Multiplying and dividing negative numbers

WEBVTT
Kind: captions
Language: en

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Welcome to the presentation
on multiplying and

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dividing negative numbers.

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Let's get started.

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I think you're going to find
that multiplying and dividing

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negative numbers are a lot
easier than it might

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look initially.

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You just have to remember a
couple rules, and I'm going to

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teach probably in the future
like I'm actually going to give

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you more intuition on
why these rules work.

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But first let me just teach
you the basic rules.

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So the basic rules are when you
multiply two negative numbers,

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so let's say I had negative
2 times negative 2.

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First you just look at each
of the numbers as if there

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was no negative sign.

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Well you say well, 2
times 2 that equals 4.

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And it turns out that if you
have a negative times a

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negative, that that
equals a positive.

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So let's write that
first rule down.

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A negative times a negative
equals a positive.

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What if it was negative
2 times positive 2?

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Well in this case, let's
first of all look at the

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two numbers without signs.

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We know that 2 times 2 is 4.

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But here we have a negative
times a positive 2, and it

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turns out that when you
multiply a negative times a

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positive you get a negative.

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So that's another rule.

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Negative times positive
is equal to negative.

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What happens if you have a
positive 2 times a negative 2?

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I think you'll probably guess
this one right, as you can tell

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that these two are pretty much
the same thing by, I believe

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it's the transitive property --
no, no I think it's the

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communicative property.

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I have to remember that.

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But 2 times negative 2, this
also equals negative 4.

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So we have the final rule that
a positive times a negative

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also equals the negative.

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And actually these second
two rules, they're kind

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of the same thing.

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A negative times a positive
is a negative, or a positive

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times a negative is negative.

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You could also say that as when
the signs are different and

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you multiply the two numbers,
you get a negative number.

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And of course, you already know
what happens when you have a

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positive times a positive.

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Well that's just a positive.

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So let's review.

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Negative times a
negative is a positive.

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A negative times a
positive is a negative.

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A positive times a
negative is a negative.

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And positive times each
other equals positive.

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I think that last little bit
completely confused you.

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Maybe I can simplify
it for you.

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What if I just told you if when
you're multiplying and they're

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the same signs that gets
you a positive result.

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And different signs gets
you a negative result.

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So that would be either, let's
say a 1 times 1 is equal to 1,

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or if I said negative 1 times
negative 1 is equal to

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positive 1 as well.

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Or if I said 1 times negative
1 is equal to negative 1, or

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negative 1 times 1 is
equal to negative 1.

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You see how on the bottom two
problems I had two different

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signs, positive 1
and negative 1?

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And the top two problems,
this one right here

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both 1s are positive.

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And this one right here
both 1s are negative.

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So let's do a bunch of problems
now, and hopefully it'll hit

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the point home, and you also
could try to do along the

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practice problems and also give
the hints and give you what

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rules to use, so that
should help you as well.

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So if I said negative 4 times
positive 3, well 4 times

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3 is 12, and we have a
negative and a positive.

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So different signs
mean negative.

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So negative 4 times
3 is a negative 12.

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That makes sense because we're
essentially saying what's

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negative 4 times itself three
times, so it's like negative 4

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plus negative 4 plus negative
4, which is negative 12.

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If you've seen the video on
adding and subtracting negative

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numbers, you probably
should watch first.

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Let's do another one.

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What if I said minus
2 times minus 7.

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And you might want to pause the
video at any time to see if you

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know how to do it and
then restart it to see

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what the answer is.

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Well, 2 times 7 is 14, and we
have the same sign here, so

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it's a positive 14 -- normally
you wouldn't have to write the

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positive but that makes it a
little bit more explicit.

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And what if I had -- let me
think -- 9 times negative 5.

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Well, 9 times 5 is 45.

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And once again, the signs are
different so it's a negative.

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And then finally what if it I
had -- let me think of some

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good numbers -- minus
6 times minus 11.

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Well, 6 times 11 is 66 and
then it's a negative and

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negative, it's a positive.

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Let me give you a
trick problem.

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What is 0 times negative 12?

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Well, you might say that the
signs are different, but

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0 is actually neither
positive nor negative.

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And 0 times anything
is still 0.

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It doesn't matter if the thing
you multiply it by is a

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negative number or
a positive number.

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0 times anything is still 0.

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So let's see if we can apply
these same rules to division.

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It actually turns out that
the same rules apply.

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If I have 9 divided
by negative 3.

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Well, first we say
what's 9 divided by 3?

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Well that's 3.

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And they have different signs,
positive 9, negative 3.

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So different signs
means a negative.

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9 divided by negative 3
is equal to negative 3.

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What is minus 16 divided by 8?

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Well, once again, 16
divided by 8 is 2, but

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the signs are different.

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Negative 16 divided by positive
8, that equals negative 2.

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Remember, different signs will
get you a negative result.

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What is minus 54
divided by minus 6?

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Well, 54 divided by 6 is 9.

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And since both terms, the
divisor and the dividend, are

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both negative -- negative 54
and negative 6 -- it turns out

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that the answer is positive.

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Remember, same signs result in
a positive quotient in this

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example we did before,
it was product.

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Let's do one more.

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Obviously, 0 divided by
anything is still 0.

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That's pretty straightforward.

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And of course, you can't
divide anything by 0

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-- that's undefined.

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Let's do one more.

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What is -- I'm just going to
think of random numbers --

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4 divided by negative 1?

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Well, 4 divided by 1 is 4,
but the signs are different.

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So it's negative 4.

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I hope that helps.

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Now what I want you to do is
actually try as many of these

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multiplying and dividing
negative numbers as you can.

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And you click on hints
and it'll remind you

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of which rule to use.

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In your own time you might want
to actually think about why

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these rules apply and what it
means to multiply a negative

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number times a positive number.

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And even more interesting, what
it means to multiply a negative

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number times a negative number.

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But I think at this point,
hopefully, you are ready to

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start doing some problems.

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Good luck.


Adding/subtracting negative numbers

WEBVTT
Kind: captions
Language: en

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Welcome to the presentation
on adding and subtracting

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negative numbers.

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So let's get started.

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So what is a negative
number, first of all?

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Well, let me draw
a number line.

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Well it's not much of a
line but I think you'll

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get the picture.

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So we're used to the positive
numbers, so if that's 0, you

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have 1, you have 2, you have 3,
you have 4, and you keep going.

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And if I were to say what's 2
plus 2, you'd start at 2 and

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then you'd add 2 and
you'd get to 4.

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I mean most of us
it's second nature.

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But if you actually drew
it on a number line

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you'd say 2 plus 2 is 4.

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And if I asked you what's
2 minus 1 or let's

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say what's 3 minus 2?

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If you start at 3 and
you subtracted 2, you

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would end up at 1.

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That's 2 plus 2 is equal to 4
and 3 minus 2 is equal to 1.

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And this is a joke for you.

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Now what if I were to
say what is 1 minus 3?

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Huh.

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Well, it's the same thing.

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You start at 1 and we're going
to go 1 -- well, now we're

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going to go below 0 --
what happens below 0?

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Well then you start going
to the negative numbers.

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Negative 1, negative 2,
negative 3, and so on.

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So if I start at 1 right here,
so 1 minus 3, so I go 1, 2,

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3, I end up at negative 2.

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So 1 minus 3 is equal
to negative 2.

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This is something that you're
probably already doing

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in your everyday life.

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If I were to tell you that boy,
it's very cold today, it's 1

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degree, but tomorrow it's going
to be 3 degrees colder, you

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might already know intuitively,
well then we're going to be

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at a temperature of
negative 2 degrees.

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So that's all a
negative number means.

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And just remember when a
negative number is big, so like

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negative 50, that's actually
colder than negative 20, right?

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So a negative 50 is actually
even a smaller number than

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negative 20 because it's even
further to the left

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of negative 20.

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That's just something you'll
get an intuitive feel for.

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Sometimes when you start you
feel like oh, 50's a bigger

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number than 20, but it's a
negative 50 as opposed

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to a positive 50.

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So let's do some problems, and
I'm going to keep using the

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number line because I
think it's useful.

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So let's do the
problem 5 minus 12.

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I think you already might
have an intuition of

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what this equals.

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But let me draw a
line, 5 minus 12.

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So let me start with minus 10,
minus 9, minus 8 --I think I'm

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going to run out of space --
minus 7, minus 6, minus 5 -- I

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should have this pre-drawn --
minus 4, minus 3, minus 2,

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minus 1, 0, 1, 2, 3, 4,
I'll put 5 right here.

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5 minus 12.

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So if we start at 5 -- let me
use a different color -- we

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start at 5 right here and we're
going to go to the left 12

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because we're subtracting 12.

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So then we go 1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12.

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Negative 7.

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That's pretty interesting.

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Because it also happens
to be that 12 minus 5

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is equal to positive 7.

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So, I want you to think a
little bit about why that is.

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Why the difference between 12
and 5 is 7, and the difference

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between -- well, I
guess it's either way.

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Because in this situation we're
also saying that the difference

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between 5 and 12 is negative 7,
but the numbers are that far

00:04:04.730 --> 00:04:07.270
apart, but now we're starting
with the lower number.

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I think that last sentence just
completely confused you, but

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we'll keep moving forward.

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We just said 5 minus 12
is equal to minus 7.

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Let's do another one.

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What's negative 3
plus 5 equals what?

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Well, let's use the
same number line.

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Let's go to negative 3 plus 5.

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So we're going to
go to the right 5.

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1, 2, 3, 4, 5.

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It's a 2.

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It equals 2.

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So negative 3 plus
5 is equal to 2.

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That's interesting because 5
minus 3 is also equal to 2.

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Well, it turns out that 5 minus
3 is the same thing, it's just

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another way of writing 5
plus negative 3 or

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negative 3 plus 5.

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A general, easy way to always
do negative numbers is it's

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just like regular subtraction
and addition and subtraction,

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but now when we subtract we
can go to the left below 0.

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Let's do another one.

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So what happens when you get
let's say 2 minus minus 3?

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Well, if you think about how
it should work out I think

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this will make sense.

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But it turns out that the
negative number, the negative

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signs actually cancel out.

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So this is the same thing
as 2 plus plus 3, and

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that just equals 5.

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Another way you could say is --
let's do another one -- what

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is negative 7 minus minus 2?

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Well that's the same thing
as negative 7 plus 2.

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And remember, so we're doing to
start at negative 7 and we're

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going to move two to the right.

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So if we move one to the right
we go to negative 6, and then

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we move two to the right
we get negative 5.

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That makes sense because
negative 7 plus 2, that's the

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same thing as 2 minus 7.

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If it's 2 degrees and it gets 7
degrees colder, it's minus 5.

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Let's do a bunch of these.

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I think the more you do the
more practice you have, and the

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modules explain it pretty well,
probably better than I do.

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So let's just do a
ton of problems.

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So if I said
negative 7 minus 3.

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Well, now we're going
to go three to the

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left of negative 7.

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We're going to get 3 less
than negative 7 so that's

00:06:41.740 --> 00:06:45.430
negative 10, right?

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That makes sense, because if we
had positive 7 plus 3 we're at

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7 to the right of 0 and we're
going to go three more to the

00:06:51.880 --> 00:06:54.045
right of 0 and we
get positive 10.

00:06:54.045 --> 00:06:57.580
So for 7 to the left of 0 and
go three more to the left we're

00:06:57.580 --> 00:06:58.690
going to get negative 10.

00:06:58.690 --> 00:06:59.470
Let's do a bunch more.

00:06:59.470 --> 00:07:02.110
I know I'm probably confusing
you, but practice is what's

00:07:02.110 --> 00:07:04.120
going to really help us.

00:07:04.120 --> 00:07:12.180
So say 3 minus minus 3, well,
these negatives cancel out

00:07:12.180 --> 00:07:14.580
so that just equals 6.

00:07:14.580 --> 00:07:16.990
What's 3 minus 3?

00:07:16.990 --> 00:07:20.410
Well, that's easy
that's just 0.

00:07:20.410 --> 00:07:23.990
What's minus 3 minus 3?

00:07:23.990 --> 00:07:26.420
Well now we're going to get
three less than minus 3,

00:07:26.420 --> 00:07:28.890
well that's minus is 6.

00:07:28.890 --> 00:07:34.040
What's minus 3 minus minus 3?

00:07:34.040 --> 00:07:35.510
Interesting.

00:07:35.510 --> 00:07:40.800
Well, the minuses cancel out
so you get minus 3 plus 3.

00:07:40.800 --> 00:07:43.530
Well, if we start three to the
left of 0 and we move three to

00:07:43.530 --> 00:07:46.070
the right we end up at 0 again.

00:07:46.070 --> 00:07:48.260
So that makes sense, right?

00:07:48.260 --> 00:07:49.370
Let me do that again.

00:07:49.370 --> 00:07:53.290
Minus 3 minus minus 3.

00:07:53.290 --> 00:07:56.400
Anything minus itself
should equal 0, right?

00:07:56.400 --> 00:07:57.960
That's why that equals 0.

00:07:57.960 --> 00:08:00.170
And that's why it makes sense
that those two negatives

00:08:00.170 --> 00:08:02.160
cancel out and that's
the same thing as this.

00:08:05.520 --> 00:08:07.430
Let's do a bunch more.

00:08:07.430 --> 00:08:12.150
Let's do 12 minus 13.

00:08:12.150 --> 00:08:13.620
That's pretty easy.

00:08:13.620 --> 00:08:17.950
Well, 12 minus 12 is 0, so
12 minus 13 is negative 1

00:08:17.950 --> 00:08:20.930
because we're going to
go one the left of 0.

00:08:20.930 --> 00:08:24.670
Let's do 8 minus 5.

00:08:24.670 --> 00:08:27.280
Well, this one is just a
normal problem, that's 3.

00:08:27.280 --> 00:08:29.650
What's 5 minus 8?

00:08:29.650 --> 00:08:33.080
Well, we're going to go all the
way to 0 and then 3 more to the

00:08:33.080 --> 00:08:35.430
left of zero, so it's minus 3.

00:08:35.430 --> 00:08:38.510
I could draw a
number line here.

00:08:38.510 --> 00:08:46.180
If this is 0 this is 5, and now
we're going to go to left 8,

00:08:46.180 --> 00:08:48.310
then we end up and negative 3.

00:08:48.310 --> 00:08:49.410
You could do that
for all of these.

00:08:49.410 --> 00:08:51.780
That actually might
be a good exercise.

00:08:51.780 --> 00:08:54.590
I think this will give you good
introduction and I recommend

00:08:54.590 --> 00:08:57.030
that you just do the modules
because the modules actually,

00:08:57.030 --> 00:08:59.450
especially if you do the hints,
it has a pretty nice graphic

00:08:59.450 --> 00:09:01.350
that's a lot nicer than
anything I could draw

00:09:01.350 --> 00:09:02.870
on this chalkboard.

00:09:02.870 --> 00:09:05.740
So try that out and I'm going
to try to record some more

00:09:05.740 --> 00:09:09.150
modules that hopefully won't
confuse you as badly.

00:09:09.150 --> 00:09:11.210
You could also attend the
seminar on adding and

00:09:11.210 --> 00:09:12.700
subtracting negative numbers.

00:09:12.700 --> 00:09:14.170
I hope you have fun.

00:09:14.170 --> 00:09:15.470
Bye.


systems of equations

WEBVTT
Kind: captions
Language: en

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

00:05:20.200 --> 00:05:23.020
So let's say I had the points--
and I'm going to write them in

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

00:07:07.456 --> 00:07:10.710
Hope this one's harder.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


Algebra: Slope 3

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:02.610
Well, I had to stop that
last presentation because

00:00:02.610 --> 00:00:03.630
I ran out of time.

00:00:03.630 --> 00:00:07.010
But now we can start up
right where we left off.

00:00:07.010 --> 00:00:09.490
So far, we had figured out the
slope and it made sense to

00:00:09.490 --> 00:00:12.350
us because we figured out
the slope was minus 4/3.

00:00:15.390 --> 00:00:16.250
You could look at it two ways.

00:00:16.250 --> 00:00:18.960
When the run is 3, the
rise is negative 4.

00:00:18.960 --> 00:00:22.730
So rise we went down by
negative 4, we ran 3, and

00:00:22.730 --> 00:00:25.310
you can keep doing that on
the line and you'll keep

00:00:25.310 --> 00:00:26.340
ending up on the line.

00:00:26.340 --> 00:00:30.330
You could've also gone down
8 and then moved over 6.

00:00:30.330 --> 00:00:32.250
Or you could move over
6 and go down 8.

00:00:32.250 --> 00:00:34.640
If you went down 8 you
would've showed up here.

00:00:34.640 --> 00:00:36.660
Then you would've
come down back here.

00:00:36.660 --> 00:00:40.880
Whatever the slope is, if you
change the y by the rise, and

00:00:40.880 --> 00:00:43.010
you change the run by the
x, you should end up

00:00:43.010 --> 00:00:44.360
on the line again.

00:00:44.360 --> 00:00:47.420
I know I'm doing this over
and over again, but I think

00:00:47.420 --> 00:00:48.730
it'll slowly sink in.

00:00:48.730 --> 00:00:52.030
But anyway, where we left off
we were going to solve for b so

00:00:52.030 --> 00:00:55.090
that we could figure out the
final equation for this line.

00:00:55.090 --> 00:00:57.780
Well, just like we did the
last time, we just have to

00:00:57.780 --> 00:01:01.240
substitute a y or an x that
we know works for this line.

00:01:01.240 --> 00:01:03.100
We know either of these
coordinates will work,

00:01:03.100 --> 00:01:04.600
and then we solve for b.

00:01:04.600 --> 00:01:07.500
Let's try this first one:
2 comma negative 3.

00:01:07.500 --> 00:01:10.585
So y is negative 3.

00:01:10.585 --> 00:01:13.070
Don't get confused
between x and y.

00:01:13.070 --> 00:01:19.980
y is negative 3 equals minus
4/3 times x, which is 2.

00:01:19.980 --> 00:01:21.570
Plus b.

00:01:21.570 --> 00:01:29.600
And then we get negative 3
equals negative 8/3 plus b.

00:01:29.600 --> 00:01:35.950
And then we get b is equal to--
well, 3 is equal to minus 9/3.

00:01:35.950 --> 00:01:38.130
And then I'm just going to
put this on the other side.

00:01:38.130 --> 00:01:41.040
So that's plus 8/3.

00:01:41.040 --> 00:01:45.270
So we get minus 1/3
is the y-intercept.

00:01:45.270 --> 00:01:51.740
So the equation of this line is
y is equal to minus 4/3 x and

00:01:51.740 --> 00:01:54.430
then this y-intercept
is minus 1/3.

00:01:57.420 --> 00:02:00.110
I know this is extremely messy,
but the equation of this line

00:02:00.110 --> 00:02:04.150
once again, is y equals
minus 4/3 x minus 1/3.

00:02:04.150 --> 00:02:07.920
Now let's look at the graph
and see if that makes sense.

00:02:07.920 --> 00:02:08.870
Well, let's see.

00:02:08.870 --> 00:02:10.810
The y-intercept is right here.

00:02:10.810 --> 00:02:13.070
That's where you intersect
the y-axis at the point.

00:02:13.070 --> 00:02:15.320
And we would know exactly
what that point is.

00:02:15.320 --> 00:02:19.700
It's 0 comma negative 1/3.

00:02:19.700 --> 00:02:21.585
That's the y-intercept,
and it makes sense.

00:02:21.585 --> 00:02:23.770
And even when you look at an
equation it's pretty obvious

00:02:23.770 --> 00:02:25.780
that this is going to be the
y-intercept because when x

00:02:25.780 --> 00:02:27.530
equals 0 this term
gets crossed out.

00:02:27.530 --> 00:02:30.090
Because 0 times negative 4/3.

00:02:30.090 --> 00:02:32.740
And then y would
equal negative 1/3.

00:02:32.740 --> 00:02:37.130
Let's do one more
just to bore you.

00:02:37.130 --> 00:02:40.790
And because my wife is a
resident and she works 30

00:02:40.790 --> 00:02:43.150
hours of time and I have
nothing better to do.

00:02:43.150 --> 00:02:44.280
All right.

00:02:44.280 --> 00:02:47.060
Let me put that
graph back there.

00:02:47.060 --> 00:02:49.000
I joke, but you don't
realize that it's true.

00:02:52.140 --> 00:02:53.810
Let's put the graph back
there and I'm going to do

00:02:53.810 --> 00:02:55.770
another random points.

00:02:55.770 --> 00:02:57.570
I'm going to go a little
faster this time.

00:02:57.570 --> 00:03:05.560
So let's say I had negative 8
comma 5 and I had-- let me

00:03:05.560 --> 00:03:11.840
think of a good one-- 2
comma-- I'm just going

00:03:11.840 --> 00:03:14.140
to make up a number.

00:03:14.140 --> 00:03:19.790
2 comma, let's say 0.

00:03:19.790 --> 00:03:20.800
That's interesting.

00:03:20.800 --> 00:03:21.560
2 comma 0.

00:03:21.560 --> 00:03:23.440
So let's graph
negative 8 comma 5.

00:03:23.440 --> 00:03:27.670
1, 2, 3, 4, 5, 6, 7, 8.

00:03:27.670 --> 00:03:31.070
1, 2, 3, 4, 5.

00:03:31.070 --> 00:03:35.240
Right there is minus 8 comma 5.

00:03:35.240 --> 00:03:39.270
And then 1, 2, and
then 0 is right here.

00:03:39.270 --> 00:03:41.940
So that's 2 comma 0.

00:03:41.940 --> 00:03:43.415
And now let me draw the line.

00:03:47.180 --> 00:03:47.740
Oh my God.

00:03:47.740 --> 00:03:48.770
I thought I was using
the line tool.

00:03:48.770 --> 00:03:50.350
That's horrendous.

00:03:50.350 --> 00:03:51.210
I wish there was an undo tool.

00:03:53.940 --> 00:03:55.160
That was horrendous.

00:03:55.160 --> 00:03:57.300
That was unacceptable.

00:03:57.300 --> 00:03:59.452
That's a little bit better.

00:03:59.452 --> 00:04:01.460
Just so you know that the
line doesn't end there.

00:04:01.460 --> 00:04:05.190
Lines go on forever and ever
in the coordinate axis.

00:04:05.190 --> 00:04:06.480
There you go.

00:04:06.480 --> 00:04:07.390
OK.

00:04:07.390 --> 00:04:09.770
So let's figure out the
slope of this line.

00:04:09.770 --> 00:04:13.420
Well, change in y
over change in x.

00:04:13.420 --> 00:04:14.785
I'm using the line tool again.

00:04:14.785 --> 00:04:15.995
I'm a spaz today.

00:04:19.090 --> 00:04:20.940
Let's take this as
a starting point.

00:04:20.940 --> 00:04:23.140
We'll get 5 minus 0.

00:04:23.140 --> 00:04:25.430
y sub 1 minus y sub 2.

00:04:25.430 --> 00:04:29.770
Over x sub 1-- minus 8 minus 2.

00:04:29.770 --> 00:04:33.650
So it equals 5 over minus 10.

00:04:33.650 --> 00:04:35.730
And that equals minus 1/2.

00:04:35.730 --> 00:04:39.372
So that means for every 2
we go over, we go down 1.

00:04:39.372 --> 00:04:43.150
Well, for every 1 we go
down, we go over 2,

00:04:43.150 --> 00:04:43.800
which makes sense.

00:04:43.800 --> 00:04:47.410
If we go down 2,
we'll go over 4.

00:04:47.410 --> 00:04:49.743
Because 2/4 is the
same thing as 1/2.

00:04:49.743 --> 00:04:51.330
I hope that makes sense to you.

00:04:51.330 --> 00:04:53.630
I know this is a
downwards sloping line.

00:04:53.630 --> 00:04:54.600
So that was fast.

00:04:54.600 --> 00:04:57.230
So we know that the equation
of the line so far is y is

00:04:57.230 --> 00:05:02.170
equal to minus 1/2 x plus b.

00:05:02.170 --> 00:05:03.520
Now we just solve for b.

00:05:03.520 --> 00:05:06.100
Let's substitute some
numbers in here.

00:05:06.100 --> 00:05:06.840
Well, let's use this one.

00:05:06.840 --> 00:05:08.610
This is interesting: 2 comma 0.

00:05:08.610 --> 00:05:10.620
So y is 0.

00:05:10.620 --> 00:05:15.280
Equals minus 1/2
times 2 plus b.

00:05:15.280 --> 00:05:19.130
Well, 0 is equal to and
this is minus 1 plus b.

00:05:19.130 --> 00:05:21.180
And we get b equals 1.

00:05:21.180 --> 00:05:23.050
This is a pretty easy problem.

00:05:23.050 --> 00:05:29.320
So now we get y is equal
to minus 1/2 x plus 1.

00:05:32.150 --> 00:05:35.110
OK, now let's see if this
actually looks right

00:05:35.110 --> 00:05:35.890
on our problem.

00:05:35.890 --> 00:05:37.210
Well, this is telling us
that the y-intercept

00:05:37.210 --> 00:05:39.310
is at the point 0, 1.

00:05:39.310 --> 00:05:41.030
0, 1 is right here.

00:05:41.030 --> 00:05:44.960
And our algebra
confirms our drawing.

00:05:44.960 --> 00:05:47.320
This is the y-intercept
and we see that the

00:05:47.320 --> 00:05:48.990
slope is negative 1/2.

00:05:48.990 --> 00:05:51.590
It makes sense because it's
a downward sloping line,

00:05:51.590 --> 00:05:52.726
but it's not too steep.

00:05:55.350 --> 00:05:58.340
For every 1 that we go
down, we go over 2.

00:05:58.340 --> 00:06:00.200
So that's negative 1/2.

00:06:00.200 --> 00:06:05.580
Or you could say for every 2
we run, we rise negative 1.

00:06:05.580 --> 00:06:07.810
Either one, we end
up on a line again.

00:06:07.810 --> 00:06:09.280
So I hope that helps.

00:06:09.280 --> 00:06:13.130
I think you are definitely
ready to do a lot of these

00:06:13.130 --> 00:06:15.450
slope problems that we
have on the Khan Academy.

00:06:15.450 --> 00:06:17.690
So I hope you have fun and if
you any questions you can

00:06:17.690 --> 00:06:20.410
attend one of the seminars.

00:06:20.410 --> 00:06:22.330
Have fun.


Algebra: Slope 2

WEBVTT
Kind: captions
Language: en

00:00:01.650 --> 00:00:02.030
Hello.

00:00:02.030 --> 00:00:04.490
We're now going to do some more
slope, and then maybe some

00:00:04.490 --> 00:00:06.550
y-intercept problems as well.

00:00:06.550 --> 00:00:08.660
Let's get started.

00:00:08.660 --> 00:00:11.090
So let me make up a problem.

00:00:11.090 --> 00:00:17.330
Let's say we have
the points 2, 5.

00:00:17.330 --> 00:00:24.790
The other point, let's make
that negative 3, negative 3.

00:00:24.790 --> 00:00:27.410
Well, first let's just
graph those two points.

00:00:27.410 --> 00:00:29.760
I'm going to graph
them in yellow.

00:00:29.760 --> 00:00:31.280
So 2, 5.

00:00:31.280 --> 00:00:33.920
Let's see that's one two.

00:00:33.920 --> 00:00:36.720
One, two, three, four, five.

00:00:36.720 --> 00:00:40.320
So 2, 5 is going to
be right over there.

00:00:47.300 --> 00:00:48.310
OK.

00:00:48.310 --> 00:00:52.080
And then let me graph
negative 3, negative 3.

00:00:52.080 --> 00:00:54.520
So it's one, two, three.

00:00:54.520 --> 00:00:56.260
One, two, three.

00:00:56.260 --> 00:00:59.820
So negative 3, negative
3 is right over there.

00:00:59.820 --> 00:01:03.750
And then now let me draw a
line that will connect them.

00:01:10.920 --> 00:01:11.760
That's my new technique.

00:01:11.760 --> 00:01:14.720
I draw it in two pieces.

00:01:14.720 --> 00:01:15.800
I think that's good enough.

00:01:15.800 --> 00:01:16.170
OK.

00:01:16.170 --> 00:01:18.963
So let's see if we can at least
first figure out the slope of

00:01:18.963 --> 00:01:20.640
the line, and then if we have
time we'll try to figure

00:01:20.640 --> 00:01:21.470
out the y-intercept.

00:01:21.470 --> 00:01:23.990
And then we'll know the whole
equation for the line.

00:01:23.990 --> 00:01:28.570
Let me pick a slightly thinner
color, and we'll get started.

00:01:28.570 --> 00:01:32.060
So the slope, if you saw
the last module that just

00:01:32.060 --> 00:01:34.040
introduces how we calculate
the slope, that's

00:01:34.040 --> 00:01:35.430
just rise over run.

00:01:35.430 --> 00:01:41.730
Or, change in y
over change in x.

00:01:41.730 --> 00:01:44.840
This is y.

00:01:44.840 --> 00:01:46.630
So let's just do
that real fast.

00:01:46.630 --> 00:01:48.490
So let's take this as
our starting point.

00:01:48.490 --> 00:01:51.430
So change in y could be
5-- remember, y is the

00:01:51.430 --> 00:01:57.930
second coordinate--
5 minus negative 3.

00:01:57.930 --> 00:01:59.860
And that's this one.

00:01:59.860 --> 00:02:05.210
Over-- now you do the change
in x-- 2 minus, this

00:02:05.210 --> 00:02:07.690
is also negative 3.

00:02:07.690 --> 00:02:11.360
Well 5 minus negative 3,
that's 5 plus plus 3.

00:02:11.360 --> 00:02:13.190
So that equals 8.

00:02:13.190 --> 00:02:15.300
And then 2 minus negative 3.

00:02:15.300 --> 00:02:19.310
Once again that's 2 plus
plus 3, so that equals 5.

00:02:19.310 --> 00:02:21.530
So we figured out the
slope of this equation.

00:02:21.530 --> 00:02:23.060
It's 8/5.

00:02:23.060 --> 00:02:24.640
And let's see if
that makes sense.

00:02:24.640 --> 00:02:27.280
Let's figure out what the
rise and the run is.

00:02:27.280 --> 00:02:31.160
If we were to start at this
point right here, let's see how

00:02:31.160 --> 00:02:34.750
much we have to rise to get to
the same y-coordinate

00:02:34.750 --> 00:02:35.940
as the other point.

00:02:35.940 --> 00:02:37.400
So let's see.

00:02:37.400 --> 00:02:41.480
We're here, and the
other point is up here.

00:02:41.480 --> 00:02:47.840
So let's figure out
what this distance is.

00:02:47.840 --> 00:02:51.830
Actually, now is a good
time to use the fat.

00:02:51.830 --> 00:02:54.110
Oh man, I have a shaky hand.

00:02:54.110 --> 00:02:54.530
OK.

00:02:54.530 --> 00:02:55.780
Let's figure out what
that distance is.

00:02:55.780 --> 00:02:59.620
That distance is delta y,
which is change in y.

00:02:59.620 --> 00:03:05.420
So it's one, two, three, four,
five, six, seven, eight.

00:03:05.420 --> 00:03:07.170
That equals 8.

00:03:07.170 --> 00:03:08.710
And that makes sense, because
if you think about it

00:03:08.710 --> 00:03:09.800
what did we just do?

00:03:09.800 --> 00:03:15.130
We just took y equals 5,
which was up here, minus

00:03:15.130 --> 00:03:17.220
y equals negative 3.

00:03:17.220 --> 00:03:19.440
And so obviously we just
calculated that distance

00:03:19.440 --> 00:03:23.240
just by looking at the two
coordinates 5 minus negative 3.

00:03:23.240 --> 00:03:25.930
When you do this calculation
it actually gives you

00:03:25.930 --> 00:03:27.590
this distance right here.

00:03:27.590 --> 00:03:29.990
So that's how we figure out
how much we have to rise.

00:03:29.990 --> 00:03:32.550
So now let's do the run.

00:03:32.550 --> 00:03:35.170
Well the run, to go from
this point to the other

00:03:35.170 --> 00:03:37.290
point, we went this far.

00:03:40.950 --> 00:03:43.300
And let's count
how far that is.

00:03:43.300 --> 00:03:47.880
Well, it's one, two,
three, four, five units.

00:03:47.880 --> 00:03:52.126
So we can say delta
x is equal to 5.

00:03:52.126 --> 00:03:55.120
And that's exactly what we did.
delta y over delta x was equal

00:03:55.120 --> 00:03:58.820
to 8/5, or rise over
run is equal to 8/5.

00:03:58.820 --> 00:04:01.540
And it would have been the same
thing if we calculated run here

00:04:01.540 --> 00:04:03.300
or if we calculated rise here.

00:04:03.300 --> 00:04:05.430
But it's the same thing.

00:04:05.430 --> 00:04:07.650
Hope that's making
sense to you.

00:04:07.650 --> 00:04:11.020
And I hope that also makes
sense that if the rise for a

00:04:11.020 --> 00:04:13.765
given run becomes more, then
the slope of the line is going

00:04:13.765 --> 00:04:16.650
to become steeper and it'll
become a bigger number.

00:04:16.650 --> 00:04:18.380
So let's see what we
have so far for the

00:04:18.380 --> 00:04:19.490
equation of this line.

00:04:19.490 --> 00:04:23.680
So so far we know the equation
of this line is equal to, y is

00:04:23.680 --> 00:04:31.710
equal to the slope
8/5 x plus b.

00:04:31.710 --> 00:04:32.740
So we're almost done.

00:04:32.740 --> 00:04:39.390
We just have to figure
out this b right here.

00:04:39.390 --> 00:04:41.200
Now that b, just so
you remember, that's

00:04:41.200 --> 00:04:42.400
the y-intercept.

00:04:42.400 --> 00:04:45.130
And that's where we
intersect the y-axis.

00:04:45.130 --> 00:04:48.200
And since this graph is pretty
neat, we can actually inspect

00:04:48.200 --> 00:04:50.530
it and see that, well, it looks
like we're intersecting

00:04:50.530 --> 00:04:51.450
the y-axis at 2.

00:04:51.450 --> 00:04:54.460
So my guess is we're going
to come up with b equals 2.

00:04:54.460 --> 00:04:56.170
But let's solve it, just in
case we didn't have this

00:04:56.170 --> 00:04:58.020
neatly drawn graph here.

00:04:58.020 --> 00:05:00.200
So how can we solve for b?

00:05:00.200 --> 00:05:02.860
Well, we can substitute
values that we know

00:05:02.860 --> 00:05:04.310
that work for x and y.

00:05:04.310 --> 00:05:07.280
Well either of these points
are on that line, so we can

00:05:07.280 --> 00:05:09.340
substitute them in for x and y.

00:05:09.340 --> 00:05:12.310
So let's use the first one.

00:05:12.310 --> 00:05:12.730
OK.

00:05:12.730 --> 00:05:21.915
So the y we get 5, will
equal 8/5 times x.

00:05:21.915 --> 00:05:24.050
Well, x there is 2.

00:05:24.050 --> 00:05:27.670
Times 2 plus b.

00:05:27.670 --> 00:05:37.080
Well, now we just get 5
is equal to 16/5 plus b.

00:05:37.080 --> 00:05:44.590
And then we get b equals--
well 5 is 25/5, right?

00:05:44.590 --> 00:05:53.080
5 is 25/5 minus
16/5 equals 9/5.

00:05:53.080 --> 00:05:53.320
All right.

00:05:53.320 --> 00:05:55.140
See, so I was actually wrong.

00:05:55.140 --> 00:05:58.150
When I looked at this graph
I said, oh that looks like

00:05:58.150 --> 00:06:00.930
almost 2, so yeah it's
probably going to be 2.

00:06:00.930 --> 00:06:03.780
But when we actually did it
using algebra, when we did it

00:06:03.780 --> 00:06:07.010
analytically, we actually
saw that b is equal to 9/5.

00:06:07.010 --> 00:06:08.340
So it's almost 2.

00:06:08.340 --> 00:06:11.480
9/5 is 1 and 4/5, or 1.8.

00:06:11.480 --> 00:06:13.430
So that's almost 2,
but it actually turns

00:06:13.430 --> 00:06:14.240
out that it's not.

00:06:14.240 --> 00:06:15.610
It's at 1.8.

00:06:15.610 --> 00:06:16.870
And I can write it
down as a decimal.

00:06:16.870 --> 00:06:18.290
1.8.

00:06:18.290 --> 00:06:20.330
So the final equation for the
line, I'm going to try to

00:06:20.330 --> 00:06:26.210
squeeze it in at the bottom of
this page, it's going to be y

00:06:26.210 --> 00:06:29.290
is equal to-- well,
we know the slope.

00:06:29.290 --> 00:06:33.560
8/5 x.

00:06:33.560 --> 00:06:36.010
Now we just add
the y-intercept.

00:06:36.010 --> 00:06:38.860
Plus 9/5.

00:06:38.860 --> 00:06:39.260
There.

00:06:39.260 --> 00:06:40.740
We solved it.

00:06:40.740 --> 00:06:41.390
Let's do another one.

00:06:41.390 --> 00:06:43.200
And so-- that's 9/5.

00:06:43.200 --> 00:06:43.840
I don't want to be
too repetitive.

00:06:43.840 --> 00:06:46.400
Let's do another problem.

00:06:46.400 --> 00:06:48.920
Time to do another problem,
and let me put that

00:06:48.920 --> 00:06:50.270
graph back there again.

00:06:53.080 --> 00:06:53.950
There you go.

00:06:53.950 --> 00:06:54.420
All right.

00:06:54.420 --> 00:06:57.180
I'm going to think of two
random numbers again.

00:06:57.180 --> 00:06:59.410
Let me try to do this fast,
because YouTube puts a

00:06:59.410 --> 00:07:01.610
10 minute limit on me.

00:07:01.610 --> 00:07:07.890
So let's say I had the
points 2, negative 3.

00:07:07.890 --> 00:07:13.890
And I had the point
negative 4, 5.

00:07:13.890 --> 00:07:15.240
So 2, negative 3.

00:07:15.240 --> 00:07:19.270
Let's plot that
sucker real fast.

00:07:19.270 --> 00:07:21.810
So x is 2, so it's here.

00:07:21.810 --> 00:07:22.690
And the negative 3.

00:07:22.690 --> 00:07:24.120
One, two, three.

00:07:24.120 --> 00:07:26.510
So 2, negative 3 is there.

00:07:26.510 --> 00:07:28.080
And negative 4, 5.

00:07:28.080 --> 00:07:31.150
So that's one,
two, three, four.

00:07:31.150 --> 00:07:33.180
One, two, three, four, five.

00:07:33.180 --> 00:07:35.180
I have to count like
this because this

00:07:35.180 --> 00:07:36.560
graph is unlabeled.

00:07:36.560 --> 00:07:38.720
But if we actually were to draw
in the coordinates you would

00:07:38.720 --> 00:07:43.750
that see this is 5, and this
is negative 4, and so on.

00:07:43.750 --> 00:07:46.460
And this is 2, and
this is negative 3.

00:07:46.460 --> 00:07:50.770
And now let's just draw a line.

00:07:50.770 --> 00:07:52.875
Let's draw it right there
with my shaky hand.

00:07:57.150 --> 00:07:57.470
OK.

00:07:57.470 --> 00:07:59.030
There you go.

00:07:59.030 --> 00:07:59.850
Good line.

00:07:59.850 --> 00:08:03.470
And another good line.

00:08:03.470 --> 00:08:03.960
All right.

00:08:03.960 --> 00:08:05.910
So first we need to
figure out the slope.

00:08:05.910 --> 00:08:09.240
Well we could just do
that doing the algebra.

00:08:09.240 --> 00:08:13.760
So its slope is just delta--
I'm still using the line tool

00:08:13.760 --> 00:08:18.270
again-- delta y over delta x.

00:08:18.270 --> 00:08:20.730
Change in y over change in x.

00:08:20.730 --> 00:08:22.620
Let's take this y as
the first point now.

00:08:22.620 --> 00:08:28.460
So we'll say 5 minus
this y, negative 3.

00:08:30.960 --> 00:08:33.930
Over-- now since we used the
5 first we have to use the

00:08:33.930 --> 00:08:35.390
negative 4 first as well.

00:08:35.390 --> 00:08:39.360
Negative 4 minus 2.

00:08:39.360 --> 00:08:42.990
Well 5 minus negative
3, that equals 8.

00:08:42.990 --> 00:08:47.305
And negative 4 minus 2, well
that equals negative 6.

00:08:47.305 --> 00:08:52.080
And negative 8/6, well
that equals-- they're

00:08:52.080 --> 00:08:53.210
both divisible by 2.

00:08:53.210 --> 00:08:55.070
So that equals minus 4/3.

00:08:57.690 --> 00:08:59.740
And let's see, does that
make sense as the slope?

00:08:59.740 --> 00:09:03.310
Well, if we were to go down
four from this point.

00:09:03.310 --> 00:09:06.660
So if the rise was negative
4-- one, two, three, four.

00:09:06.660 --> 00:09:09.630
So if we go down--
woops, I'm using white.

00:09:09.630 --> 00:09:13.060
So that's why you can't see it.

00:09:13.060 --> 00:09:18.940
We go down by four here, and
then we go to the right

00:09:18.940 --> 00:09:20.350
three, positive 3.

00:09:20.350 --> 00:09:21.580
We still end up on the line.

00:09:21.580 --> 00:09:23.190
So it works.

00:09:23.190 --> 00:09:23.950
Looks good to me.

00:09:23.950 --> 00:09:27.600
Let's see if I can solve the
y-intercept in 30 seconds.

00:09:27.600 --> 00:09:29.170
Otherwise, I'll start
it on the next module.

00:09:29.170 --> 00:09:35.970
So we get y is equal to
minus 4/3 x, plus b.

00:09:35.970 --> 00:09:38.160
And actually what we'll do is
we'll leave off here, and I'm

00:09:38.160 --> 00:09:40.540
going to solve for b-- and you
could try to do it on your

00:09:40.540 --> 00:09:43.990
own-- in the next installment
of this presentation.


Algebra: Slope

WEBVTT
Kind: captions
Language: en

00:00:01.050 --> 00:00:04.020
Welcome to the presentation
on figuring out the slope.

00:00:04.020 --> 00:00:05.320
Let's get started.

00:00:05.320 --> 00:00:07.780
So, let's say I
have two points.

00:00:07.780 --> 00:00:10.270
And, as we learned in previous
presentations, that all

00:00:10.270 --> 00:00:12.080
you need to define a
line is two points.

00:00:12.080 --> 00:00:15.200
And I think if you think about
that, that makes sense.

00:00:15.200 --> 00:00:16.200
Let's say we have two points.

00:00:16.200 --> 00:00:19.450
And let me write down the two
points we're going to have.

00:00:19.450 --> 00:00:25.020
Let's say one point is,
why isn't it writing.

00:00:25.020 --> 00:00:26.846
Sometimes this thing
acts a little finicky.

00:00:26.846 --> 00:00:30.080
Oh, that's because I was
trying to write in black.

00:00:30.080 --> 00:00:38.160
Let's say that one point
is, negative 1, 3.

00:00:38.160 --> 00:00:38.660
So, let's see.

00:00:38.660 --> 00:00:40.500
Where do we graph that?

00:00:40.500 --> 00:00:42.470
So, this is 0, 0.

00:00:42.470 --> 00:00:47.490
We go negative 1, this
is negative 1 here.

00:00:47.490 --> 00:00:48.970
And then we're
going to go 3 up.

00:00:48.970 --> 00:00:50.530
1, 2, 3.

00:00:50.530 --> 00:00:52.560
Because this is 3 right here.

00:00:52.560 --> 00:00:56.580
So, negative 1, 3 is going
to be right over there.

00:00:56.580 --> 00:00:58.310
OK, so that's the first point.

00:00:58.310 --> 00:01:01.210
The second point, I'm going to
do it in a different color.

00:01:01.210 --> 00:01:06.890
The second point is 2, 1.

00:01:06.890 --> 00:01:08.290
Let's see where we
would put that.

00:01:08.290 --> 00:01:11.080
We would count 1, 2.

00:01:11.080 --> 00:01:13.610
This is 2, 1.

00:01:13.610 --> 00:01:14.670
Because this is 1.

00:01:14.670 --> 00:01:17.530
So the point's
going to be here.

00:01:17.530 --> 00:01:19.710
So we've graphed
our two points.

00:01:19.710 --> 00:01:23.600
And now the line that connects
them, it's going to look

00:01:23.600 --> 00:01:25.450
something thing like this.

00:01:25.450 --> 00:01:27.060
And I hope I can draw it well.

00:01:36.300 --> 00:01:39.078
Through that point.

00:01:39.078 --> 00:01:40.430
Like that.

00:01:40.430 --> 00:01:41.070
Then I'm going to do it.

00:01:41.070 --> 00:01:43.170
And then I'm just going to try
to continue the line from here.

00:01:43.170 --> 00:01:47.400
That might be the
best technique.

00:01:47.400 --> 00:01:48.293
Something like that.

00:01:57.680 --> 00:01:58.570
So, let's look at that line.

00:01:58.570 --> 00:02:02.270
So what we want to do in this
presentation is, figure out

00:02:02.270 --> 00:02:04.000
the slope of that line.

00:02:04.000 --> 00:02:06.225
So let's write out a
couple of things that

00:02:06.225 --> 00:02:07.120
I think will help you.

00:02:07.120 --> 00:02:09.410
So, there's a couple
ways to view slope.

00:02:09.410 --> 00:02:11.930
I think, intuitively, you
know that the slope is the

00:02:11.930 --> 00:02:13.170
inclination of this line.

00:02:13.170 --> 00:02:14.470
And we can already
see that this is a

00:02:14.470 --> 00:02:15.880
downward sloping line.

00:02:15.880 --> 00:02:18.820
Because it comes from the top
left to the bottom right.

00:02:18.820 --> 00:02:20.590
So it's going to be a
negative number, the slope.

00:02:20.590 --> 00:02:22.200
So you know that immediately.

00:02:22.200 --> 00:02:24.770
And we'll have -- what we're
going to do is figure out how

00:02:24.770 --> 00:02:26.750
to figure out the slope.

00:02:26.750 --> 00:02:32.280
So the slope, let me write this
down, slope and -- oftentimes

00:02:32.280 --> 00:02:35.700
they'll use the variable m, for
slope, I have no idea why.

00:02:35.700 --> 00:02:39.060
Because m, clearly, does
not stand for slope.

00:02:39.060 --> 00:02:41.370
That is equal to -- there's
a couple of things

00:02:41.370 --> 00:02:42.300
you might hear.

00:02:42.300 --> 00:02:45.270
Change in y over change in x.

00:02:45.270 --> 00:02:48.920
That triangle, which is
pronounced, delta just a Greek

00:02:48.920 --> 00:02:50.560
letter, that means change.

00:02:50.560 --> 00:02:52.560
The change in y
over change in x.

00:02:52.560 --> 00:02:57.970
And that also is equal
to rise over run.

00:02:57.970 --> 00:02:59.800
And I'm going to explain what
all of this means in a second.

00:02:59.800 --> 00:03:01.650
So let's start at one
of these points.

00:03:01.650 --> 00:03:05.130
Let's start at this green
point, negative 1, 3.

00:03:05.130 --> 00:03:09.760
So how much do we have to rise
and how much do we have to run

00:03:09.760 --> 00:03:12.710
to get to the second
point, 2, 1?

00:03:12.710 --> 00:03:14.130
So let's do the rise first.

00:03:14.130 --> 00:03:21.640
Well, we have to go minus
2, so that's the rise.

00:03:21.640 --> 00:03:25.400
So the rise is
equal to minus 2.

00:03:25.400 --> 00:03:28.120
Because we have to go down
2 to get to the same y

00:03:28.120 --> 00:03:29.290
as this yellow point.

00:03:29.290 --> 00:03:33.450
And then we have to
run right there.

00:03:33.450 --> 00:03:36.830
We have to run plus 3.

00:03:36.830 --> 00:03:42.140
So rise divided by run is
equal to minus 2 over 3.

00:03:42.140 --> 00:03:44.340
Well, how would we do that if
we didn't have this nice graph

00:03:44.340 --> 00:03:46.790
here to actually draw on?

00:03:46.790 --> 00:03:51.400
Well, what we can do is, we
can say let's take this

00:03:51.400 --> 00:03:53.690
as a starting point.

00:03:53.690 --> 00:04:00.360
Change in y, change in y, over
change in x, is equal to

00:04:00.360 --> 00:04:04.110
we take the first y
point, which is 3.

00:04:04.110 --> 00:04:06.010
And we subtract the
second y point, which

00:04:06.010 --> 00:04:07.700
is 1, you see that?

00:04:07.700 --> 00:04:10.590
We just took 3 minus 1.

00:04:10.590 --> 00:04:16.620
So that's the change in y over,
and we take the first x point.

00:04:16.620 --> 00:04:22.250
Negative 1, minus the
second x point, minus

00:04:22.250 --> 00:04:25.345
2, so 3 minus 1 is 2.

00:04:25.345 --> 00:04:30.590
And negative 1 minus 2
is equal to minus 3.

00:04:30.590 --> 00:04:30.990
So, same thing.

00:04:30.990 --> 00:04:33.880
We got minus 2 over 3.

00:04:33.880 --> 00:04:35.070
Now we could have done
it the other way.

00:04:35.070 --> 00:04:36.850
And I'm running out
of space here.

00:04:36.850 --> 00:04:41.380
But we could've made
this the first point.

00:04:41.380 --> 00:04:43.770
If we made that the first
point, then the change in y

00:04:43.770 --> 00:04:47.100
would have been -- I want to
make it really cluttered,

00:04:47.100 --> 00:04:48.460
so to confuse you.

00:04:48.460 --> 00:04:50.300
Change in y would be this y.

00:04:50.300 --> 00:04:57.380
1 minus 3 over change in x,
would be 2, minus minus 1.

00:04:57.380 --> 00:05:00.960
Well, 1 minus 3 is minus 2.

00:05:00.960 --> 00:05:03.330
And 2 minus negative 1 is 3.

00:05:03.330 --> 00:05:06.620
So, once again, we got minus
2/3, So it doesn't matter which

00:05:06.620 --> 00:05:10.030
point we start with, as long
as, if we use the y in this

00:05:10.030 --> 00:05:12.310
coordinate first, then we have
to use the x in that

00:05:12.310 --> 00:05:13.480
coordinate first.

00:05:13.480 --> 00:05:14.900
Let's do some more problems.

00:05:14.900 --> 00:05:17.280
Actually, I'm going to do a
couple just so you see the

00:05:17.280 --> 00:05:19.565
algebra without even
graphing it first.

00:05:22.450 --> 00:05:24.560
So, let's say I wanted to
figure out the slope between

00:05:24.560 --> 00:05:33.100
the points 5, 2, and 3, 5.

00:05:33.100 --> 00:05:35.760
Well, let's take this
as our starting point.

00:05:35.760 --> 00:05:40.930
So, change in y over change in
x, or rise over run, well,

00:05:40.930 --> 00:05:43.430
change in y would be this 5.

00:05:43.430 --> 00:05:46.950
5 minus this 2.

00:05:46.950 --> 00:05:52.471
Over this 3 minus this 5.

00:05:52.471 --> 00:05:59.110
And that gets us 3, this
is a 5, over minus 2.

00:05:59.110 --> 00:06:01.890
Equals minus 3/2.

00:06:01.890 --> 00:06:04.370
Let's do another one.

00:06:04.370 --> 00:06:06.070
This time I'm going to try to
make it color-coded so it'll

00:06:06.070 --> 00:06:08.070
more self-explanatory.

00:06:08.070 --> 00:06:09.410
Say, it's 1, 2.

00:06:09.410 --> 00:06:10.990
That's the first point.

00:06:10.990 --> 00:06:17.390
And then the second
point is 4, 3.

00:06:17.390 --> 00:06:25.120
So, once again, we say slope
is equal to change in

00:06:25.120 --> 00:06:28.790
y over change in x.

00:06:28.790 --> 00:06:29.980
Well, in y.

00:06:29.980 --> 00:06:31.180
We take the first y.

00:06:31.180 --> 00:06:32.270
Let's start here.

00:06:32.270 --> 00:06:33.850
And we'll call that y1.

00:06:33.850 --> 00:06:42.250
So that's 3 minus the
second y, which is that 2.

00:06:42.250 --> 00:06:47.240
And then all of that over,
once again, the first x.

00:06:47.240 --> 00:06:54.045
Which is 4, minus the
second x, which is that 1.

00:06:54.045 --> 00:07:00.230
And this equals 3
minus 2, is 1.

00:07:00.230 --> 00:07:02.620
And 4 minus 1 is 3.

00:07:02.620 --> 00:07:05.540
So the slope in this
example is 1/3.

00:07:05.540 --> 00:07:06.860
And we could have actually
switched it around.

00:07:06.860 --> 00:07:08.440
We could have also
done it other way.

00:07:08.440 --> 00:07:22.380
We could have said, 2
minus 3 over 1 minus 4.

00:07:22.380 --> 00:07:24.820
In which case we would
have gotten negative

00:07:24.820 --> 00:07:26.800
1 over negative 3.

00:07:26.800 --> 00:07:28.170
Well, that just
equals 1/3 again.

00:07:28.170 --> 00:07:29.750
Because the negatives
cancel out.

00:07:29.750 --> 00:07:32.660
So I'll let you think about
why this and this come

00:07:32.660 --> 00:07:34.110
out to the same thing.

00:07:34.110 --> 00:07:36.810
But the important thing to
realize is, if we use the 3

00:07:36.810 --> 00:07:40.290
first, if we use the 3 first
for the y, we also have to

00:07:40.290 --> 00:07:42.320
use the 4 first for the x.

00:07:42.320 --> 00:07:43.550
That's a common mistake.

00:07:43.550 --> 00:07:45.810
And also, you always have to be
very careful with the negative

00:07:45.810 --> 00:07:48.240
signs when you do these
type of problems.

00:07:48.240 --> 00:07:51.240
But I think that will give you
at least enough of a sense that

00:07:51.240 --> 00:07:53.740
you could start the
slope problems.

00:07:53.740 --> 00:07:55.450
The next module, I'll actually
show you how to figure

00:07:55.450 --> 00:07:56.380
out the y intercept.

00:07:56.380 --> 00:07:59.250
Because, as we said, before
the equation of any line is,

00:07:59.250 --> 00:08:02.880
y is equal to m x plus b.

00:08:02.880 --> 00:08:04.680
And I'm going to go
into some more detail.

00:08:04.680 --> 00:08:06.280
Where m is the slope.

00:08:06.280 --> 00:08:08.310
So if you know the
slope of a line.

00:08:08.310 --> 00:08:11.403
And you know the y intercept of
a line, you know everything you

00:08:11.403 --> 00:08:13.350
need to know about the line,
and you can actually write down

00:08:13.350 --> 00:08:15.180
the equation of a line, and
figure out other points

00:08:15.180 --> 00:08:15.990
that are on it.

00:08:15.990 --> 00:08:18.400
So I'm going to do that
in future modules.

00:08:18.400 --> 00:08:20.660
I hope I haven't
confused you too much.

00:08:20.660 --> 00:08:22.620
And try some of those
the slope modules.

00:08:22.620 --> 00:08:23.410
You should be able to do them.

00:08:23.410 --> 00:08:25.710
And I hope you have fun.


Algebra: Slope and Y-intercept intuition

WEBVTT
Kind: captions
Language: en

00:00:02.120 --> 00:00:02.770
Good morning.

00:00:02.770 --> 00:00:04.660
Actually I don't know
what time it is for you,

00:00:04.660 --> 00:00:05.680
it's morning for me.

00:00:05.680 --> 00:00:09.490
Welcome to the presentation
on slope and y-intercept.

00:00:09.490 --> 00:00:12.740
This presentation isn't going
to teach you how to solve for

00:00:12.740 --> 00:00:15.760
slope and y-intercept, but
hopefully it will give you

00:00:15.760 --> 00:00:18.660
a good intuition of what
slope and y-intercept is.

00:00:18.660 --> 00:00:21.380
And we're going to do something
a little bit different

00:00:21.380 --> 00:00:22.800
this time as opposed to
what we normally do.

00:00:22.800 --> 00:00:23.880
We're not going to use the
chalkboard, we're actually

00:00:23.880 --> 00:00:27.490
going to go on to the Khan
academy website and use the

00:00:27.490 --> 00:00:30.210
graph of a line exercise to get
a little bit of an intuition

00:00:30.210 --> 00:00:32.810
for what slope and
y-intercept is.

00:00:32.810 --> 00:00:35.420
So when the application starts
off, it starts with the

00:00:35.420 --> 00:00:38.360
equation y equals 1x plus 1.

00:00:38.360 --> 00:00:40.820
So that's the same thing
as y equals x plus 1.

00:00:40.820 --> 00:00:42.790
But we see that the
slope here is 1.

00:00:42.790 --> 00:00:44.580
If you looked at the
introduction to graphing that

00:00:44.580 --> 00:00:46.430
I talked about, the slope is
the same thing is the

00:00:46.430 --> 00:00:49.310
coefficient on the x term.

00:00:49.310 --> 00:00:52.910
And if you see here,
whenever we move over

00:00:52.910 --> 00:00:55.600
by 1, we move up by 1.

00:00:55.600 --> 00:00:57.710
And I'm going to do another
module on that slope is

00:00:57.710 --> 00:01:00.070
actually rise over run.

00:01:00.070 --> 00:01:03.510
So it's for every amount
you rise, how much do you

00:01:03.510 --> 00:01:04.990
have to run to get that?

00:01:04.990 --> 00:01:07.940
And rise just means how much
do you change in y, run means

00:01:07.940 --> 00:01:09.310
how much do you change in x.

00:01:09.310 --> 00:01:14.070
So here rise over run is just
1, and y-intercept is where

00:01:14.070 --> 00:01:16.550
you intercept the y-axis.

00:01:16.550 --> 00:01:20.160
Now, as I change the slope and
the y-intercept for this graph,

00:01:20.160 --> 00:01:21.770
I think it's going to make a
little bit more sense to you.

00:01:21.770 --> 00:01:25.440
Watch what happens when the
slope goes from 1 to 3/2.

00:01:25.440 --> 00:01:27.970
So 3/2 is the same
thing is 1 and 1/2.

00:01:27.970 --> 00:01:29.830
So notice it got steeper.

00:01:29.830 --> 00:01:32.460
And if I increase the slope
more it gets steeper even

00:01:32.460 --> 00:01:35.530
more, and y equals 2x.

00:01:35.530 --> 00:01:37.960
If I increase any more
5/2 is 2 and 1/2.

00:01:37.960 --> 00:01:40.200
So the more I increase
the slope, I think you

00:01:40.200 --> 00:01:40.940
see what's happening.

00:01:40.940 --> 00:01:42.380
This thing jumps around.

00:01:42.380 --> 00:01:44.030
I should fix that.

00:01:44.030 --> 00:01:45.100
Let me move it back.

00:01:45.100 --> 00:01:50.400
And actually, the goal is to
make the line go through

00:01:50.400 --> 00:01:51.490
those two blue points.

00:01:51.490 --> 00:01:54.900
That's the goal of I guess
you'd call it the game.

00:01:54.900 --> 00:01:57.070
I don't like how this thing
jumps around though.

00:01:57.070 --> 00:01:58.835
That was interesting,
let me go back there.

00:02:01.420 --> 00:02:04.460
y equals 0x plus 1.

00:02:04.460 --> 00:02:07.800
We could have rewritten this as
just y equals 1, because 0x

00:02:07.800 --> 00:02:08.700
is the same things as 0.

00:02:08.700 --> 00:02:10.730
And notice it's a
completely flat line.

00:02:10.730 --> 00:02:13.190
No matter what x is y is 1.

00:02:13.190 --> 00:02:14.580
And that makes sense because
this equation would

00:02:14.580 --> 00:02:17.750
just be y equals 1.

00:02:17.750 --> 00:02:20.190
Now I've been showing you
what happens to the slope.

00:02:20.190 --> 00:02:22.395
Now notice we have
a negative slope.

00:02:22.395 --> 00:02:24.560
The slope is now
downwards sloping.

00:02:24.560 --> 00:02:27.610
It's downward sloping
at a slope of 1/2.

00:02:27.610 --> 00:02:31.020
Because let's say the rise in
this situation is negative

00:02:31.020 --> 00:02:32.810
1, and the run is 2.

00:02:32.810 --> 00:02:36.500
So that's why we get
negative 1 over 2.

00:02:36.500 --> 00:02:38.880
And we had just been doing
slope so far and I think you

00:02:38.880 --> 00:02:42.170
get the idea that as we
decrease slope, it's going to

00:02:42.170 --> 00:02:45.640
push the line further and
further-- it's going to

00:02:45.640 --> 00:02:46.720
slope downward even more.

00:02:46.720 --> 00:02:50.490
I hate to use a word in its own
definition, but I think you

00:02:50.490 --> 00:02:52.030
see that now in the picture.

00:02:52.030 --> 00:02:53.520
Now let's put up the
y-intercept a little bit.

00:02:53.520 --> 00:02:54.460
And this is even
more interesting.

00:02:54.460 --> 00:02:59.980
So y-intercept-- oh boy, how
did that happen, that was

00:02:59.980 --> 00:03:03.760
strange --y-intercept--

00:03:03.760 --> 00:03:08.820
Notice, negative 1x plus 2, so
the slope is negative 1 but it

00:03:08.820 --> 00:03:10.420
intersects the y-axis at 2.

00:03:10.420 --> 00:03:13.615
Now if we increase y-intercept
by 1 more it's just going

00:03:13.615 --> 00:03:15.570
to push this line up 1.

00:03:15.570 --> 00:03:17.670
Let's do that.

00:03:17.670 --> 00:03:17.940
See.

00:03:17.940 --> 00:03:21.110
Oh, this is actually increasing
it by increments of 1/2.

00:03:21.110 --> 00:03:23.690
Let's do another one, I just
want to see what happens

00:03:23.690 --> 00:03:24.530
on another graph.

00:03:24.530 --> 00:03:27.670
It actually depends on
the actual problem.

00:03:27.670 --> 00:03:29.350
OK, this is interesting.

00:03:29.350 --> 00:03:30.140
OK, this is the same thing.

00:03:30.140 --> 00:03:31.600
We start at the same point.

00:03:31.600 --> 00:03:37.030
Let's actually try to figure
out the equation of a line that

00:03:37.030 --> 00:03:39.120
goes through these two points.

00:03:39.120 --> 00:03:40.150
Well, let's see.

00:03:40.150 --> 00:03:41.890
It looks like the y-intercept
if is going to have to

00:03:41.890 --> 00:03:43.970
be a little bit lower.

00:03:43.970 --> 00:03:45.980
I do not get why
it would do that.

00:03:49.770 --> 00:03:54.240
It just brings the line down
as we lower the y-intercept.

00:03:54.240 --> 00:03:56.520
And let's see I think the slope
needs to be higher, because

00:03:56.520 --> 00:03:58.100
those two points, the line that
goes through them is

00:03:58.100 --> 00:04:00.270
definitely steeper.

00:04:00.270 --> 00:04:01.980
I apologize for this
thing acting up like.

00:04:01.980 --> 00:04:04.530
That looks like about
the right slope.

00:04:04.530 --> 00:04:08.320
The slope is like that, and
these two points are connected.

00:04:08.320 --> 00:04:10.240
Yeah, I think that looks like
the right slope, but the

00:04:10.240 --> 00:04:11.470
y-intercept has to be lower.

00:04:16.130 --> 00:04:17.875
Almost there, I think.

00:04:20.570 --> 00:04:21.690
There you go!

00:04:21.690 --> 00:04:24.970
So the equation of
this line is 7/4x.

00:04:24.970 --> 00:04:28.410
So 7/4, that's the same
thing as like 1.75.

00:04:28.410 --> 00:04:33.360
So the slope of this line
slopes faster than 1/1 and

00:04:33.360 --> 00:04:34.540
you can kind of see that.

00:04:34.540 --> 00:04:36.300
I'll show you how to figure out
all this, I just want to give

00:04:36.300 --> 00:04:38.980
you an intuitive sense of what
sloping and y-intercept is.

00:04:38.980 --> 00:04:42.520
And it intersects the
y-axis at negative 13/4.

00:04:42.520 --> 00:04:45.210
That's a little more than 3,
which you can-- negative

00:04:45.210 --> 00:04:47.860
3 --which you can
see right there.

00:04:47.860 --> 00:04:50.890
Let's see if we can
do another one.

00:04:50.890 --> 00:04:53.120
And if you want, we can assign
this module to you and you can

00:04:53.120 --> 00:04:55.380
play with it just like
I'm doing right here.

00:04:55.380 --> 00:04:58.290
So let's see, the line that
we want to get will go

00:04:58.290 --> 00:04:59.320
something like that.

00:04:59.320 --> 00:05:02.540
Looks like the current line's
slope is a little too high.

00:05:02.540 --> 00:05:04.130
Let me lower the
slope a little bit.

00:05:04.130 --> 00:05:05.420
That looks about right.

00:05:05.420 --> 00:05:09.140
7/8, so that means for every
8 you move to the right

00:05:09.140 --> 00:05:10.580
you're going to move 7 up.

00:05:10.580 --> 00:05:12.790
And I'm going to draw that
better in another module.

00:05:12.790 --> 00:05:16.470
This module I'm kind of doing
on the fly, so I apologize.

00:05:16.470 --> 00:05:18.920
I do every model on the
fly so I guess I really

00:05:18.920 --> 00:05:20.390
should apologize.

00:05:20.390 --> 00:05:24.350
But you're not paying for this,
so I shouldn't apologize.

00:05:24.350 --> 00:05:27.240
Oh, I distracted very easily.

00:05:27.240 --> 00:05:29.180
Let's see, let's
move this line up.

00:05:29.180 --> 00:05:30.660
And you do that just
by the y-intercept.

00:05:30.660 --> 00:05:32.860
You can see shifting the
y-intercept up just shifts

00:05:32.860 --> 00:05:33.890
the line straight up.

00:05:33.890 --> 00:05:36.200
It doesn't change the
inclination of the line.

00:05:36.200 --> 00:05:38.100
The slope changes the
inclination of the line.

00:05:41.140 --> 00:05:42.190
There we go.

00:05:42.190 --> 00:05:46.390
The equation of this
line is 7/8x plus 13/4.

00:05:46.390 --> 00:05:49.060
Let's see if what I said about
slope is right if we move.

00:05:49.060 --> 00:05:51.640
If we run 8, we should rise 7.

00:05:51.640 --> 00:05:52.040
So let's see.

00:05:52.040 --> 00:05:53.080
Run 8.

00:05:53.080 --> 00:06:02.090
1, 2, 3, 4, 5, 6, 7, 8.

00:06:02.090 --> 00:06:03.790
So that gets us right there.

00:06:03.790 --> 00:06:06.670
And then we should rise 7.

00:06:06.670 --> 00:06:14.930
1, 2, 3, 4, 5, 6, 7.

00:06:14.930 --> 00:06:16.740
Well that actually gets
us those exact points.

00:06:16.740 --> 00:06:18.250
And we're back on
the line again.

00:06:18.250 --> 00:06:21.170
I'm going to draw another thing
like that for you so if you get

00:06:21.170 --> 00:06:23.240
confused don't lose heart.

00:06:23.240 --> 00:06:26.370
Let's do one more.

00:06:26.370 --> 00:06:27.800
OK.

00:06:27.800 --> 00:06:29.090
Where's the other dot?

00:06:29.090 --> 00:06:31.560
I don't know.

00:06:31.560 --> 00:06:33.610
Let me see.

00:06:33.610 --> 00:06:34.870
The other dot doesn't exist.

00:06:34.870 --> 00:06:37.170
I gotta fix all these
bugs in this thing.

00:06:37.170 --> 00:06:37.540
Oh there.

00:06:37.540 --> 00:06:37.830
Good.

00:06:37.830 --> 00:06:38.440
It showed up.

00:06:38.440 --> 00:06:39.020
It showed up.

00:06:39.020 --> 00:06:39.850
Excellent.

00:06:39.850 --> 00:06:41.090
OK, so look.

00:06:41.090 --> 00:06:43.530
We have to make the line go
through these two points.

00:06:43.530 --> 00:06:45.910
It looks like the slope
is negative, definitely.

00:06:45.910 --> 00:06:49.120
Not that negative, it's like
a fractional negative slope.

00:06:49.120 --> 00:06:51.720
And it'll intercept the y-axis
somewhere around here.

00:06:51.720 --> 00:06:54.400
The y-intercept is going to
be like 7 and something.

00:06:54.400 --> 00:06:55.360
7 and change.

00:06:55.360 --> 00:06:56.910
So first of all let's
get this slope down.

00:06:56.910 --> 00:06:57.110
Oh boy.

00:06:57.110 --> 00:07:00.410
This thing is going to
jump around again.

00:07:00.410 --> 00:07:02.410
Notice y equals 0, x plus 1.

00:07:02.410 --> 00:07:05.020
If we increase the slope.

00:07:05.020 --> 00:07:07.300
This thing is doing all sorts--
I haven't seen this application

00:07:07.300 --> 00:07:11.130
in a while, so I must've
written it when I had inferior

00:07:11.130 --> 00:07:15.180
programming skills, let me keep
--OK, that slope

00:07:15.180 --> 00:07:16.160
might be right.

00:07:16.160 --> 00:07:17.960
Let's bring the line up higher.

00:07:21.550 --> 00:07:26.200
No, it still seems like my
slope-- see the y-intercept,

00:07:26.200 --> 00:07:27.215
I'm raising the line.

00:07:30.160 --> 00:07:31.830
Oh good, I got it
exactly right.

00:07:31.830 --> 00:07:32.620
And I was right.

00:07:32.620 --> 00:07:36.310
The slope is negative, because
you can see it slopes downward.

00:07:36.310 --> 00:07:38.710
But it's not sloping
downward that fast.

00:07:38.710 --> 00:07:41.130
And that make sense, that
the slope is negative 1/3.

00:07:41.130 --> 00:07:49.850
And that makes sense because if
we run 3, 1, 2, 3, we rise

00:07:49.850 --> 00:07:53.060
negative 1, we rise negative 1.

00:07:53.060 --> 00:07:53.540
Right there.

00:07:53.540 --> 00:07:55.940
So that's why the slope
is negative 1/3.

00:07:55.940 --> 00:07:58.530
And then the
y-intercept is 22/3.

00:07:58.530 --> 00:08:00.170
Well that's 7 and 1/3.

00:08:00.170 --> 00:08:03.650
And right there, we intercept
the y-axis 1/3 of the

00:08:03.650 --> 00:08:06.850
way between 7 and 8.

00:08:06.850 --> 00:08:10.020
Well I think that should at
least give you a little bit of

00:08:10.020 --> 00:08:14.600
an intuition on what slope and
y-intercept are and you can

00:08:14.600 --> 00:08:16.840
have this module assigned for
you, so you could play

00:08:16.840 --> 00:08:17.540
with it yourself.

00:08:17.540 --> 00:08:19.940
And I'm going to do some more
models where you actually

00:08:19.940 --> 00:08:22.650
calculate slope and y-intercept
and hopefully give you even

00:08:22.650 --> 00:08:24.820
though further intuition
on what they are.

00:08:24.820 --> 00:08:28.850
So I hope you have fun playing
around with this stuff.

00:08:28.850 --> 00:08:31.000
I remember I was very excited
when I first learned

00:08:31.000 --> 00:08:32.890
this stuff, because
it's very visual.

00:08:32.890 --> 00:08:34.870
So, have fun.


Algebra: graphing lines 1

WEBVTT
Kind: captions
Language: en

00:00:01.220 --> 00:00:04.090
Welcome to the presentation
on graphing lines.

00:00:04.090 --> 00:00:06.200
Let's get started.

00:00:06.200 --> 00:00:10.530
So let's say I had the
equation-- let me make sure

00:00:10.530 --> 00:00:14.170
that this line doesn't
show up too thick.

00:00:14.170 --> 00:00:18.370
Let's say I had the equation--
why isn't that showing up?

00:00:18.370 --> 00:00:18.760
Let's see.

00:00:18.760 --> 00:00:19.990
Oh, there you go.

00:00:19.990 --> 00:00:26.692
y is equal to 2x plus 1.

00:00:26.692 --> 00:00:29.670
So this is giving a
relationship between x and y.

00:00:29.670 --> 00:00:35.930
So say x equals 1, then y would
be 2 times 1 plus 1 or 3.

00:00:35.930 --> 00:00:39.140
So for every x that we can
think of we can think

00:00:39.140 --> 00:00:41.350
of a corresponding y.

00:00:41.350 --> 00:00:42.710
So let's do that.

00:00:42.710 --> 00:00:46.520
If we said that-- put
a little table here.

00:00:49.850 --> 00:00:50.400
x and y.

00:00:50.400 --> 00:00:53.630
And let's just throw out
some random numbers for x.

00:00:53.630 --> 00:00:58.940
If x was let's say, negative
1, then y would be 2 times

00:00:58.940 --> 00:01:00.940
negative 1, which
is negative 2.

00:01:00.940 --> 00:01:04.810
Plus 1, which would
be negative 1.

00:01:04.810 --> 00:01:07.480
If x was 0 that's easy.

00:01:07.480 --> 00:01:09.310
It'd be 2 times 0, which is 0.

00:01:09.310 --> 00:01:11.672
Plus 1, which is 1.

00:01:11.672 --> 00:01:17.830
If x was 1, y would be
2 times 1, which is 2.

00:01:17.830 --> 00:01:21.700
Plus 1, which is 3.

00:01:21.700 --> 00:01:26.520
If x was 2, then I think
you get the idea here.

00:01:26.520 --> 00:01:27.730
y would be 5.

00:01:27.730 --> 00:01:29.530
And we could keep on going.

00:01:29.530 --> 00:01:31.540
Obviously, there are an
infinite number of x's we

00:01:31.540 --> 00:01:33.820
could choose and we could
pick a corresponding y.

00:01:33.820 --> 00:01:35.410
So now you see we have a
little table that gives the

00:01:35.410 --> 00:01:37.470
relationships between x and y.

00:01:37.470 --> 00:01:40.540
What we can do now is actually
graph those points on

00:01:40.540 --> 00:01:42.500
a coordinate axis.

00:01:42.500 --> 00:01:46.870
So let me see if I can draw
this somewhat neatly.

00:01:46.870 --> 00:01:50.425
I'll use this line so
I get straight lines.

00:01:55.670 --> 00:01:56.970
That's pretty good.

00:01:56.970 --> 00:02:00.905
Again, let me draw some
coordinate points.

00:02:00.905 --> 00:02:08.210
So let's say that's 1,
that's 2, that's 3.

00:02:08.210 --> 00:02:13.170
This is negative 1,
negative 2, negative 3.

00:02:13.170 --> 00:02:14.180
So this is the x-axis.

00:02:16.950 --> 00:02:22.340
We have 1, 2, 3.

00:02:22.340 --> 00:02:24.020
Notice we could keep going.

00:02:24.020 --> 00:02:28.730
1, 2, 3, and this
is the y-axis.

00:02:32.255 --> 00:02:35.990
And this would be 1,
2, 3, and so on.

00:02:35.990 --> 00:02:37.220
This would be negative 1.

00:02:37.220 --> 00:02:38.360
I think you get the idea.

00:02:38.360 --> 00:02:40.550
So we can graph each
of these points.

00:02:40.550 --> 00:02:45.440
So if we have the point x is
negative 1, y is negative 1.

00:02:45.440 --> 00:02:48.710
So x, we go along the x-axis
here, and we go to x is

00:02:48.710 --> 00:02:49.745
equal to negative 1.

00:02:49.745 --> 00:02:52.990
Then we go to y is equal to
negative 1, so the point

00:02:52.990 --> 00:02:53.775
would be right here.

00:02:53.775 --> 00:02:56.520
Hope that makes sense to you.

00:02:56.520 --> 00:02:57.670
That's the point.

00:02:57.670 --> 00:03:00.650
I'll label it: negative
1 comma negative 1.

00:03:00.650 --> 00:03:01.140
It's a little messy.

00:03:01.140 --> 00:03:02.920
That says negative 1
comma negative 1.

00:03:02.920 --> 00:03:05.330
That point I just
x'ed right there.

00:03:05.330 --> 00:03:06.820
Let's do another one.

00:03:06.820 --> 00:03:07.880
That's this point.

00:03:07.880 --> 00:03:10.280
I'll do it in a different
color this time.

00:03:10.280 --> 00:03:14.960
Let's say we had the
point 0 comma 1.

00:03:14.960 --> 00:03:17.410
Well, x is 0, which is here.

00:03:17.410 --> 00:03:21.110
And y is 1, so that
point is right there.

00:03:21.110 --> 00:03:21.885
Let's do one more.

00:03:21.885 --> 00:03:25.350
If we have the point 1 comma 3.

00:03:25.350 --> 00:03:33.140
Well, 1 comma 3, x is
1 and we have y is 3.

00:03:33.140 --> 00:03:34.730
So we have the
point right there.

00:03:34.730 --> 00:03:36.750
Hope that's making
sense for you.

00:03:36.750 --> 00:03:39.180
And we could keep graphing
them, but I think you see here,

00:03:39.180 --> 00:03:41.550
and especially if I had drawn
this a little bit neater, that

00:03:41.550 --> 00:03:43.090
these points are
forming a line.

00:03:43.090 --> 00:03:46.980
Let me draw that line in.

00:03:46.980 --> 00:03:49.370
The line looks
something like this.

00:03:53.810 --> 00:03:54.860
That's not a good line.

00:03:54.860 --> 00:03:56.280
Let me do it better than that.

00:03:56.280 --> 00:03:58.350
The line looks
something like this.

00:04:03.340 --> 00:04:03.860
You see that?

00:04:03.860 --> 00:04:06.720
Well, that's actually a pretty
bad line that I just drew.

00:04:06.720 --> 00:04:11.670
So it would be a line that goes
through-- let me change tools.

00:04:11.670 --> 00:04:13.960
It'd be a line that goes
through here, through

00:04:13.960 --> 00:04:16.960
here, and through here.

00:04:16.960 --> 00:04:19.010
I don't know if I'm making
this clear at all.

00:04:19.010 --> 00:04:22.630
Let me make these
points a little bit.

00:04:22.630 --> 00:04:24.420
You see the line will go
through all of these points,

00:04:24.420 --> 00:04:27.200
but it will also go through the
point 2 comma 5, which will

00:04:27.200 --> 00:04:30.800
be up here some place.

00:04:30.800 --> 00:04:34.600
For any x that you can think
of, if you had x is equal to

00:04:34.600 --> 00:04:38.510
10,380,000,000 the
corresponding y will

00:04:38.510 --> 00:04:39.770
also be on this line.

00:04:39.770 --> 00:04:44.180
So this pink line, and it
keeps going on forever, that

00:04:44.180 --> 00:04:49.850
represents every possible
combination of x's and y's that

00:04:49.850 --> 00:04:51.580
will satisfy this equation.

00:04:51.580 --> 00:04:53.770
And of course, x doesn't
have to just be whole

00:04:53.770 --> 00:04:54.880
numbers or integers.

00:04:54.880 --> 00:04:59.670
x could be pi-- 3.14159.

00:04:59.670 --> 00:05:02.090
In which case it would be
someplace here and in which

00:05:02.090 --> 00:05:05.240
case y would be 2 pi plus 1.

00:05:05.240 --> 00:05:09.130
So every number that x could
be there's a corresponding y.

00:05:09.130 --> 00:05:09.990
Let's do another 1.

00:05:14.060 --> 00:05:21.535
So if I had the equation y is
equal to-- that's an ugly y. y

00:05:21.535 --> 00:05:29.020
is equal to negative 3x plus 5.

00:05:29.020 --> 00:05:32.180
Well, I'm going to draw it
quick and dirty this time.

00:05:32.180 --> 00:05:34.180
So that's the x-axis.

00:05:34.180 --> 00:05:36.020
That's the y-axis.

00:05:36.020 --> 00:05:39.334
Let's put some values here.

00:05:39.334 --> 00:05:41.630
x and y.

00:05:41.630 --> 00:05:44.860
Let's say if x is negative
1, then negative 1 times

00:05:44.860 --> 00:05:49.245
negative 3 is 3 plus y is 8.

00:05:49.245 --> 00:05:52.750
If x is 0, then y is 5.

00:05:52.750 --> 00:05:54.020
That's pretty easy.

00:05:54.020 --> 00:05:58.690
If x is 1, negative 3
times 1 is negative 3.

00:05:58.690 --> 00:06:00.620
Then y is 2.

00:06:00.620 --> 00:06:04.990
If x is 2, negative 3
times 2 is negative 6.

00:06:04.990 --> 00:06:06.730
Then y is 1.

00:06:06.730 --> 00:06:08.070
Is that right?

00:06:08.070 --> 00:06:09.610
Negative 6-- no, no.

00:06:09.610 --> 00:06:10.315
Negative 1.

00:06:10.315 --> 00:06:12.810
I knew something
was wrong there.

00:06:12.810 --> 00:06:14.610
So let's graph some
of these points.

00:06:14.610 --> 00:06:18.000
So when x is negative 1 and I'm
just kind of approximating.

00:06:18.000 --> 00:06:21.710
When x is negative
1, y is negative 8.

00:06:21.710 --> 00:06:23.610
So that point would be
someplace around here.

00:06:23.610 --> 00:06:25.680
And there's a whole module I'm
graphing coordinates if you're

00:06:25.680 --> 00:06:28.540
finding the graphing a
coordinate pair to be

00:06:28.540 --> 00:06:31.240
a little confusing.

00:06:31.240 --> 00:06:31.700
Oh, wait.

00:06:31.700 --> 00:06:32.790
I just made a mistake.

00:06:32.790 --> 00:06:34.830
When x is negative 1, y is 9.

00:06:34.830 --> 00:06:37.070
Not negative 8, so
ignore this right here.

00:06:37.070 --> 00:06:42.270
When x is negative
1, y is positive 8.

00:06:42.270 --> 00:06:44.610
So y being up here someplace.

00:06:44.610 --> 00:06:47.540
When x is 0, y is 5.

00:06:47.540 --> 00:06:50.440
So it'd be here someplace.

00:06:50.440 --> 00:06:53.720
When x is 1, y is 2.

00:06:53.720 --> 00:06:54.650
So it's like here.

00:06:57.160 --> 00:07:02.310
When x is 2, y is negative 1.

00:07:02.310 --> 00:07:04.550
So as you can see-- and
I've approximated it.

00:07:04.550 --> 00:07:08.740
If I had graphing paper or if I
had a better drawn chart you

00:07:08.740 --> 00:07:11.210
could have seen it and it would
have been exactly right.

00:07:11.210 --> 00:07:14.950
I think this line
will do the job.

00:07:14.950 --> 00:07:19.430
That every point that satisfies
this equation actually

00:07:19.430 --> 00:07:21.470
falls on this line.

00:07:21.470 --> 00:07:23.700
And something interesting
here I'll point out.

00:07:23.700 --> 00:07:26.790
You notice that this line
it slopes downwards.

00:07:26.790 --> 00:07:29.510
It goes from the top left
to the bottom right.

00:07:29.510 --> 00:07:31.800
While the line we had drawn
before had gone from the

00:07:31.800 --> 00:07:35.160
bottom left to the top right.

00:07:35.160 --> 00:07:37.740
Is there anything about this
equation that seems a little

00:07:37.740 --> 00:07:40.320
bit different than the last?

00:07:40.320 --> 00:07:43.240
I'll give you a little
bit of a hint.

00:07:43.240 --> 00:07:47.230
This number-- the negative 3,
or you could say that the

00:07:47.230 --> 00:07:52.280
coefficient on x-- that
determines whether the line

00:07:52.280 --> 00:07:55.040
slopes upward, or the line
slows downward, and it tells

00:07:55.040 --> 00:07:56.725
you also how steep the line is.

00:07:56.725 --> 00:07:58.900
And that actually,
negative 3 is the slope.

00:07:58.900 --> 00:08:02.350
And I'm going to do a whole
nother module on slope.

00:08:02.350 --> 00:08:05.400
And this number here is
called the y-intercept.

00:08:05.400 --> 00:08:06.960
And that actually tells
you where you're going

00:08:06.960 --> 00:08:08.970
to intersect the y-axis.

00:08:08.970 --> 00:08:10.460
And it turns out here,
that you intersect the

00:08:10.460 --> 00:08:13.250
axis at 0 comma 5.

00:08:15.920 --> 00:08:18.470
Let's do one more real fast.

00:08:21.770 --> 00:08:26.130
y is equal to 2--
we already did 2x.

00:08:26.130 --> 00:08:35.374
y is equal to 1/2 x
plus 2 So real fast.

00:08:35.374 --> 00:08:37.180
x and y.

00:08:37.180 --> 00:08:39.270
And you only need two
points for a line, really.

00:08:39.270 --> 00:08:41.120
So you could just say
let's say, x equals 0.

00:08:41.120 --> 00:08:43.410
That's easy. y equals 2.

00:08:43.410 --> 00:08:46.580
And if x equals 2
then y equals 3.

00:08:46.580 --> 00:08:51.660
So before when we were doing 3
and 4 points that was just to

00:08:51.660 --> 00:08:53.650
kind of show you, but you
really just need two

00:08:53.650 --> 00:08:54.230
points for a line.

00:08:54.230 --> 00:08:57.930
So 0 comma 1 2.

00:08:57.930 --> 00:08:58.730
So that's on there.

00:08:58.730 --> 00:09:03.320
And then 1, 2 comma 3.

00:09:03.320 --> 00:09:05.950
So it's there.

00:09:05.950 --> 00:09:08.110
So the line is going to
look something like this.

00:09:12.190 --> 00:09:14.440
So notice here, once again,
we're upward sloping and that's

00:09:14.440 --> 00:09:16.870
because this 1/2 is positive.

00:09:16.870 --> 00:09:20.120
But we're not sloping-- we're
not moving up as quickly as

00:09:20.120 --> 00:09:22.870
when we had y equals
2x. y equals 2x looked

00:09:22.870 --> 00:09:24.452
something like this.

00:09:24.452 --> 00:09:26.390
It was sloping up much,
much, much faster.

00:09:26.390 --> 00:09:27.620
I hope I'm not confusing you.

00:09:27.620 --> 00:09:30.710
And then the y intercept of
course is at 0 comma 2,

00:09:30.710 --> 00:09:32.160
which is right here.

00:09:32.160 --> 00:09:35.300
So if you ever want to graph
a line it's really easy.

00:09:35.300 --> 00:09:37.760
You have to just try out some
points and you can graph it.

00:09:37.760 --> 00:09:39.440
And now in the next module I'm
going to show you a little bit

00:09:39.440 --> 00:09:41.360
more about slope and
y-intercept and you won't

00:09:41.360 --> 00:09:42.480
even have to do this.

00:09:42.480 --> 00:09:45.490
But this gives you good
intuitive feel, I think,

00:09:45.490 --> 00:09:47.320
what a graph of a line is.

00:09:47.320 --> 00:09:49.250
I hope you have fun.