows, it must be July.

00:09:03.295 --> 00:09:05.184
Because if he was given the month July,

00:09:05.184 --> 00:09:08.029
there's only one option now for the date.

00:09:08.029 --> 00:09:13.029
So therefore, the date must be July 16th.

00:09:13.647 --> 00:09:15.924
Albert must have been given July,

00:09:15.924 --> 00:09:17.934
Albert must have been given July.

00:09:17.934 --> 00:09:20.108
And Bernard must have been given,

00:09:20.108 --> 00:09:22.322
must have been given the 16th.

00:09:22.322 --> 00:09:25.474
So Cheryl was born on July 16th.

00:09:25.474 --> 00:09:27.347
Lots of fun.