Zero, negative, and fractional exponents

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Kind: captions
Language: en

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So a few videos ago, I told
you that anything to the

00:00:03.320 --> 00:00:05.260
zeroth power is equal to 1.

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So x to the zeroth power
is equal to 1.

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And I gave you one argument
why this is the case.

00:00:13.140 --> 00:00:16.910
I used the example of, if we
have 3 to the first power,

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that is equal to 3.

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3 to the second power
is equal to 9.

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3 to the third power
is equal to 27.

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So every time we decrease by a
power, we're dividing by 3.

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27 divided by 3 is 9.

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9 divided by 3 is 3.

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Then 3 divided by 3 is 1.

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And that should be what 3
to the zeroth power is.

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So that's one way to
think about it.

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The other way to think about
it is that we need this for

00:00:43.640 --> 00:00:45.920
the exponent properties
to work.

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For example, I told you that a
to the b times a to the c is

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equal to a to the b plus c.

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Now, what happens if c is 0?

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What happens if we have a to
the b times a to the 0?

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Well, by this property, this
needs to be equal to a to the

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b plus 0, which is equal
to a to the b.

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So a to the b times a to the 0
must be equal to a to the b.

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If you divide both sides of this
times a-- let me rewrite

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this-- a to the b times a to the
0, if we use this property

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up here, must be equal to a to
the b, right? b plus 0 is b.

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If you divide both sides by a
to the b, what do you get?

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On the left-hand side,
you're left with

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just a to the 0, right?

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These cancel out.

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a to the 0 is equal to 1.

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And you can use a similar
argument in pretty much all of

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the exponent properties, that we
need anything to the zeroth

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power to be equal to 1.

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And it also makes sense as we
divide by 3, each step as we

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decrement our exponent.

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It keeps working.

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When you take 3 to the negative
1 power, we saw on

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the last video that that's equal
to 1 over 3 to the first

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power, or 1/3 .

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So once again, from 3 to
the 0 to 1/3, you're

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dividing by 3 again.

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So it really makes sense on
some level that 3 to the

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zeroth power is equal to 1.

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But that leaves a little
bit of a gap.

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What about 0 to the
zeroth power?

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This is a very strange notion.

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0 multiplied by itself
0 times.

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And it depends what context
you're using.

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Sometimes people will say that
this is undefined, but many

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more times, at least in my
experience, this'll be

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defined to be 1.

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And the reason why-- even though
this is completely not

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intuitive, and you could type
in 0 to the zeroth power in

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Google, and it'll give you 1.

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Even though this is completely
not intuitive, the reason why

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this is defined to be this way
is that makes a lot of

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formulas work.

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One in particular, the binomial
formula works for

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your binomial coefficients,
which I'm not going to go over

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right here, when 0 to the zeroth
power is equal to 1.

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So that's an interesting thing
for you to think about, what

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that might even mean.

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So let's talk about some of
the other properties.

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And then we can put them all
together with a couple of

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example problems. I told you
in the last video what it

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means to raise to a
negative power.

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a to the negative 1 power, or
maybe I should say a to the

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negative b power is equal to
1 over a to the b power.

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So just to do that with a couple
of concrete examples, 3

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to the negative 3 power is equal
to 1 over 3 to the third

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power, which is equal to 1 over
3 times 3, times 3, which

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is equal to 1 over 27.

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If I were to ask you what 1/3
to the negative 2 power is--

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well, this is going to be equal
to 1 over 1/3 to the

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second power.

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You get rid of the negative
and you inverse it.

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So this is going to be
equal to 1 over--

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what's 1/3 times 1/3?

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1/9.

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Which is equal to-- this is 1
divided by 1/9 is the same

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thing is 1 times 9, so
this is equal to 9.

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And this makes complete sense,
because 1/3, remember, 1/3 is

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the same thing as 3 to the
negative 1 power, right?

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3 to the negative 1 is equal
to 1 over 3 to the 1 power,

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which is the same
thing is 1/3.

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So if we replace 1/3 with 3 to
the negative 1, this is 3 to

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the negative 1 to the
negative 2 power.

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These two things are equivalent
statements.

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And if we use one of the
properties we learned in the

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first video, we can
take the product

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of these two exponents.

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So this is equal to 3 to the
negative 1, times negative 2,

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which is just positive 2,
which is equal to 9.

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So it's really neat how all of
these exponent properties

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really fit together in a nice,
neat puzzle, that they don't

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contradict each other.

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And it doesn't matter which
property you use, you'll get

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the right answer in the end,
as long as you don't do

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something crazy.

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Now, the last thing I want to
define is the notion of a

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fractional exponent.

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So if I have something to a
fractional power-- so let's

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say I have a to the
1 over b power.

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I'm going to define this.

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This is going to be equal
to the bth root of a.

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So let me be very clear here.

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Let me make it with
some numbers here.

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If I said 4 to the 1/2 power
right there, this means this

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is equivalent to the
square root of 4.

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Which is equal to, if we're
taking the principal root,

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this is equal to 2.

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So if I were to take, let's be
clear, 8 to the 1/3 power,

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this is taking the
cube root of 8.

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And this is, on some level,
one of the most sometimes

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confusing things in exponents.

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Here I'm saying, what number
times itself 3 times

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is equal to 8 ?

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So if I said that x is equal to
8 to the 1/3 power, this is

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the exact same thing as
saying x to the third

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power is equal to 8.

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And how do I know that these
are equivalent statements?

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Well, I could take both
sides of this equation

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to the third power.

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If I take the left-hand side
of the third power and the

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right-hand side of the third
power, what do I get?

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On the left-hand side,
I get x to the third.

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On the right-hand side, I get 8
to the 1/3 times 3, which is

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just 3 over 3, which
is just 1.

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So if x is equal to 8 to
the 1/3, what is x?

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Well, 2 times 2, times
2 is equal to 8.

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And there's no really easy way,
especially once you go to

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the fourth root, or the fifth
root, and you have decimals of

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calculating these.

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You probably need a calculator
most of the time to do these.

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But things like 8 to the 1/3, or
16 to the 1/4, or 27 to the

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1/3 , they're not too
hard to calculate.

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So this right here, let
me be clear, is 2.

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Now, let's make it a little
bit more confusing.

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What is 27 to the negative
1/3 power?

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Well, don't get too worried.

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We're just going to take
it step by step.

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When you take the negative
power, this is completely

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equivalent to 1 over 27
to the 1/3 power.

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These two are equivalent.

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You get rid of the negative
and take 1

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over the whole thing.

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And then what is 27
to the 1/3 power?

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Well, what number times itself
3 times is equal to 27?

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Well, that's equal to 3.

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So this is going to be
equal to 1 over 3.

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Not too bad.

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Now I'm going to take it even to
another level, make it even

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more confusing, even
more daunting.

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Now, let me do something
interesting.

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What is 8 to the 2/3 power?

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Now that seems a little
bit scary.

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And all you have to remember
is this is the same thing,

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using our exponent rules
really, as 8

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squared to the 1/3 power.

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How do I know that?

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Well, if I multiply these two
exponents, this is 2/3.

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So 8 to the 2/3 is the same
thing is 8 squared, and then

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the third root of that.

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But you could view
it the other way.

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This should also be equal to
8 to the 1/3 power squared.

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Because either way, when I
multiply these exponents, I

00:09:06.540 --> 00:09:08.760
get 8 to the 2/3.

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Let's verify for ourselves
that we really do

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get the same value.

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So 8 squared is 64.

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And we're going to take
that to the 1/3 power.

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Down here, we have
8 to the 1/3.

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We already figured
out what that is.

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That's 2, because 2 to
the third power is 8.

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So this is 2 squared.

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Now, what is 64 to the 1/3?

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What times itself 3 times
is equal to 64?

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Well, 4 times 4, times 4 is
equal to 64, or 4 to the third

00:09:39.830 --> 00:09:46.230
is equal to 64, which means
that 4 is equal

00:09:46.230 --> 00:09:49.360
to 64 to the 1/3.

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So this is equal to 4.

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And, lucky for us, 2 squared
is also equal to 4.

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So it doesn't matter which
way you do it.

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You could take the square and
then the third root, or you

00:09:59.760 --> 00:10:02.200
could take the third root
and then square it.

00:10:02.200 --> 00:10:05.740
You're going to get the
exact same answer.

00:10:05.740 --> 00:10:06.830
Now, everything I've
been doing has

00:10:06.830 --> 00:10:07.920
been with actual numbers.

00:10:07.920 --> 00:10:11.020
Let me do a couple of problems
that just bring everything

00:10:11.020 --> 00:10:14.662
we've done together
using variables.

00:10:14.662 --> 00:10:18.180
So let's say we wanted to do a
few expressions and we want to

00:10:18.180 --> 00:10:19.220
make sure there are no negative

00:10:19.220 --> 00:10:20.980
exponents in the answer.

00:10:20.980 --> 00:10:28.140
So let's add x to the negative
3 over x to the negative 7.

00:10:28.140 --> 00:10:30.410
There's a bunch of ways
we could view this.

00:10:30.410 --> 00:10:35.460
We could view this as equal to
x to the negative 3, times 1

00:10:35.460 --> 00:10:38.390
over x to the negative 7.

00:10:38.390 --> 00:10:42.150
And what is 1 over x
to the negative 7?

00:10:42.150 --> 00:10:49.890
This is the same thing as x to
the seventh power, right?

00:10:49.890 --> 00:10:52.570
If you have 1 over something,
you can get rid of the 1 over,

00:10:52.570 --> 00:10:54.370
and put a negative in front
of the exponent.

00:10:54.370 --> 00:10:55.630
But if you're putting a
negative in front of a

00:10:55.630 --> 00:10:58.020
negative 7, you're going
to get x to a seventh.

00:10:58.020 --> 00:11:02.550
So this thing can simplify to
x to the negative 3, times x

00:11:02.550 --> 00:11:04.540
to the seventh power.

00:11:04.540 --> 00:11:07.480
And then we can add the
exponents, and that is x to

00:11:07.480 --> 00:11:08.870
the fourth power.

00:11:08.870 --> 00:11:12.150
Now, another way, a completely
legitimate way we could have

00:11:12.150 --> 00:11:16.300
done this, is we could have just
subtracted the exponents.

00:11:16.300 --> 00:11:18.520
We could have said, well, gee,
this is the same base.

00:11:18.520 --> 00:11:22.660
This is going to be x to the
negative 3, minus negative

00:11:22.660 --> 00:11:24.390
seventh power.

00:11:24.390 --> 00:11:27.390
Well, negative 3 minus negative
7, that's a negative

00:11:27.390 --> 00:11:33.920
3 plus 7 which is equal to
x to the fourth power.

00:11:33.920 --> 00:11:36.600
And then one final way-- I mean,
actually, there's more

00:11:36.600 --> 00:11:38.275
than one final way we could
have done this.

00:11:38.275 --> 00:11:41.280
We could have said x to the
negative 3 over x to the

00:11:41.280 --> 00:11:46.140
negative 7-- sorry, not negative
x-- over x to the

00:11:46.140 --> 00:11:47.370
negative 7.

00:11:47.370 --> 00:11:50.220
Well, x to the negative 3 is the
same thing as 1 over x to

00:11:50.220 --> 00:11:55.710
the third-- that's that term
right there-- times 1 over x

00:11:55.710 --> 00:11:59.330
to the negative 7, so this would
have been equal to 1

00:11:59.330 --> 00:12:03.350
over x to the third times
x to the negative 7.

00:12:03.350 --> 00:12:07.470
You could add the exponents,
so that's equal to 1 over 3

00:12:07.470 --> 00:12:10.350
minus 7 is x to the
negative 4.

00:12:10.350 --> 00:12:13.480
And then this-- if we just get
rid of the inverse, we take

00:12:13.480 --> 00:12:15.890
the inverse of it, we can put
a negative in front of this

00:12:15.890 --> 00:12:18.480
negative, making it a positive--
this is going to be

00:12:18.480 --> 00:12:20.510
equal to x to the 4.

00:12:20.510 --> 00:12:22.960
So no matter how we did it, as
long as we're consistent with

00:12:22.960 --> 00:12:27.150
the rules, we got
x to the fourth.

00:12:27.150 --> 00:12:29.770
Let's do one more slightly
hairy one.

00:12:29.770 --> 00:12:32.710
And then I think we'll
be done for now.

00:12:32.710 --> 00:12:43.690
Let's say we have 3x squared
times y to the 3/2 power.

00:12:43.690 --> 00:12:51.250
And we're going to divide it by
x times y to the 1/2 power.

00:12:51.250 --> 00:12:56.130
Well, once again, this is the
same thing as 3 times the x

00:12:56.130 --> 00:13:03.070
terms right here, so 3 times x
squared over x, times y to the

00:13:03.070 --> 00:13:04.690
3/2 over y to the 1/2.

00:13:10.000 --> 00:13:14.260
Well, this is going to be equal
to 3 times-- what's x

00:13:14.260 --> 00:13:15.920
squared over x?

00:13:15.920 --> 00:13:18.910
Or x squared over x to
the first power?

00:13:18.910 --> 00:13:24.080
That's going to be equal
to x to the 2 minus 1.

00:13:24.080 --> 00:13:31.520
And then this is going to be
times y to the 3/2 minus 1/2.

00:13:31.520 --> 00:13:32.790
So what does the whole
thing become?

00:13:32.790 --> 00:13:35.990
It becomes 3 times x.

00:13:35.990 --> 00:13:41.520
2 minus 1 is just 1-- I can just
write x there-- times 3/2

00:13:41.520 --> 00:13:43.980
minus 1/2 is 2/2.

00:13:43.980 --> 00:13:45.670
So that's y to the 2/2.

00:13:45.670 --> 00:13:49.570
2/2, or 2 2ths-- that's just
the same thing is y.

00:13:49.570 --> 00:13:53.040
so this is equal to 3xy.

00:13:53.040 --> 00:13:54.580
Anyway, I encourage you
to do many, many

00:13:54.580 --> 00:13:55.800
more examples of that.

00:13:55.800 --> 00:13:57.880
But, you'll see that just using
the rules that we've

00:13:57.880 --> 00:14:00.350
been exposed to in the last
few videos, you can pretty

00:14:00.350 --> 00:14:03.760
much simplify any exponent
expression.


Exponent properties involving quotients (examples)

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.480
Let's do some exponent examples

00:00:02.480 --> 00:00:04.420
that involve division.

00:00:04.420 --> 00:00:08.340
Let's say I were to ask you
what 5 to the sixth power

00:00:08.340 --> 00:00:12.370
divided by 5 to the
second power is?

00:00:12.370 --> 00:00:15.700
Well, we can just go to the
basic definition of what an

00:00:15.700 --> 00:00:19.420
exponent represents and say 5
to the sixth power, that's

00:00:19.420 --> 00:00:26.470
going to be 5 times 5 times
5 times 5 times 5-- one

00:00:26.470 --> 00:00:27.730
more 5-- times 5.

00:00:27.730 --> 00:00:30.030
5 times itself six times.

00:00:30.030 --> 00:00:33.720
And 5 squared, that's just 5
times itself two times, so

00:00:33.720 --> 00:00:36.770
it's just going to
be 5 times 5.

00:00:36.770 --> 00:00:40.380
Well, we know how to simplify
a fraction or a rational

00:00:40.380 --> 00:00:41.620
expression like this.

00:00:41.620 --> 00:00:45.470
We can divide the numerator and
the denominator by one 5,

00:00:45.470 --> 00:00:47.900
and then these will cancel out,
and then we can do it by

00:00:47.900 --> 00:00:51.110
another 5, or this 5 and
this 5 will cancel out.

00:00:51.110 --> 00:00:52.960
And what are we going
to be left with?

00:00:52.960 --> 00:00:58.220
5 times 5 times 5 times 5 over
1, or you could say that this

00:00:58.220 --> 00:01:01.540
is just 5 to the fourth power.

00:01:01.540 --> 00:01:03.320
Now, notice what happens.

00:01:03.320 --> 00:01:07.060
Essentially we started with six
in the numerator, six 5's

00:01:07.060 --> 00:01:10.580
multiplied by themselves in
the numerator, and then we

00:01:10.580 --> 00:01:11.720
subtracted out.

00:01:11.720 --> 00:01:14.520
We were able to cancel out
the 2 in the denominator.

00:01:14.520 --> 00:01:21.450
So this really was equal to 5
to the sixth power minus 2.

00:01:21.450 --> 00:01:24.020
So we were able to subtract
the exponent in the

00:01:24.020 --> 00:01:26.620
denominator from the exponent
in the numerator.

00:01:26.620 --> 00:01:29.610
Let's remember how this relates
to multiplication.

00:01:29.610 --> 00:01:32.790
If I had 5 to the-- let me do
this in a different color.

00:01:32.790 --> 00:01:41.570
5 to the sixth times 5 to the
second power, we saw in the

00:01:41.570 --> 00:01:48.330
last video that this is equal
to 5 to the 6 plus-- I'm

00:01:48.330 --> 00:01:51.970
trying to make it color coded
for you-- 6 plus 2 power.

00:01:51.970 --> 00:01:53.670
Now, we see a new property.

00:01:53.670 --> 00:01:55.370
And in the next video, we're
going see that these aren't

00:01:55.370 --> 00:01:56.940
really different properties.

00:01:56.940 --> 00:01:59.210
They're really kind of same
sides of the same coin when we

00:01:59.210 --> 00:02:01.520
learn about negative
exponents.

00:02:01.520 --> 00:02:09.250
But now in this video, we just
saw that 5 to the sixth power

00:02:09.250 --> 00:02:17.040
divided by 5 to the second
power-- let me do it in a

00:02:17.040 --> 00:02:26.610
different color-- is going to
be equal to 5 to the-- it's

00:02:26.610 --> 00:02:29.830
time consuming to make it color
coded for you-- 6 minus

00:02:29.830 --> 00:02:35.580
2 power or 5 to the
fourth power.

00:02:35.580 --> 00:02:37.420
Here it's going to be
5 to the eighth.

00:02:37.420 --> 00:02:40.095
So when you multiply exponents
with the same base, you add

00:02:40.095 --> 00:02:40.830
the exponents.

00:02:40.830 --> 00:02:45.270
When you divide with the same
base, you subtract the

00:02:45.270 --> 00:02:48.870
denominator exponent from
the numerator exponent.

00:02:48.870 --> 00:02:54.250
Let's do a bunch more of these
examples right here.

00:02:54.250 --> 00:02:59.470
What is 6 to the seventh
power divided by 6

00:02:59.470 --> 00:03:01.710
to the third power?

00:03:01.710 --> 00:03:04.060
Well, once again, we can
just use this property.

00:03:04.060 --> 00:03:08.190
This going to be 6 to the 7
minus 3 power, which is equal

00:03:08.190 --> 00:03:11.430
to 6 to the fourth power.

00:03:11.430 --> 00:03:13.540
And you can multiply it out this
way like we did in the

00:03:13.540 --> 00:03:17.680
first problem and verify that
it indeed will be 6 to the

00:03:17.680 --> 00:03:18.840
fourth power.

00:03:18.840 --> 00:03:22.460
Now let's try something
interesting.

00:03:22.460 --> 00:03:26.350
This will be a good segue
into the next video.

00:03:26.350 --> 00:03:32.140
Let's say we have 3 to the
fourth power divided by 3 to

00:03:32.140 --> 00:03:33.960
the tenth power.

00:03:33.960 --> 00:03:37.240
Well, if we just go from basic
principles, this would be 3

00:03:37.240 --> 00:03:43.100
times 3 times 3 times 3, all of
that over 3 times 3-- we're

00:03:43.100 --> 00:03:47.040
going to have ten of these--
3 times 3 times 3 times 3

00:03:47.040 --> 00:03:48.920
times 3 times 3.

00:03:48.920 --> 00:03:49.310
How many is that?

00:03:49.310 --> 00:03:54.960
One, two, three, four, five,
six, seven, eight, nine, ten.

00:03:54.960 --> 00:03:58.160
Well, if we do what we did in
the last video, this 3 cancels

00:03:58.160 --> 00:03:58.910
with that 3.

00:03:58.910 --> 00:03:59.790
Those 3's cancel.

00:03:59.790 --> 00:04:00.570
Those 3's cancel.

00:04:00.570 --> 00:04:01.740
Those 3's cancel.

00:04:01.740 --> 00:04:05.590
And we're left with 1 over--
one, two, three,

00:04:05.590 --> 00:04:07.730
four, five, six 3's.

00:04:07.730 --> 00:04:12.090
So 1 over 3 to the sixth
power, right?

00:04:12.090 --> 00:04:14.970
We have 1 over all of
these 3's down here.

00:04:14.970 --> 00:04:18.029
But that property that I just
told you, would have told you

00:04:18.029 --> 00:04:24.820
that this should also be equal
to 3 to the 4 minus 10 power.

00:04:24.820 --> 00:04:25.010
Well.

00:04:25.010 --> 00:04:26.590
What's 4 minus 10?

00:04:26.590 --> 00:04:27.630
Well, you're going to get
a negative number.

00:04:27.630 --> 00:04:30.690
This is 3 to the negative
sixth power.

00:04:30.690 --> 00:04:33.350
So using the property we just
saw, you'd get 3 to the

00:04:33.350 --> 00:04:34.670
negative sixth power.

00:04:34.670 --> 00:04:37.400
Just multiplying them out,
you get 1 over 3

00:04:37.400 --> 00:04:38.680
to the sixth power.

00:04:38.680 --> 00:04:41.690
And the fun part about
all of this is these

00:04:41.690 --> 00:04:43.100
are the same quantity.

00:04:43.100 --> 00:04:44.930
So now you're learning a little
bit about what it means

00:04:44.930 --> 00:04:46.520
to take a negative exponent.

00:04:46.520 --> 00:04:52.040
3 to the negative sixth power
is equal to 1 over 3 to the

00:04:52.040 --> 00:04:52.630
sixth power.

00:04:52.630 --> 00:04:54.990
And I'm going do many, many more
examples of this in the

00:04:54.990 --> 00:04:55.790
next video.

00:04:55.790 --> 00:04:58.520
But if you take anything to the
negative power, so a to

00:04:58.520 --> 00:05:04.110
the negative b power is equal
to 1 over a to the b.

00:05:04.110 --> 00:05:07.230
That's one thing that we just
established just now.

00:05:07.230 --> 00:05:11.770
And earlier in this video, we
saw that if I have a to the b

00:05:11.770 --> 00:05:17.670
over a to the c, that this is
equal to a to the b minus c.

00:05:17.670 --> 00:05:21.010
That's the other property
we've been using.

00:05:21.010 --> 00:05:24.510
Now, using what we've just
learned and what we learned in

00:05:24.510 --> 00:05:32.160
the last video, let's do some
more complicated problems.

00:05:32.160 --> 00:05:37.630
Let's say I have a to the third,
b to the fourth power

00:05:37.630 --> 00:05:44.180
over a squared b, and all of
that to the third power.

00:05:44.180 --> 00:05:47.290
Well, we can use the property
we just learned to simplify

00:05:47.290 --> 00:05:48.010
the inside.

00:05:48.010 --> 00:05:50.800
This is going to be equal
to-- a to the third

00:05:50.800 --> 00:05:52.650
divided by a squared.

00:05:52.650 --> 00:05:56.110
That's a to the 3 minus
2 power, right?

00:05:56.110 --> 00:05:59.190
So this would simplify
to just an a.

00:05:59.190 --> 00:06:01.150
And you could imagine, this
is a times a times a

00:06:01.150 --> 00:06:02.690
divided by a times a.

00:06:02.690 --> 00:06:04.260
You'll just have an a on top.

00:06:04.260 --> 00:06:09.080
And then the b, b to the fourth
divided by b, well,

00:06:09.080 --> 00:06:12.630
that's just going to be
b to the third, right?

00:06:12.630 --> 00:06:13.990
This is b to the first power.

00:06:13.990 --> 00:06:18.810
4 minus 1 is 3, and then all of
that in parentheses to the

00:06:18.810 --> 00:06:20.140
third power.

00:06:20.140 --> 00:06:22.060
We don't want to forget about
this third power out here.

00:06:22.060 --> 00:06:24.220
This third power is this one.

00:06:24.220 --> 00:06:25.110
Let me color code it.

00:06:25.110 --> 00:06:28.485
That third power is that one
right there, and then this a

00:06:28.485 --> 00:06:30.990
in orange is that
a right there.

00:06:30.990 --> 00:06:33.280
I think we understand
what maps to what.

00:06:33.280 --> 00:06:35.850
And now we can use the property
that when we multiply

00:06:35.850 --> 00:06:41.730
something and take it to the
third power, this is equal to

00:06:41.730 --> 00:06:48.820
a to the third power times
b to the third

00:06:48.820 --> 00:06:52.140
to the third power.

00:06:52.140 --> 00:06:57.700
And then this is going to be
equal to a to the third power.

00:07:03.770 --> 00:07:09.880
times b to the 3 times 3 power,
times b to the ninth.

00:07:09.880 --> 00:07:11.830
And we would have simplified
this about as

00:07:11.830 --> 00:07:14.680
far as you can go.

00:07:14.680 --> 00:07:16.240
Let's do one more of these.

00:07:16.240 --> 00:07:18.980
I think they're good practice
and super-valuable

00:07:18.980 --> 00:07:21.150
experience later on.

00:07:21.150 --> 00:07:33.220
Let's say I have 25xy to
the sixth over 20y

00:07:33.220 --> 00:07:40.190
to the fifth x squared.

00:07:40.190 --> 00:07:42.315
So once again, we can rearrange
the numerators and

00:07:42.315 --> 00:07:43.400
the denominators.

00:07:43.400 --> 00:07:50.890
So this you could rewrite as
25 over 20 times x over x

00:07:50.890 --> 00:07:52.940
squared, right?

00:07:52.940 --> 00:07:56.170
We could have made this bottom
20x squared y to the fifth--

00:07:56.170 --> 00:07:59.850
it doesn't matter the order we
do it in-- times y to the

00:07:59.850 --> 00:08:03.140
sixth over y to the fifth.

00:08:03.140 --> 00:08:06.470
And let's use our newly learned
exponent properties in

00:08:06.470 --> 00:08:08.150
actually just simplify
fractions.

00:08:08.150 --> 00:08:13.220
25 over 20, if you divide
them both by 5, this is

00:08:13.220 --> 00:08:17.240
equal to 5 over 4.

00:08:17.240 --> 00:08:20.990
x divided by x squared-- well,
there's two ways you could

00:08:20.990 --> 00:08:21.880
think about it.

00:08:21.880 --> 00:08:24.890
That you could view as
x to the negative 1.

00:08:24.890 --> 00:08:26.080
You have a first power here.

00:08:26.080 --> 00:08:28.240
1 minus 2 is negative 1.

00:08:28.240 --> 00:08:34.169
So this right here is equal to
x to the negative 1 power.

00:08:34.169 --> 00:08:36.169
Or it could also be
equal to 1 over x.

00:08:36.169 --> 00:08:37.620
These are equivalent.

00:08:37.620 --> 00:08:40.690
So let's say that this
is equal into 1 over

00:08:40.690 --> 00:08:42.510
x, just like that.

00:08:42.510 --> 00:08:44.900
And it would be. x
over x times x.

00:08:44.900 --> 00:08:47.050
One of those sets of x's would
cancel out and you're just

00:08:47.050 --> 00:08:48.560
left with 1 over x.

00:08:48.560 --> 00:08:52.160
And then finally, y to the sixth
over y to the fifth,

00:08:52.160 --> 00:08:57.540
that's y to the 6 minus 5 power,
which is just y to the

00:08:57.540 --> 00:09:02.230
first power, or just
y, so times y.

00:09:02.230 --> 00:09:04.730
So if you want to write it all
out as just one combined

00:09:04.730 --> 00:09:08.410
rational expression, you have 5
times 1 times y, which would

00:09:08.410 --> 00:09:13.690
be 5y, all of that over
4 times x, right?

00:09:13.690 --> 00:09:18.930
This is y over 1, so 4 times x
times 1, all of that over 4x,

00:09:18.930 --> 00:09:22.240
and we have successfully
simplified it.


Exponent properties involving products

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:03.840
In this video, I want to do a
bunch of examples involving

00:00:03.840 --> 00:00:05.200
exponent properties.

00:00:05.200 --> 00:00:07.860
But, before I even do that,
let's have a little bit of a

00:00:07.860 --> 00:00:10.970
review of what an exponent
even is.

00:00:10.970 --> 00:00:15.380
So let's say I had 2
to the third power.

00:00:15.380 --> 00:00:17.630
You might be tempted to
say, oh is that 6?

00:00:17.630 --> 00:00:20.010
And I would say no,
it is not 6.

00:00:20.010 --> 00:00:23.580
This means 2 times itself,
three times.

00:00:23.580 --> 00:00:29.940
So this is going to be equal to
2 times 2 times 2, which is

00:00:29.940 --> 00:00:32.549
equal to 2 times 2 is 4.

00:00:32.549 --> 00:00:35.220
4 times 2 is equal to 8.

00:00:35.220 --> 00:00:40.090
If I were to ask you what 3 to
the second power is, or 3

00:00:40.090 --> 00:00:45.110
squared, this is equal to 3
times itself two times.

00:00:45.110 --> 00:00:48.220
This is equal to 3 times 3.

00:00:48.220 --> 00:00:50.030
Which is equal to 9.

00:00:50.030 --> 00:00:51.060
Let's do one more of these.

00:00:51.060 --> 00:00:54.080
I think you're getting the
general sense, if you've never

00:00:54.080 --> 00:00:56.080
seen these before.

00:00:56.080 --> 00:01:03.020
Let's say I have 5 to
the seventh power.

00:01:03.020 --> 00:01:07.120
That's equal to 5 times
itself, seven times.

00:01:07.120 --> 00:01:16.280
5 times 5 times 5 times 5
times 5 times 5 times 5.

00:01:16.280 --> 00:01:16.960
That's seven, right?

00:01:16.960 --> 00:01:19.200
One, two, three, four,
five, six, seven.

00:01:19.200 --> 00:01:21.560
This is going to be a really,
really, really, really, large

00:01:21.560 --> 00:01:23.710
number and I'm not going to
calculate it right now.

00:01:23.710 --> 00:01:26.470
If you want to do it by hand,
feel free to do so.

00:01:26.470 --> 00:01:28.620
Or use a calculator, but this
is a really, really, really,

00:01:28.620 --> 00:01:29.320
large number.

00:01:29.320 --> 00:01:32.400
So one thing that you might
appreciate very quickly is

00:01:32.400 --> 00:01:35.000
that exponents increase
very rapidly.

00:01:35.000 --> 00:01:39.700
5 to the 17th would be even a
way, way more massive number.

00:01:39.700 --> 00:01:41.920
But anyway, that's a review
of exponents.

00:01:41.920 --> 00:01:47.130
Let's get a little bit steeped
in algebra, using exponents.

00:01:47.130 --> 00:01:52.610
So what would 3x-- let me do
this in a different color--

00:01:52.610 --> 00:01:59.670
what would 3x times
3x times 3x be?

00:01:59.670 --> 00:02:02.340
Well, one thing you need to
remember about multiplication

00:02:02.340 --> 00:02:04.990
is, it doesn't matter what
order you do the

00:02:04.990 --> 00:02:06.060
multiplication in.

00:02:06.060 --> 00:02:10.440
So this is going to be the same
thing as 3 times 3 times

00:02:10.440 --> 00:02:16.550
3 times x times x times x.

00:02:16.550 --> 00:02:20.620
And just based on what we
reviewed just here, that part

00:02:20.620 --> 00:02:24.160
right there, 3 times 3,
three times, that's 3

00:02:24.160 --> 00:02:25.780
to the third power.

00:02:25.780 --> 00:02:30.210
And this right here, x times
itself three times.

00:02:30.210 --> 00:02:32.750
that's x to the third power.

00:02:32.750 --> 00:02:36.050
So this whole thing can be
rewritten as 3 to the third

00:02:36.050 --> 00:02:37.570
times x to the third.

00:02:37.570 --> 00:02:40.960
Or if you know what 3 to the
third is, this is 9 times 3,

00:02:40.960 --> 00:02:42.240
which is 27.

00:02:42.240 --> 00:02:46.180
This is 27 x to the
third power.

00:02:46.180 --> 00:02:49.840
Now you might have said, hey,
wasn't 3x times 3x times 3x.

00:02:49.840 --> 00:02:52.830
Wasn't that 3x to
the third power?

00:02:52.830 --> 00:02:53.140
Right?

00:02:53.140 --> 00:02:55.480
You're multiplying 3x times
itself three times.

00:02:55.480 --> 00:02:57.560
And I would say, yes it is.

00:02:57.560 --> 00:03:03.210
So this, right here, you could
interpret that as 3x to the

00:03:03.210 --> 00:03:04.730
third power.

00:03:04.730 --> 00:03:06.950
And just like that, we stumbled
on one of our

00:03:06.950 --> 00:03:08.280
exponent properties.

00:03:08.280 --> 00:03:09.290
Notice this.

00:03:09.290 --> 00:03:11.600
When I have something times
something, and the whole thing

00:03:11.600 --> 00:03:15.430
is to the third power, that
equals each of those things to

00:03:15.430 --> 00:03:17.490
the third power times
each other.

00:03:17.490 --> 00:03:21.290
So 3x to the third is the same
thing is 3 to the third times

00:03:21.290 --> 00:03:27.860
x to the third, which is
27 to the third power.

00:03:27.860 --> 00:03:30.730
Let's do a couple
more examples.

00:03:30.730 --> 00:03:39.050
What if I were to ask you what
6 to the third times 6 to the

00:03:39.050 --> 00:03:40.410
sixth power is?

00:03:40.410 --> 00:03:42.860
And this is going to be a really
huge number, but I want

00:03:42.860 --> 00:03:45.480
to write it as a power of 6.

00:03:45.480 --> 00:03:50.290
Let me write the 6 to the sixth
in a different color.

00:03:50.290 --> 00:03:53.460
6 to the third times 6 to the
sixth power, what is this

00:03:53.460 --> 00:03:55.540
going to be equal to?

00:03:55.540 --> 00:03:58.200
Well, 6 to the third, we
know that's 6 times

00:03:58.200 --> 00:03:59.880
itself three times.

00:03:59.880 --> 00:04:03.400
So it's 6 times 6 times 6.

00:04:03.400 --> 00:04:07.440
And then that's going to be
times-- the times here is in

00:04:07.440 --> 00:04:09.060
green, so I'll do it in green.

00:04:09.060 --> 00:04:11.430
Maybe I'll make both
of them in orange.

00:04:11.430 --> 00:04:15.990
That is going to be times
6 to the sixth power.

00:04:15.990 --> 00:04:17.430
Well, what's 6 to
the sixth power?

00:04:17.430 --> 00:04:20.540
That's 6 times itself
six times.

00:04:20.540 --> 00:04:27.180
So, it's 6 times 6 times
6 times 6 times 6.

00:04:27.180 --> 00:04:29.380
Then you get one
more, times 6.

00:04:29.380 --> 00:04:31.620
So what is this whole
number going to be?

00:04:31.620 --> 00:04:34.740
Well, this whole thing-- we're
multiplying 6 times itself--

00:04:34.740 --> 00:04:35.560
how many times?

00:04:35.560 --> 00:04:38.650
One, two, three, four,
five, six, seven,

00:04:38.650 --> 00:04:40.640
eight, nine times, right?

00:04:40.640 --> 00:04:44.450
Three times here and then
another six times here.

00:04:44.450 --> 00:04:47.450
So we're multiplying 6 times
itself nine times.

00:04:47.450 --> 00:04:48.910
3 plus 6.

00:04:48.910 --> 00:04:55.360
So this is equal to 6 to
the 3 plus 6 power or 6

00:04:55.360 --> 00:04:57.100
to the ninth power.

00:04:57.100 --> 00:04:58.650
And just like that,
we/ve stumbled on

00:04:58.650 --> 00:05:01.130
another exponent property.

00:05:01.130 --> 00:05:04.100
When we take exponents, in this
case, 6 to the third, the

00:05:04.100 --> 00:05:05.760
number 6 is the base.

00:05:05.760 --> 00:05:09.190
We're taking the base to
the exponent of 3.

00:05:09.190 --> 00:05:11.900
When you have the same base,
and you're multiplying two

00:05:11.900 --> 00:05:17.360
exponents with the same base,
you can add the exponents.

00:05:17.360 --> 00:05:19.850
Let me do several more
examples of this.

00:05:19.850 --> 00:05:23.020
Let's do it in magenta.

00:05:23.020 --> 00:05:29.420
Let's say I had 2 squared times
2 to the fourth times 2

00:05:29.420 --> 00:05:31.030
to the sixth.

00:05:31.030 --> 00:05:33.840
Well, I have the same base
in all of these, so

00:05:33.840 --> 00:05:35.340
I can add the exponents.

00:05:35.340 --> 00:05:41.490
This is going to be equal to 2
to the 2 plus 4 plus 6, which

00:05:41.490 --> 00:05:44.970
is equal to 2 to
the 12th power.

00:05:44.970 --> 00:05:46.680
And hopefully that makes sense,
because this is going

00:05:46.680 --> 00:05:51.440
to be 2 times itself two times,
2 times itself four

00:05:51.440 --> 00:05:53.880
times, 2 times itself
six times.

00:05:53.880 --> 00:05:56.000
When you multiply them all out,
it's going to be 2 times

00:05:56.000 --> 00:06:00.350
itself, 12 times or 2
to the 12th power.

00:06:00.350 --> 00:06:04.740
Let's do it in a little bit more
abstract way, using some

00:06:04.740 --> 00:06:08.150
variables, but it's the
same exact idea.

00:06:08.150 --> 00:06:14.570
What is x to the squared or x
squared times x to the fourth?

00:06:14.570 --> 00:06:16.610
Well, we could use the property
we just learned.

00:06:16.610 --> 00:06:19.060
We have the exact
same base, x.

00:06:19.060 --> 00:06:22.300
So it's going to be x to
the 2 plus 4 power.

00:06:22.300 --> 00:06:25.100
It's going to be x to
the sixth power.

00:06:25.100 --> 00:06:29.380
And if you don't believe
me, what is x squared?

00:06:29.380 --> 00:06:33.210
x squared is equal
to x times x.

00:06:33.210 --> 00:06:37.110
And if you were going to
multiply that times x to the

00:06:37.110 --> 00:06:41.630
fourth, you're multiplying it by
x times itself four times.

00:06:41.630 --> 00:06:46.200
x times x times x times x.

00:06:46.200 --> 00:06:49.560
So how many times are you now
multiplying x by itself?

00:06:49.560 --> 00:06:53.470
Well, one, two, three, four,
five, six times.

00:06:53.470 --> 00:06:56.280
x to the sixth power.

00:06:56.280 --> 00:06:58.470
Let's do another one of these.

00:06:58.470 --> 00:07:02.280
The more examples you see,
I figure, the better.

00:07:02.280 --> 00:07:05.760
So let's do the other
property, just to

00:07:05.760 --> 00:07:06.590
mix and match it.

00:07:06.590 --> 00:07:14.320
Let's say I have a to the third
to the fourth power.

00:07:14.320 --> 00:07:16.610
So I'll tell you the property
here, and I'll show you why it

00:07:16.610 --> 00:07:17.450
makes sense.

00:07:17.450 --> 00:07:19.880
When you add something to an
exponent, and then you raise

00:07:19.880 --> 00:07:22.680
that to an exponent, you can
multiply the exponents.

00:07:22.680 --> 00:07:28.160
So this is going to be a to the
3 times 4 power or a to

00:07:28.160 --> 00:07:30.020
the 12th power.

00:07:30.020 --> 00:07:31.730
And why does that make sense?

00:07:31.730 --> 00:07:35.340
Well this right here
is a to the third

00:07:35.340 --> 00:07:36.820
times itself four times.

00:07:36.820 --> 00:07:41.330
So this is equal to a to the
third times a to the third

00:07:41.330 --> 00:07:45.300
times a to the third times
a to the third.

00:07:45.300 --> 00:07:48.430
Well, we have the same base, so
we can add the exponents.

00:07:48.430 --> 00:07:52.620
So there's going to be a to
the 3 times 4, right?

00:07:52.620 --> 00:07:57.910
This is equal to a to the 3
plus 3 plus 3 plus 3

00:07:57.910 --> 00:08:03.600
power, which is the same thing
is a the 3 times 4 power or a

00:08:03.600 --> 00:08:05.010
to the 12th power.

00:08:05.010 --> 00:08:07.580
So just to review the properties
we've learned so

00:08:07.580 --> 00:08:11.560
far in this video, besides
just a review of what an

00:08:11.560 --> 00:08:19.830
exponent is, if I have x to the
a power times x to the b

00:08:19.830 --> 00:08:22.550
power, this is going
to be equal to x to

00:08:22.550 --> 00:08:25.120
the a plus b power.

00:08:25.120 --> 00:08:27.950
We saw that right here.

00:08:27.950 --> 00:08:30.615
x squared times x to the fourth
is equal to x to the

00:08:30.615 --> 00:08:32.490
sixth, 2 plus 4.

00:08:32.490 --> 00:08:39.500
We also saw that if I have x
times y to the a power, this

00:08:39.500 --> 00:08:42.440
is the same thing is
x to the a power

00:08:42.440 --> 00:08:44.680
times y to the a power.

00:08:44.680 --> 00:08:46.810
We saw that early on
in this video.

00:08:46.810 --> 00:08:49.570
We saw that over here.

00:08:49.570 --> 00:08:53.000
3x to the third is the same
thing as 3 to the third times

00:08:53.000 --> 00:08:55.130
x to the third.

00:08:55.130 --> 00:08:56.410
That's what this is
saying right here.

00:08:56.410 --> 00:09:00.460
3x to the third is the same
thing is 3 to the third times

00:09:00.460 --> 00:09:01.730
x to the third.

00:09:01.730 --> 00:09:05.720
And then the last property,
which we just stumbled upon

00:09:05.720 --> 00:09:10.710
is, if you have x to the a and
then you raise that to the bth

00:09:10.710 --> 00:09:15.430
power, that's equal to
x to the a times b.

00:09:15.430 --> 00:09:18.460
And we saw that right there. a
to the third and then raise

00:09:18.460 --> 00:09:21.690
that to the fourth power is the
same thing is a to the 3

00:09:21.690 --> 00:09:25.280
times 4 or a to the
12th power.

00:09:25.280 --> 00:09:32.970
So let's use these properties
to do a handful of more

00:09:32.970 --> 00:09:48.070
complex problems. Let's say
we have 2xy squared times

00:09:48.070 --> 00:09:55.060
negative x squared
y squared times

00:09:55.060 --> 00:09:57.900
three x squared y squared.

00:09:57.900 --> 00:09:59.330
And we wanted to
simplify this.

00:10:04.980 --> 00:10:07.550
This you can view as
negative 1 times x

00:10:07.550 --> 00:10:08.930
squared times y squared.

00:10:08.930 --> 00:10:13.350
So if we take this whole thing
to the squared power, this is

00:10:13.350 --> 00:10:15.670
like raising each of these
to the second power.

00:10:15.670 --> 00:10:19.590
So this part right here could
be simplified as negative 1

00:10:19.590 --> 00:10:26.320
squared times x squared squared,
times y squared.

00:10:26.320 --> 00:10:28.270
And then if we were to simplify
that, negative 1

00:10:28.270 --> 00:10:31.890
squared is just 1, x squared
squared-- remember you can

00:10:31.890 --> 00:10:35.100
just multiply the exponents-- so
that's going to be x to the

00:10:35.100 --> 00:10:37.770
fourth y squared.

00:10:37.770 --> 00:10:40.345
That's what this middle
part simplifies to.

00:10:40.345 --> 00:10:43.760
And let's see if we can merge
it with the other parts.

00:10:43.760 --> 00:10:47.410
The other parts, just to
remember, were 2 xy squared,

00:10:47.410 --> 00:10:51.350
and then 3x squared y squared.

00:10:51.350 --> 00:10:53.380
Well now we're just going ahead
and just straight up

00:10:53.380 --> 00:10:54.400
multiplying everything.

00:10:54.400 --> 00:10:56.710
And we learned in multiplication
that it doesn't

00:10:56.710 --> 00:10:58.860
matter which order you
multiply things in.

00:10:58.860 --> 00:10:59.930
So I can just rearrange.

00:10:59.930 --> 00:11:02.150
We're just going and multiplying
2 times x times y

00:11:02.150 --> 00:11:04.600
squared times x to the fourth
times y squared times 3 times

00:11:04.600 --> 00:11:05.550
x squared times y squared.

00:11:05.550 --> 00:11:10.290
So I can rearrange this, and I
will rearrange it so that it's

00:11:10.290 --> 00:11:11.800
in a way that's easy
to simplify.

00:11:11.800 --> 00:11:16.995
So I can multiply 2 times 3, and
then I can worry about the

00:11:16.995 --> 00:11:18.610
x terms.

00:11:18.610 --> 00:11:19.960
Let me do it in this color.

00:11:19.960 --> 00:11:30.330
Then I have times x times x to
the fourth times x squared.

00:11:30.330 --> 00:11:36.080
And then I have to worry about
the y terms, times y squared

00:11:36.080 --> 00:11:40.036
times another y squared times
another y squared.

00:11:42.950 --> 00:11:44.600
And now what are
these equal to?

00:11:44.600 --> 00:11:45.880
Well, 2 times 3.

00:11:45.880 --> 00:11:47.200
You knew how to do that.

00:11:47.200 --> 00:11:49.330
That's equal to 6.

00:11:49.330 --> 00:11:53.450
And what is x times x to the
fourth times x squared.

00:11:53.450 --> 00:11:56.310
Well, one thing to remember is
x is the same thing as x to

00:11:56.310 --> 00:11:57.340
the first power.

00:11:57.340 --> 00:12:00.480
Anything to the first power
is just that number.

00:12:00.480 --> 00:12:03.130
So you know, 2 to the first
power is just 2.

00:12:03.130 --> 00:12:05.970
3 to the first power
is just 3.

00:12:05.970 --> 00:12:08.780
So what is this going
to be equal to?

00:12:08.780 --> 00:12:13.190
This is going to be equal to--
we have the same base, x.

00:12:13.190 --> 00:12:19.760
We can add the exponents, x to
the 1 plus 4 plus 2 power, and

00:12:19.760 --> 00:12:21.250
I'll add it in the next step.

00:12:21.250 --> 00:12:25.510
And then on the y's, this
is times y to the 2

00:12:25.510 --> 00:12:28.100
plus 2 plus 2 power.

00:12:28.100 --> 00:12:29.120
And what does that give us?

00:12:29.120 --> 00:12:36.840
That gives us 6 x to the
seventh power, y

00:12:36.840 --> 00:12:39.250
to the sixth power.

00:12:39.250 --> 00:12:41.310
And I'll just leave you with
some thing that you might

00:12:41.310 --> 00:12:43.290
already know, but it's
pretty interesting.

00:12:43.290 --> 00:12:44.900
And that's the question of what
happens when you take

00:12:44.900 --> 00:12:46.710
something to the zeroth power?

00:12:46.710 --> 00:12:50.360
So if I say 7 to the zeroth
power, What does that equal?

00:12:50.360 --> 00:12:52.350
And I'll tell you right now--
and this might seem very

00:12:52.350 --> 00:12:56.820
counterintuitive-- this is equal
to 1, or 1 to the zeroth

00:12:56.820 --> 00:12:58.740
power is also equal to 1.

00:12:58.740 --> 00:13:01.600
Anything that the zeroth power,
any non-zero number to

00:13:01.600 --> 00:13:06.060
the zero power is going
to be equal to 1.

00:13:06.060 --> 00:13:07.230
And just to give you
a little bit of

00:13:07.230 --> 00:13:09.320
intuition on why that is.

00:13:09.320 --> 00:13:12.460
Think about it this way.

00:13:12.460 --> 00:13:17.100
3 to the first power-- let me
write the powers-- 3 to the

00:13:17.100 --> 00:13:19.260
first, second, third.

00:13:19.260 --> 00:13:23.800
We'll just do it the
with the number 3.

00:13:23.800 --> 00:13:25.420
So 3 to the first power is 3.

00:13:25.420 --> 00:13:26.340
I think that makes sense.

00:13:26.340 --> 00:13:28.760
3 to the second power is 9.

00:13:28.760 --> 00:13:31.610
3 to the third power is 27.

00:13:31.610 --> 00:13:33.590
And of course, we're trying to
figure out what should 3 to

00:13:33.590 --> 00:13:35.700
the zeroth power be?

00:13:35.700 --> 00:13:36.410
Well, think about it.

00:13:36.410 --> 00:13:39.040
Every time you decrement
the exponent.

00:13:39.040 --> 00:13:41.680
Every time you take the exponent
down by 1, you are

00:13:41.680 --> 00:13:43.160
dividing by 3.

00:13:43.160 --> 00:13:45.640
To go from 27 to 9,
you divide by 3.

00:13:45.640 --> 00:13:48.000
To go from 9 to 3,
you divide by 3.

00:13:48.000 --> 00:13:50.400
So to go from this exponent to
that exponent, maybe we should

00:13:50.400 --> 00:13:51.630
divide by 3 again.

00:13:51.630 --> 00:13:54.950
And that's why, anything to
the zeroth power, in this

00:13:54.950 --> 00:13:58.050
case, 3 to the zeroth
power is 1.

00:13:58.050 --> 00:13:59.940
See you in the next video.


Systems of Linear Inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:02.180
Everything we've been
doing so far has

00:00:02.180 --> 00:00:04.830
involved systems of equations.

00:00:04.830 --> 00:00:08.460
In this video, we're going to
tackle the problem of systems

00:00:08.460 --> 00:00:12.100
of linear inequalities.

00:00:12.100 --> 00:00:14.480
What we're going to do is,
essentially, figure out all of

00:00:14.480 --> 00:00:18.960
the points on the xy plane
that satisfy a system of

00:00:18.960 --> 00:00:21.240
linear inequalities.

00:00:21.240 --> 00:00:25.680
So let's start with our first
linear inequality.

00:00:25.680 --> 00:00:34.600
Let's say we have 2x minus 3y
is less than or equal to 21.

00:00:34.600 --> 00:00:37.060
And we've seen how to
graph this before.

00:00:37.060 --> 00:00:40.600
You can just graph this line,
and then figure out what the

00:00:40.600 --> 00:00:42.180
less than part means.

00:00:42.180 --> 00:00:45.610
So let's just rearrange this
into slope-intercept form.

00:00:45.610 --> 00:00:56.810
So if you subtract 2x from both
sides, we get negative

00:00:56.810 --> 00:01:00.090
3y-- these cancel out
--is less than or

00:01:00.090 --> 00:01:01.850
equal to 21 minus 2x.

00:01:01.850 --> 00:01:06.010
Or we could say negative
2x plus 21.

00:01:06.010 --> 00:01:09.410
And then we can divide both
sides by negative 3.

00:01:09.410 --> 00:01:12.330
And remember, when you divide
by a negative number, or you

00:01:12.330 --> 00:01:17.180
multiply by a negative number,
you swap the inequality.

00:01:17.180 --> 00:01:23.430
So you get negative 3y over
negative 3 is greater than or

00:01:23.430 --> 00:01:29.500
equal to negative 2 over
negative 3x plus 21 over

00:01:29.500 --> 00:01:30.680
negative 3.

00:01:30.680 --> 00:01:35.080
And so this equation becomes y
is greater than or equal to

00:01:35.080 --> 00:01:39.420
2/3 x minus 7.

00:01:39.420 --> 00:01:42.150
So that's the first inequality
right there.

00:01:42.150 --> 00:01:43.430
And let's graph it.

00:01:43.430 --> 00:01:45.700
I'll graph it in orange.

00:01:45.700 --> 00:01:53.270
y-intercept is negative 7,
so 1, 2, 3, 4, 5, 6, 7.

00:01:53.270 --> 00:01:55.660
And it's slope is
positive 2/3.

00:01:55.660 --> 00:01:58.325
So when you run 3,
you go up 2.

00:02:01.450 --> 00:02:04.970
When you run 3, you go up 2.

00:02:04.970 --> 00:02:09.320
And in this direction, if you
go back 3, you go down 2, so

00:02:09.320 --> 00:02:12.390
the line will look something
like this.

00:02:12.390 --> 00:02:15.550
The line will look like that.

00:02:15.550 --> 00:02:16.820
Keep going.

00:02:16.820 --> 00:02:21.450
And since we have y is greater
than, or equal to,

00:02:21.450 --> 00:02:25.900
essentially, this line, the
area that satisfies this

00:02:25.900 --> 00:02:28.990
inequality is going to
be above the line.

00:02:28.990 --> 00:02:33.290
At any point, for any x
you pick, this is the

00:02:33.290 --> 00:02:36.240
value 2/3 x minus 7.

00:02:36.240 --> 00:02:38.395
So if y is going to be greater
than that, it'll be all the

00:02:38.395 --> 00:02:39.920
points above that.

00:02:39.920 --> 00:02:43.560
So just this constraint right
here, y is greater than 2/3 x

00:02:43.560 --> 00:02:47.020
minus 7, or this constraint,
which is the same thing, 2x

00:02:47.020 --> 00:02:51.220
minus 3y is less than or equal
to 21, will give you all of

00:02:51.220 --> 00:02:52.410
this area up here.

00:02:52.410 --> 00:02:55.400
I don't want to draw it
too dark just yet.

00:02:55.400 --> 00:02:57.150
So it'll give you all
of this area all the

00:02:57.150 --> 00:02:58.950
way above this line.

00:02:58.950 --> 00:03:02.860
Now let's add another inequality
to the mix.

00:03:02.860 --> 00:03:05.840
Let me do that other
inequality in blue.

00:03:05.840 --> 00:03:11.210
Let's say I have x plus 4y.

00:03:11.210 --> 00:03:12.770
Let's make it less than 6.

00:03:12.770 --> 00:03:14.730
The problem I have written here,
has less than or equal

00:03:14.730 --> 00:03:18.820
to, but I want to mix it up
with an equal sign and a

00:03:18.820 --> 00:03:20.590
couple of them without
an equal sign.

00:03:20.590 --> 00:03:23.590
Just so we remember how
to graph these things.

00:03:23.590 --> 00:03:26.970
And so let's put this again into
slope-intercept form, so

00:03:26.970 --> 00:03:30.490
we can subtract x
from both sides.

00:03:30.490 --> 00:03:33.280
Then the left-hand side
just becomes 4y.

00:03:33.280 --> 00:03:36.930
The right-hand side becomes
a negative x plus 6.

00:03:36.930 --> 00:03:39.865
And then we can divide
both sides by 4.

00:03:42.560 --> 00:03:50.350
And we get y is less than
negative 1/4 x plus 6.

00:03:50.350 --> 00:03:51.960
So just like the
first equation,

00:03:51.960 --> 00:03:54.940
let's graph the line.

00:03:54.940 --> 00:03:56.910
The y-intercept is positive 6.

00:03:56.910 --> 00:04:01.670
So 1, 2, 3, 4, 5, 6.

00:04:01.670 --> 00:04:04.320
And then the slope is
a negative 1/4.

00:04:04.320 --> 00:04:07.550
So every time we run by
4, 1, 2, 3, 4, we're

00:04:07.550 --> 00:04:09.360
going to go down 1.

00:04:09.360 --> 00:04:13.590
So you run by 4, 1, 2,
3, 4, you go down 1.

00:04:13.590 --> 00:04:18.620
You run by 4, 1, 2, 3, 4, you're
going to go down 1.

00:04:18.620 --> 00:04:20.519
So this line is actually
going off the table.

00:04:20.519 --> 00:04:24.090
To go back by 4, 1, 2, 3, 4,
you're going to go up by 1.

00:04:24.090 --> 00:04:26.366
And since we're not actually
including the line, we're

00:04:26.366 --> 00:04:28.200
going to have to draw
a dotted line here.

00:04:28.200 --> 00:04:31.580
This is going to be
a dotted line.

00:04:31.580 --> 00:04:34.110
Try my best to draw
a dotted line.

00:04:34.110 --> 00:04:36.040
And actually, let me continue
that previous line there

00:04:36.040 --> 00:04:39.190
because I actually want show
where they intersect.

00:04:39.190 --> 00:04:41.280
Goes a little bit off
of our graph paper.

00:04:41.280 --> 00:04:43.270
But the graph of this equation
is that right.

00:04:43.270 --> 00:04:45.700
I did a dotted line because it's
not less than or equal

00:04:45.700 --> 00:04:47.510
to, it's just less than.

00:04:47.510 --> 00:04:50.440
So this inequality, if I were to
graph it, would be the area

00:04:50.440 --> 00:04:59.790
below the line right there.

00:04:59.790 --> 00:05:02.110
That's what satisfies
this inequality.

00:05:02.110 --> 00:05:04.840
For any x, this is the
value of that line.

00:05:04.840 --> 00:05:07.430
But the y's that satisfy
it are the y's that

00:05:07.430 --> 00:05:09.210
are below the line.

00:05:09.210 --> 00:05:12.810
Now if I were to ask you, if I
were to have a system, where I

00:05:12.810 --> 00:05:21.465
have 2x minus 3y is less
than, or equal to 21.

00:05:21.465 --> 00:05:22.180
These aren't equations.

00:05:22.180 --> 00:05:23.690
These are inequalities,
they're not

00:05:23.690 --> 00:05:25.050
equal to each other.

00:05:25.050 --> 00:05:31.270
The other one is x plus
4y is less than 6.

00:05:31.270 --> 00:05:35.440
And if I were to ask you what
x's and y's satisfy both of

00:05:35.440 --> 00:05:38.450
these any inequalities, well you
say, it's just going to be

00:05:38.450 --> 00:05:41.000
the overlap of their solution
sets, right?

00:05:41.000 --> 00:05:44.690
It's going to be the area in
the xy plane that satisfies

00:05:44.690 --> 00:05:45.330
both of them.

00:05:45.330 --> 00:05:47.100
So in this case, it'll
be the part that's

00:05:47.100 --> 00:05:48.430
above the orange line.

00:05:48.430 --> 00:05:50.250
Remember I shaded that in
orange right there.

00:05:54.470 --> 00:05:57.420
So all of this above the orange
line, and the part

00:05:57.420 --> 00:06:00.260
that's below the blue line,
and the part that overlaps

00:06:00.260 --> 00:06:06.320
that satisfies them both of
these-- I'll do it in yellow

00:06:06.320 --> 00:06:11.160
--the part that satisfies both
of these is right in between

00:06:11.160 --> 00:06:12.590
the lines there.

00:06:12.590 --> 00:06:16.260
Now what if, just for fun, we
were to add a third equation.

00:06:16.260 --> 00:06:18.720
Let's add a third equation
in green.

00:06:18.720 --> 00:06:24.270
3x plus y is greater than
or equal to negative 4.

00:06:24.270 --> 00:06:25.550
Well, it's the same drill.

00:06:25.550 --> 00:06:27.290
We put this in the
slope-intercept form.

00:06:27.290 --> 00:06:29.430
So you get y is greater
than or equal to

00:06:29.430 --> 00:06:31.910
negative 3x minus 4.

00:06:31.910 --> 00:06:34.090
I just subtracted 3x
from both sides.

00:06:34.090 --> 00:06:35.350
We can graph this line.

00:06:35.350 --> 00:06:39.430
It's y-intercept is 1,
2, 3, negative 4.

00:06:39.430 --> 00:06:42.860
And it has a slope
of negative 3.

00:06:42.860 --> 00:06:45.450
So every time you go one to
the right, you go down 3.

00:06:45.450 --> 00:06:47.800
1, 2, 3.

00:06:47.800 --> 00:06:50.500
If you go one to the left, if
you go negative 1 in the

00:06:50.500 --> 00:06:52.860
x-direction, you go up 3.

00:06:52.860 --> 00:06:56.370
So you go negative 1, 1, 2, 3.

00:06:56.370 --> 00:06:58.690
Negative 1, 1, 2, 3.

00:06:58.690 --> 00:07:01.170
Negative 1, 1, 2, 3.

00:07:01.170 --> 00:07:05.820
So this line right there's
going to look like this.

00:07:05.820 --> 00:07:07.840
It's going to look like that.

00:07:07.840 --> 00:07:09.520
Trying my best to draw it.

00:07:09.520 --> 00:07:10.300
That's that right there.

00:07:10.300 --> 00:07:12.010
And I can have it in bold.

00:07:12.010 --> 00:07:13.850
I don't have to do a dotted line
because I have a greater

00:07:13.850 --> 00:07:16.450
than or equal to and so what
are all the y's that are

00:07:16.450 --> 00:07:18.330
greater than this line?

00:07:18.330 --> 00:07:24.540
Well, for any x value you pick,
the value of negative 3x

00:07:24.540 --> 00:07:26.420
minus 4 is going to be this.

00:07:26.420 --> 00:07:28.970
And we care about the y values
that are greater than that, so

00:07:28.970 --> 00:07:31.090
it'll be above the line.

00:07:31.090 --> 00:07:33.160
So for just this inequality
it'll be

00:07:33.160 --> 00:07:35.630
everything above the line.

00:07:35.630 --> 00:07:37.540
And then if someone were to come
along and say, all right

00:07:37.540 --> 00:07:42.350
, tell me, show me, graph me,
all of the points on the xy

00:07:42.350 --> 00:07:46.660
plane that satisfy this system
of inequalities.

00:07:46.660 --> 00:07:47.850
What would you tell them?

00:07:47.850 --> 00:07:50.500
Well, it would be the overlap
of their solution sets?

00:07:50.500 --> 00:07:53.210
So the part of the xy plane that
exists in all of their

00:07:53.210 --> 00:07:55.850
solutions, so what's above
the orange line, right?

00:07:55.850 --> 00:07:58.390
We have to be above the
orange line, right?

00:07:58.390 --> 00:07:59.490
And we graphed that before.

00:07:59.490 --> 00:08:03.370
It was all in orange before,
because we're above the line

00:08:03.370 --> 00:08:06.600
to 2/3x minus 7.

00:08:06.600 --> 00:08:07.660
It might trick you a
little bit because

00:08:07.660 --> 00:08:08.550
there's a less than here.

00:08:08.550 --> 00:08:10.962
But when you put it in the
slope-intercept form, you have

00:08:10.962 --> 00:08:11.720
a greater than.

00:08:11.720 --> 00:08:14.710
So it's everything above this
line, which is all of that

00:08:14.710 --> 00:08:15.530
over there.

00:08:15.530 --> 00:08:19.650
Everything below this line--
below the blue line and not

00:08:19.650 --> 00:08:21.510
including the blue line
--and everything

00:08:21.510 --> 00:08:23.520
above the green line.

00:08:23.520 --> 00:08:31.610
So it's going to be this area
right over here, is the

00:08:31.610 --> 00:08:32.340
solution set.

00:08:32.340 --> 00:08:35.640
And we're going to include this
boundary right here, and

00:08:35.640 --> 00:08:38.960
we're going to include that
boundary right there.

00:08:38.960 --> 00:08:42.169
And since I engineered the
problem to have a less than

00:08:42.169 --> 00:08:44.580
there, not a less than or equal
to, up here is going to

00:08:44.580 --> 00:08:46.190
be a dotted line.

00:08:46.190 --> 00:08:49.130
Because you're not actually
going to include that line.

00:08:49.130 --> 00:08:51.870
Anyway, hopefully you
found that useful.


Special Types of Linear Systems

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:04.050
Let's do some more systems
of equations problem.

00:00:04.050 --> 00:00:08.210
In this video, we're going to
encounter systems that might

00:00:08.210 --> 00:00:11.860
have no solutions, or that might
have an infinite many

00:00:11.860 --> 00:00:14.440
solutions, and we'll label
them with words.

00:00:14.440 --> 00:00:15.700
So let's start with one.

00:00:15.700 --> 00:00:23.680
Let's say we have 3x minus
4y is equal to 13.

00:00:23.680 --> 00:00:30.250
Let's say my other equation in
my system is y is equal to

00:00:30.250 --> 00:00:34.320
negative 3x minus y.

00:00:34.320 --> 00:00:36.620
So the first thing-- this is
kind of in a strange form

00:00:36.620 --> 00:00:37.100
right here.

00:00:37.100 --> 00:00:39.940
I want to get into the standard
form and maybe I'll

00:00:39.940 --> 00:00:43.070
do elimination for this systems.
Let me rewrite the

00:00:43.070 --> 00:00:44.140
top equation.

00:00:44.140 --> 00:00:48.620
We have 3x minus 4y
is equal to 13.

00:00:48.620 --> 00:00:52.060
And let me rearrange this
bottom equation here.

00:00:52.060 --> 00:00:56.740
So if I were to add 2y-- well,
let me subtract y from both

00:00:56.740 --> 00:00:57.840
sides of this equation.

00:00:57.840 --> 00:01:01.820
So if I subtract y from both
sides of this equation, it

00:01:01.820 --> 00:01:09.820
becomes 0 is equal to negative
3x minus 2y, or negative 3x

00:01:09.820 --> 00:01:11.860
minus 2y is equal to 0.

00:01:11.860 --> 00:01:13.190
So let me write that
over here.

00:01:13.190 --> 00:01:15.735
And it looks nice because I
have a negative 3x here, I

00:01:15.735 --> 00:01:17.070
have a positive 3x here.

00:01:17.070 --> 00:01:19.076
Looks well suited
for elimination.

00:01:19.076 --> 00:01:26.950
So, I have negative 3x minus
2y is equal to 0.

00:01:26.950 --> 00:01:29.440
So let's add the left-hand side
of this equation to the

00:01:29.440 --> 00:01:33.270
left-hand side of the
yellow equation.

00:01:33.270 --> 00:01:36.200
And we're going to add 0 to
the right-hand side of the

00:01:36.200 --> 00:01:37.850
yellow equation and
we're essentially

00:01:37.850 --> 00:01:38.890
adding 0 to both sides.

00:01:38.890 --> 00:01:41.300
We're adding the same quantity
to both sides, which we can

00:01:41.300 --> 00:01:43.210
always do with an equation.

00:01:43.210 --> 00:01:47.350
So, the left hant-side, the 3x
cancels out with a 3x and

00:01:47.350 --> 00:01:50.410
we're left with negative
4y minus 2y.

00:01:50.410 --> 00:01:55.950
You get negative 6y
is equal to 13.

00:01:55.950 --> 00:01:58.715
Divide both sides
by negative 6.

00:02:01.780 --> 00:02:07.640
We are left with y is equal
to negative 13/6.

00:02:07.640 --> 00:02:10.330
Now let's solve for x.

00:02:10.330 --> 00:02:13.570
And we can solve for x using
either of these equations.

00:02:13.570 --> 00:02:16.050
Let's use that top one,
just for fun.

00:02:16.050 --> 00:02:21.210
So we have 3 times x minus
4 times negative

00:02:21.210 --> 00:02:26.540
13/6 is equal to 13.

00:02:26.540 --> 00:02:28.800
Now we have a negative times a
negative, so those are both

00:02:28.800 --> 00:02:30.420
going to become positives.

00:02:30.420 --> 00:02:35.560
And then the 4/6, that's
the same thing as 2/3.

00:02:35.560 --> 00:02:44.430
So this becomes 3x plus 2 times
13 which is 26/3, is

00:02:44.430 --> 00:02:45.840
equal to 13.

00:02:45.840 --> 00:02:49.870
Instead of 13, since I'm about
to subtract 26/3 from both

00:02:49.870 --> 00:02:55.520
sides, let me rewrite
13 as 39/3.

00:02:55.520 --> 00:02:55.830
right?

00:02:55.830 --> 00:02:58.750
39 divided by 3 is 13.

00:02:58.750 --> 00:03:01.785
So, let me subtract 26/3
from both sides.

00:03:08.420 --> 00:03:12.500
The left-hand side becomes 3x--
these cancel out-- is

00:03:12.500 --> 00:03:18.770
equal to 39 minus 26 is 13/3.

00:03:18.770 --> 00:03:21.360
And then we're going to want
to divide both sides by 3.

00:03:21.360 --> 00:03:23.920
Or you can view it as
multiplying both sides by 1/3.

00:03:28.600 --> 00:03:31.240
The left-hand side, we're
just left with an x.

00:03:31.240 --> 00:03:36.850
The right-hand side, x is going
to be equal to 13/9.

00:03:36.850 --> 00:03:41.010
So this system had a well
defined solution.

00:03:41.010 --> 00:03:45.050
The solution is x is equal
to 13/9 and y is equal to

00:03:45.050 --> 00:03:46.480
negative 13/6.

00:03:46.480 --> 00:03:48.250
It only has one solution.

00:03:48.250 --> 00:03:50.810
So if you think of these as
lines, these two lines

00:03:50.810 --> 00:03:53.300
intersect in exactly
one point.

00:03:53.300 --> 00:03:57.220
And a system like this, where
it has exactly one solution,

00:03:57.220 --> 00:04:07.300
is called a consistent
system of equations.

00:04:07.300 --> 00:04:09.740
And everything we've been
doing so far has been

00:04:09.740 --> 00:04:12.910
consistent systems. Let's see
if we can stumble upon

00:04:12.910 --> 00:04:16.120
something that's maybe a
little less consistent.

00:04:16.120 --> 00:04:18.690
Let's say we have a system.

00:04:18.690 --> 00:04:25.190
Let's say it's 5x minus
4y is equal to 1.

00:04:25.190 --> 00:04:32.340
And let's say we have negative
10x plus 8y is equal to

00:04:32.340 --> 00:04:34.020
negative 30.

00:04:34.020 --> 00:04:37.180
Once again, I'm tempted to do
elimination here, because I

00:04:37.180 --> 00:04:39.010
have a negative 10x
and I have a 5x.

00:04:39.010 --> 00:04:48.310
If I take this top equation and
I multiply it by 2, I'll

00:04:48.310 --> 00:04:58.410
get 10x minus 8y is
equal to 2, right?

00:04:58.410 --> 00:05:00.990
I just multiplied
both sides by 2.

00:05:00.990 --> 00:05:09.940
And if we add the left-hand
sides, we'll get 0x plus 0y is

00:05:09.940 --> 00:05:13.180
equal to negative 28.

00:05:13.180 --> 00:05:15.360
So we essentially get 0 is
equal to negative 28.

00:05:15.360 --> 00:05:16.080
That's crazy.

00:05:16.080 --> 00:05:17.260
We know that that's not true.

00:05:17.260 --> 00:05:18.370
This can never be true.

00:05:18.370 --> 00:05:21.430
We're getting an inconsistent
statement.

00:05:21.430 --> 00:05:24.190
We're getting a weirdo
statement, and that's because

00:05:24.190 --> 00:05:25.440
this has no solution.

00:05:31.350 --> 00:05:33.760
When you solve a system of
equation, it doesn't matter

00:05:33.760 --> 00:05:36.550
how you do it, whether it's
through substitution or

00:05:36.550 --> 00:05:38.830
whether it's through
elimination, like I did here.

00:05:38.830 --> 00:05:41.620
When you get one of these
statements where 0 equals

00:05:41.620 --> 00:05:44.630
negative 28 or 5 is equal to 7
or two things that clearly

00:05:44.630 --> 00:05:47.420
don't equal each other, when
they essentially have to equal

00:05:47.420 --> 00:05:50.020
each other in order for the
system to work, we call that

00:05:50.020 --> 00:05:51.510
an inconsistent system.

00:05:57.670 --> 00:06:00.010
And it will have no solution.

00:06:00.010 --> 00:06:01.960
So what does it mean
for both of these

00:06:01.960 --> 00:06:04.800
equations to have no solution?

00:06:04.800 --> 00:06:06.040
Let's actually graph these.

00:06:06.040 --> 00:06:09.030
I think you'll have a better
feel for what it means not to

00:06:09.030 --> 00:06:09.840
have a solution.

00:06:09.840 --> 00:06:16.060
So the first equation is 5x
minus 4y is equal to 1.

00:06:16.060 --> 00:06:18.730
Let me put it into
slope-intercept form.

00:06:18.730 --> 00:06:25.690
So if we subtract 5x from both
sides, we get negative 4y is

00:06:25.690 --> 00:06:29.870
equal to negative 5x plus 1.

00:06:29.870 --> 00:06:33.750
Now if we divide both sides by
negative 4, you get y is equal

00:06:33.750 --> 00:06:41.450
to negative 5/4x minus
1/4, right?

00:06:41.450 --> 00:06:42.500
1 divided by negative 4.

00:06:42.500 --> 00:06:46.100
So this is the first equation,
right over here, in

00:06:46.100 --> 00:06:47.690
slope-intercept form.

00:06:47.690 --> 00:06:49.250
Now let me write the
second equation in

00:06:49.250 --> 00:06:50.870
slope-intercept form.

00:06:50.870 --> 00:06:57.390
We have negative 10x plus 8y
is equal to negative 30.

00:06:57.390 --> 00:06:59.570
Let's add 10x to both sides.

00:06:59.570 --> 00:07:04.770
You get 8y is equal to 10x minus
30, and let's divide

00:07:04.770 --> 00:07:06.450
both sides by 8.

00:07:06.450 --> 00:07:13.870
You'll get y is equal to 10/8,
is the same thing as 5/4.

00:07:13.870 --> 00:07:18.020
5/4x minus 30/8.

00:07:18.020 --> 00:07:25.030
30/8 is the same
thing as 15/4.

00:07:25.030 --> 00:07:26.230
Oh, and actually I made
a mistake here.

00:07:26.230 --> 00:07:29.610
When we divide both sides of
this equation by negative 4,

00:07:29.610 --> 00:07:34.020
negative 5 divided by negative
4 is positive 5/4.

00:07:34.020 --> 00:07:35.580
So I shouldn't have had
a negative there.

00:07:35.580 --> 00:07:37.650
I almost made a blunder.

00:07:37.650 --> 00:07:39.170
So there should be no
negative there.

00:07:39.170 --> 00:07:42.280
Then a 1 divided by negative
4, is negative 1/4.

00:07:42.280 --> 00:07:47.090
So going back to the two
equations, what do you notice?

00:07:47.090 --> 00:07:49.350
Well, when you put them in
slope-intercept form, they

00:07:49.350 --> 00:07:53.400
both have the exact
same slope, 5/4.

00:07:53.400 --> 00:07:55.210
but they have different
y-intercepts.

00:07:55.210 --> 00:07:57.160
So what would their
graphs look like?

00:07:57.160 --> 00:07:58.800
Let me graph them.

00:07:58.800 --> 00:08:01.860
Let's say that that
is my y-axis.

00:08:01.860 --> 00:08:05.170
That is my x-axis.

00:08:05.170 --> 00:08:09.260
This equation, its y-intercept
is at 0 negative 1, 4.

00:08:09.260 --> 00:08:11.080
Maybe that's that point
right there.

00:08:11.080 --> 00:08:13.370
And it goes up at 5/4x.

00:08:13.370 --> 00:08:14.586
That's a little bit
more than one.

00:08:14.586 --> 00:08:16.110
It's a 1.25x.

00:08:16.110 --> 00:08:20.050
Every time you go to the right
1, you're going up 1.25.

00:08:20.050 --> 00:08:22.490
So this line is going to look
something like this.

00:08:22.490 --> 00:08:23.590
I'm just drawing it rough.

00:08:23.590 --> 00:08:24.930
I want you to get the
general idea.

00:08:24.930 --> 00:08:27.150
That's what that line
looks like.

00:08:27.150 --> 00:08:31.800
Now this line, its y-intercept
is at negative 15/4.

00:08:31.800 --> 00:08:33.980
15/4 is what?

00:08:33.980 --> 00:08:35.299
3 3/4.

00:08:35.299 --> 00:08:41.020
So if this is negative 1/4, its
y-intercept is going to be

00:08:41.020 --> 00:08:42.270
way down here someplace.

00:08:45.560 --> 00:08:47.590
I'll do it in the same color.

00:08:47.590 --> 00:08:49.060
Way down here someplace.

00:08:49.060 --> 00:08:51.420
Let me continue my
x-axis down.

00:08:51.420 --> 00:08:54.220
This would be at
negative 15/4.

00:08:54.220 --> 00:08:56.400
But its slope is the
exact same thing.

00:08:56.400 --> 00:08:58.610
Every time you go to the right,
1, you're going to go

00:08:58.610 --> 00:09:02.200
up by 5/4, so its slope
is going to be

00:09:02.200 --> 00:09:03.800
the exact same thing.

00:09:03.800 --> 00:09:05.700
So what do you notice about
these two lines?

00:09:05.700 --> 00:09:07.000
They are parallel.

00:09:07.000 --> 00:09:10.350
They have the same slope,
different y-intercepts, so

00:09:10.350 --> 00:09:12.690
they will never intersect.

00:09:12.690 --> 00:09:18.350
These two lines will
never intersect.

00:09:18.350 --> 00:09:22.500
Which means that there is no
point on the coordinate plane

00:09:22.500 --> 00:09:25.250
on the x-, y-coordinate
plane that satisfies

00:09:25.250 --> 00:09:27.000
both of these equations.

00:09:27.000 --> 00:09:29.220
Remember, this line represents
all of the points that

00:09:29.220 --> 00:09:30.910
satisfied this equation.

00:09:30.910 --> 00:09:32.790
This line represents all
of the points that

00:09:32.790 --> 00:09:34.190
satisfied that equation.

00:09:34.190 --> 00:09:36.820
Notice, no point
satisfies both.

00:09:36.820 --> 00:09:39.990
There's no point of intersection
and that's why

00:09:39.990 --> 00:09:43.300
this was an inconsistent
system.

00:09:43.300 --> 00:09:44.550
Let's do one more.

00:09:48.420 --> 00:10:00.990
Let's say I have 4x plus 5y is
equal to 0, and I have 3x is

00:10:00.990 --> 00:10:05.520
equal to 6y plus 4.5.

00:10:08.890 --> 00:10:10.725
Actually, let me do a slightly
different one, because I want

00:10:10.725 --> 00:10:13.460
to show you all of the different
types that we can

00:10:13.460 --> 00:10:15.410
see in systems of equations.

00:10:15.410 --> 00:10:16.660
Let me clear this.

00:10:19.810 --> 00:10:30.240
Let's say my first system is
3x minus 7y is equal to 1.

00:10:30.240 --> 00:10:35.880
And let's say my other equation
in my system is

00:10:35.880 --> 00:10:44.540
negative 6x plus 14y is
equal to negative 2.

00:10:44.540 --> 00:10:47.860
So, let's try to find the x's
and y's that satisfy this

00:10:47.860 --> 00:10:48.570
equation here.

00:10:48.570 --> 00:10:51.550
And just for a change of pace,
let's do some substitution.

00:10:54.360 --> 00:10:58.210
Although this is very tempting
to do elimination here, let's

00:10:58.210 --> 00:10:59.080
do substitution.

00:10:59.080 --> 00:11:01.470
You get 3x.

00:11:01.470 --> 00:11:03.490
Let's solve it for x.

00:11:03.490 --> 00:11:06.120
Actually, let's just do
elimination, because this is

00:11:06.120 --> 00:11:09.860
just so glaringly prepared
for elimination, let's

00:11:09.860 --> 00:11:10.710
just do it that way.

00:11:10.710 --> 00:11:13.090
So let's multiply this
top equation by 2.

00:11:16.200 --> 00:11:17.340
And what do we get?

00:11:17.340 --> 00:11:23.140
We get 6x minus 14y is
equal to 2 right?

00:11:23.140 --> 00:11:26.750
I just multiplied every term
on both sides by 2.

00:11:26.750 --> 00:11:28.810
And now let's add the
left sides together.

00:11:28.810 --> 00:11:31.866
You get 0 plus 0.

00:11:31.866 --> 00:11:34.870
And on the right hand side,
you get is equal to 0.

00:11:34.870 --> 00:11:39.270
You get 0 is equal to 0, which
is always going to be true.

00:11:39.270 --> 00:11:42.410
This type of system is called
a dependent system.

00:11:45.500 --> 00:11:48.550
So remember, when you get a nice
clean solution, that's a

00:11:48.550 --> 00:11:49.780
consistent system.

00:11:49.780 --> 00:11:52.660
When you get something crazy
like 0 equals 1, that's an

00:11:52.660 --> 00:11:53.730
inconsistent system.

00:11:53.730 --> 00:11:55.630
That means the lines
are parallel.

00:11:55.630 --> 00:11:58.980
When you get 0 equals 0, or 1 is
equal to 1 or anything like

00:11:58.980 --> 00:12:01.210
that, you're dealing with
the dependent system.

00:12:01.210 --> 00:12:04.510
Which really means that these
are the exact same lines, even

00:12:04.510 --> 00:12:06.450
though they might look a
little bit different.

00:12:06.450 --> 00:12:08.730
And to verify that, let's
put them both into

00:12:08.730 --> 00:12:10.430
slope-intercept form.

00:12:10.430 --> 00:12:15.940
So this top line, you have 3x
minus 7y is equal to 1.

00:12:15.940 --> 00:12:18.080
Let's subtract 3x
from both sides.

00:12:18.080 --> 00:12:23.360
You get negative 7y is equal
to negative 3x plus 1.

00:12:23.360 --> 00:12:25.510
Now let's divide both
sides by negative 7.

00:12:25.510 --> 00:12:33.020
You get y is equal to positive
3/7x minus 1/7.

00:12:33.020 --> 00:12:34.930
That's that first equation.

00:12:34.930 --> 00:12:36.510
Now let's put the second
equation into

00:12:36.510 --> 00:12:38.320
slope-intercept form.

00:12:38.320 --> 00:12:44.570
You have negative 6x plus 14y
is equal to negative 2.

00:12:44.570 --> 00:12:48.550
Let's add 6x to both sides
of the equation.

00:12:48.550 --> 00:12:54.140
So you get 14y is equal
to 6x minus 2.

00:12:54.140 --> 00:12:56.430
Then divide both sides of
the equation by 14.

00:12:56.430 --> 00:13:03.490
You get y is equal to
6/14x minus 2/14.

00:13:03.490 --> 00:13:04.690
Well, this is the same thing.

00:13:04.690 --> 00:13:10.900
6/14 is the same thing
is 3/7x minus 1/7.

00:13:10.900 --> 00:13:15.390
Notice, they are really the
exact same equation.

00:13:15.390 --> 00:13:18.890
So if you want to find x's and
y's that satisfy both, let's

00:13:18.890 --> 00:13:20.470
think about it.

00:13:20.470 --> 00:13:21.750
Let's graph it.

00:13:21.750 --> 00:13:26.380
So if that is my coordinate
plane, that's the y-axis, that

00:13:26.380 --> 00:13:28.580
is the x-axis.

00:13:28.580 --> 00:13:30.600
This graph is going to look
something like this.

00:13:30.600 --> 00:13:31.790
I'm going to draw
it very roughly.

00:13:31.790 --> 00:13:36.880
It might look something like
that, where its slope is 3/7.

00:13:36.880 --> 00:13:38.470
So this line is going to look
something like that.

00:13:38.470 --> 00:13:39.970
That line looks exactly
like that.

00:13:39.970 --> 00:13:41.730
It's the same exact line.

00:13:41.730 --> 00:13:44.010
So when you say, well, what
are the x's and y's that

00:13:44.010 --> 00:13:46.500
satisfy both of these
equations?

00:13:46.500 --> 00:13:49.110
Well, it's every x and y
that's on these points.

00:13:49.110 --> 00:13:50.070
It's that x and y.

00:13:50.070 --> 00:13:50.750
That coordinate.

00:13:50.750 --> 00:13:51.310
That coordinate.

00:13:51.310 --> 00:13:52.110
That coordinate.

00:13:52.110 --> 00:13:54.260
There are an infinite
number of solutions.

00:13:54.260 --> 00:13:57.140
And when we use the word
dependent -- because you can

00:13:57.140 --> 00:14:00.200
get to one of these equations
from the other.

00:14:00.200 --> 00:14:02.240
These equations are dependent
on each other.

00:14:02.240 --> 00:14:04.950
You can just scale one or the
other and rearrange it, and

00:14:04.950 --> 00:14:05.930
they equal each other.

00:14:05.930 --> 00:14:08.020
So here you have infinite
solutions.

00:14:11.980 --> 00:14:14.950
Anything that satisfies one line
will satisfy the other.

00:14:14.950 --> 00:14:16.350
So you just pick an x.

00:14:16.350 --> 00:14:21.080
When x is 1, you get 3/7
minus 1/7 that's 2/7.

00:14:21.080 --> 00:14:25.470
So 1, 2/7 satisfies
both equations.

00:14:25.470 --> 00:14:26.910
If you pick x is equal to 0.

00:14:26.910 --> 00:14:30.490
0, negative 1/7 satisfies
both equations.

00:14:30.490 --> 00:14:33.210
And you could pick an infinite
number of values for x, solve

00:14:33.210 --> 00:14:38.400
for y, and those coordinates
will satisfy both equations.

00:14:38.400 --> 00:14:40.040
Let me review this
a little bit.

00:14:40.040 --> 00:14:42.380
So we started off just with
the plain vanilla.

00:14:42.380 --> 00:14:45.120
When you actually get a solution
that is consistent.

00:14:45.120 --> 00:14:49.240
These lines actually intersect
in one point.

00:14:49.240 --> 00:14:51.650
Then you have the situation
where you get something crazy,

00:14:51.650 --> 00:14:53.070
when you solve your system
of equations.

00:14:53.070 --> 00:14:54.890
0 is equal to negative 28.

00:14:54.890 --> 00:14:56.330
Definitely not true.

00:14:56.330 --> 00:14:58.180
This is an inconsistent
system.

00:14:58.180 --> 00:14:59.890
It has no solution,
which means that

00:14:59.890 --> 00:15:01.690
these lines are parallel.

00:15:01.690 --> 00:15:03.570
They never intersect.

00:15:03.570 --> 00:15:06.520
And then finally, if you get
something that's always true--

00:15:06.520 --> 00:15:09.920
that's just kind of silly how
true it is-- this is a

00:15:09.920 --> 00:15:10.910
dependent system.

00:15:10.910 --> 00:15:12.700
These are going to
be the same line.

00:15:12.700 --> 00:15:15.340
Then you can verify it by
putting both of them into

00:15:15.340 --> 00:15:17.840
slope-intercept form.


Systems of equations with elimination (and manipulation)

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.530
Let's solve a few more systems
of equations using

00:00:03.530 --> 00:00:07.520
elimination, but in these it
won't be kind of a one-step

00:00:07.520 --> 00:00:08.070
elimination.

00:00:08.070 --> 00:00:11.350
We're going to have to massage
the equations a little bit in

00:00:11.350 --> 00:00:13.530
order to prepare them
for elimination.

00:00:13.530 --> 00:00:20.480
So let's say that we have an
equation, 5x minus 10y is

00:00:20.480 --> 00:00:23.130
equal to 15.

00:00:23.130 --> 00:00:32.159
And we have another equation,
3x minus 2y is equal to 3.

00:00:32.159 --> 00:00:34.350
And I said we want to do
this using elimination.

00:00:34.350 --> 00:00:37.440
Once again, we could use
substitution, we could graph

00:00:37.440 --> 00:00:40.430
both of these lines and figure
out where they intersect.

00:00:40.430 --> 00:00:41.930
But we're going to
use elimination.

00:00:41.930 --> 00:00:43.740
But the first thing you might
say, hey, Sal, you know, with

00:00:43.740 --> 00:00:46.450
elimination, you were
subtracting the left-hand side

00:00:46.450 --> 00:00:49.160
of one equation from another,
or adding the two, and then

00:00:49.160 --> 00:00:50.580
adding the two right-hand
sides.

00:00:50.580 --> 00:00:52.610
And I could do that, because it
was essentially adding the

00:00:52.610 --> 00:00:55.180
same thing to both sides
of the equation.

00:00:55.180 --> 00:00:57.010
But here, it's not
obvious that that

00:00:57.010 --> 00:00:57.870
would be of any help.

00:00:57.870 --> 00:01:00.970
If we added these two left-hand
sides, you would get

00:01:00.970 --> 00:01:03.750
8x minus 12y.

00:01:03.750 --> 00:01:05.580
That wouldn't eliminate
any variables.

00:01:05.580 --> 00:01:06.930
And on the right-hand
side, you would just

00:01:06.930 --> 00:01:08.030
be left with a number.

00:01:08.030 --> 00:01:09.400
And if you subtracted,
that wouldn't

00:01:09.400 --> 00:01:10.420
eliminate any variables.

00:01:10.420 --> 00:01:12.710
So how is elimination
going to help here?

00:01:12.710 --> 00:01:16.370
And the answer is, we can
multiply both of these

00:01:16.370 --> 00:01:20.070
equations in such a way that
maybe we can get one of these

00:01:20.070 --> 00:01:24.250
terms to cancel out with
one of the others.

00:01:24.250 --> 00:01:26.210
And you could really
pick which term you

00:01:26.210 --> 00:01:27.890
want to cancel out.

00:01:27.890 --> 00:01:31.690
Let's say we want to cancel out
the y terms. So I'll just

00:01:31.690 --> 00:01:34.510
rewrite this 5x minus
10y here.

00:01:34.510 --> 00:01:39.860
5x minus 10y is equal to 15.

00:01:39.860 --> 00:01:43.260
Now, is there anything that
I can multiply this green

00:01:43.260 --> 00:01:47.420
equation by so that this
negative 2y term becomes a

00:01:47.420 --> 00:01:50.200
term that will cancel out
with the negative 10y?

00:01:50.200 --> 00:01:52.980
So I essentially want to make
this negative 2y into a

00:01:52.980 --> 00:01:54.690
positive 10y.

00:01:54.690 --> 00:01:54.950
Right?

00:01:54.950 --> 00:01:57.660
Because if this is a positive
10y, it'll cancel out when I

00:01:57.660 --> 00:02:00.400
add the left-hand sides
of this equation.

00:02:00.400 --> 00:02:03.280
So what can I multiply
this equation by?

00:02:03.280 --> 00:02:06.210
Well, if I multiply it by
negative 5, negative 5 times

00:02:06.210 --> 00:02:08.880
negative 2 right here would
be positive 10.

00:02:08.880 --> 00:02:09.949
So let's do that.

00:02:09.949 --> 00:02:15.870
Let's multiply this equation
times negative 5.

00:02:15.870 --> 00:02:18.820
So you multiply the left-hand
side by negative 5, and

00:02:18.820 --> 00:02:22.280
multiply the right-hand
side by negative 5.

00:02:22.280 --> 00:02:23.310
And what do you get?

00:02:23.310 --> 00:02:25.110
Remember, we're not
fundamentally

00:02:25.110 --> 00:02:26.490
changing the equation.

00:02:26.490 --> 00:02:28.700
We're not changing the
information in the equation.

00:02:28.700 --> 00:02:31.750
We're doing the same thing
to both sides of it.

00:02:31.750 --> 00:02:34.950
So the left-hand side of the
equation becomes negative 5

00:02:34.950 --> 00:02:40.310
times 3x is negative 15x.

00:02:40.310 --> 00:02:46.010
And then negative 5 times
negative 2y is plus 10y, is

00:02:46.010 --> 00:02:51.210
equal to 3 times negative
5 is negative 15.

00:02:51.210 --> 00:02:54.820
And now, we're ready to
do our elimination.

00:02:54.820 --> 00:02:57.980
If we add this to the left-hand
side of the yellow

00:02:57.980 --> 00:03:01.120
equation, and we add the
negative 15 to the right-hand

00:03:01.120 --> 00:03:04.690
side of the yellow equation, we
are adding the same thing

00:03:04.690 --> 00:03:06.460
to both sides of the equation.

00:03:06.460 --> 00:03:08.230
Because this is equal to that.

00:03:08.230 --> 00:03:09.480
So let's do that.

00:03:13.260 --> 00:03:17.430
So 5x minus 15y-- we have this
little negative sign there, we

00:03:17.430 --> 00:03:21.020
don't want to lose that--
that's negative 10x.

00:03:21.020 --> 00:03:22.320
The y's cancel out.

00:03:22.320 --> 00:03:25.430
Negative 10y plus
10y, that's 0y.

00:03:25.430 --> 00:03:28.480
That was the whole point behind
multiplying this by

00:03:28.480 --> 00:03:29.720
negative 5.

00:03:29.720 --> 00:03:35.230
Is going to be equal to--
15 minus 15 is 0.

00:03:35.230 --> 00:03:37.450
So negative 10x is equal to 0.

00:03:37.450 --> 00:03:43.250
Divide both sides by negative
10, and you get

00:03:43.250 --> 00:03:46.010
x is equal to 0.

00:03:46.010 --> 00:03:49.880
And now we can substitute back
into either of these equations

00:03:49.880 --> 00:03:51.920
to figure out what y
must be equal to.

00:03:51.920 --> 00:03:53.960
Let's substitute into
the top equation.

00:03:53.960 --> 00:04:01.280
So we get 5 times 0, minus
10y, is equal to 15.

00:04:01.280 --> 00:04:04.560
Or negative 10y is
equal to 15.

00:04:04.560 --> 00:04:05.240
Let me write that.

00:04:05.240 --> 00:04:08.290
Negative 10y is equal to 15.

00:04:08.290 --> 00:04:13.660
Divide both sides
by negative 10.

00:04:13.660 --> 00:04:16.670
And we are left with
y is equal to

00:04:16.670 --> 00:04:21.620
15/10, is negative 3/2.

00:04:21.620 --> 00:04:25.100
So if you were to graph it,
the point of intersection

00:04:25.100 --> 00:04:29.430
would be the point
0, negative 3/2.

00:04:29.430 --> 00:04:32.370
And you can verify that it also
satisfies this equation.

00:04:32.370 --> 00:04:37.020
The original equation over
here was 3x minus

00:04:37.020 --> 00:04:39.290
2y is equal to 3.

00:04:39.290 --> 00:04:48.370
3 times 0, which is 0, minus 2
times negative 3/2 is, this is

00:04:48.370 --> 00:04:51.810
0, this is positive 3.

00:04:51.810 --> 00:04:52.080
Right?

00:04:52.080 --> 00:04:54.160
These cancel out, these
become positive.

00:04:54.160 --> 00:04:56.130
Plus positive 3 is equal to 3.

00:04:56.130 --> 00:05:00.040
So this does indeed satisfy
both equations.

00:05:00.040 --> 00:05:03.160
Let's do another one of these
where we have to multiply, and

00:05:03.160 --> 00:05:06.890
to massage the equations, and
then we can eliminate one of

00:05:06.890 --> 00:05:09.000
the variables.

00:05:09.000 --> 00:05:11.550
Let's do another one.

00:05:11.550 --> 00:05:21.330
Let's say we have 5x plus
7y is equal to 15.

00:05:21.330 --> 00:05:29.150
And we have 7-- let me do
another color-- 7x minus 3y is

00:05:29.150 --> 00:05:31.350
equal to 5.

00:05:31.350 --> 00:05:33.810
Now once again, if you just
added or subtracted both the

00:05:33.810 --> 00:05:35.020
left-hand sides, you're
not going to

00:05:35.020 --> 00:05:36.170
eliminate any variables.

00:05:36.170 --> 00:05:39.350
These aren't in any way kind of
have the same coefficient

00:05:39.350 --> 00:05:41.180
or the negative of their
coefficient.

00:05:41.180 --> 00:05:42.940
So let's pick a variable
to eliminate.

00:05:42.940 --> 00:05:46.740
Let's say we want to eliminate
the x's this time.

00:05:46.740 --> 00:05:49.180
And you could literally
pick on one of the

00:05:49.180 --> 00:05:49.930
variables or another.

00:05:49.930 --> 00:05:50.480
It doesn't matter.

00:05:50.480 --> 00:05:52.415
You can say let's eliminate the
y's first. But I'm going

00:05:52.415 --> 00:05:56.060
to choose to eliminate the x's
first. And so what I need to

00:05:56.060 --> 00:06:00.840
do is massage one or both of
these equations in a way that

00:06:00.840 --> 00:06:03.330
these guys have the same
coefficients, or their

00:06:03.330 --> 00:06:05.510
coefficients are the negatives
of each other, so that when I

00:06:05.510 --> 00:06:08.400
add the left-hand sides, they're
going to eliminate

00:06:08.400 --> 00:06:09.600
each other.

00:06:09.600 --> 00:06:12.960
Now, there's nothing obvious--
I can multiply this by a

00:06:12.960 --> 00:06:15.630
fraction to make it equal
to negative 5.

00:06:15.630 --> 00:06:17.950
Or I can multiply this by a
fraction to make it equal to

00:06:17.950 --> 00:06:18.930
negative 7.

00:06:18.930 --> 00:06:22.420
But even a more fun thing to do
is I can try to get both of

00:06:22.420 --> 00:06:24.220
them to be their least
common multiple.

00:06:24.220 --> 00:06:27.980
I could get both
of these to 35.

00:06:27.980 --> 00:06:30.130
And the way I can do it is by
multiplying by each other.

00:06:30.130 --> 00:06:32.775
So I can multiply this
top equation by 7.

00:06:38.170 --> 00:06:41.560
And I'm picking 7 so that
this becomes a 35.

00:06:41.560 --> 00:06:44.835
And I can multiply this bottom
equation by negative 5.

00:06:49.740 --> 00:06:52.230
And the reason why I'm doing
that is so this becomes a

00:06:52.230 --> 00:06:53.270
negative 35.

00:06:53.270 --> 00:06:55.980
Remember, my point is I want
to eliminate the x's.

00:06:55.980 --> 00:07:00.670
So if I make this a 35, and if
I make this a negative 35,

00:07:00.670 --> 00:07:01.630
then I'm going to be all set.

00:07:01.630 --> 00:07:04.480
I can add the left-hand
and the right-hand

00:07:04.480 --> 00:07:06.070
sides of the equations.

00:07:06.070 --> 00:07:09.996
So this top equation, when you
multiply it by 7, it becomes--

00:07:09.996 --> 00:07:13.790
let me scroll up a little bit--
we multiply it by 7, it

00:07:13.790 --> 00:07:22.960
becomes 35x plus 49y is equal
to-- let's see, this is 70

00:07:22.960 --> 00:07:28.950
plus 35 is equal to 105.

00:07:28.950 --> 00:07:29.250
Right?

00:07:29.250 --> 00:07:31.930
15 and 70, plus 35,
is equal to 105.

00:07:31.930 --> 00:07:33.960
That's what the top
equation becomes.

00:07:33.960 --> 00:07:38.000
This bottom equation becomes
negative 5 times 7x, is

00:07:38.000 --> 00:07:47.110
negative 35x, negative 5 times
negative 3y is plus 15y.

00:07:47.110 --> 00:07:48.520
The negatives cancel out.

00:07:48.520 --> 00:07:51.640
And then 5-- this isn't a
minus 5-- this is times

00:07:51.640 --> 00:07:53.060
negative 5.

00:07:53.060 --> 00:07:57.640
5 times negative 5 is equal
to negative 25.

00:07:57.640 --> 00:08:01.760
Now, we can start with this top
equation and add the same

00:08:01.760 --> 00:08:05.220
thing to both sides, where that
same thing is negative

00:08:05.220 --> 00:08:08.130
25, which is also equal
to this expression.

00:08:08.130 --> 00:08:09.960
So let's add the left-hand
sides and

00:08:09.960 --> 00:08:10.710
the right-hand sides.

00:08:10.710 --> 00:08:13.250
Because we're really adding the
same thing to both sides

00:08:13.250 --> 00:08:15.200
of the equation.

00:08:15.200 --> 00:08:17.430
So the left-hand side,
the x's cancel out.

00:08:17.430 --> 00:08:20.050
35x minus 35x.

00:08:20.050 --> 00:08:22.020
That was the whole point.

00:08:22.020 --> 00:08:27.760
They cancel out, and on the
y's, you get 49y plus 15y,

00:08:27.760 --> 00:08:30.360
that is 64y.

00:08:30.360 --> 00:08:37.470
64y is equal to 105 minus
25 is equal to 80.

00:08:37.470 --> 00:08:49.060
Divide both sides by 64, and you
get y is equal to 80/64.

00:08:49.060 --> 00:08:50.820
And let's see, if you divide
the numerator and the

00:08:50.820 --> 00:08:55.570
denominator by 8-- actually
you could probably do 16.

00:08:55.570 --> 00:08:56.780
16 would be better.

00:08:56.780 --> 00:08:58.590
But let's do 8 first,
just because we

00:08:58.590 --> 00:08:59.860
know our 8 times tables.

00:08:59.860 --> 00:09:03.720
So that becomes 10/8, and then
you can divide this by 2, and

00:09:03.720 --> 00:09:06.200
you get 5/4.

00:09:06.200 --> 00:09:08.020
If you divided just straight
up by 16, you would've gone

00:09:08.020 --> 00:09:09.600
straight to 5/4.

00:09:09.600 --> 00:09:11.340
So y is equal to 5/4.

00:09:11.340 --> 00:09:13.390
Let's figure out what x is.

00:09:13.390 --> 00:09:16.340
So we can substitute either into
one of these equations,

00:09:16.340 --> 00:09:18.300
or into one of the original
equations.

00:09:18.300 --> 00:09:21.140
Let's substitute into the
second of the original

00:09:21.140 --> 00:09:26.250
equations, where we had 7x
minus 3y is equal to 5.

00:09:26.250 --> 00:09:29.450
That was the original version of
the second equation that we

00:09:29.450 --> 00:09:31.790
later transformed into this.

00:09:31.790 --> 00:09:42.070
So we get 7x minus 3 times y,
times 5/4, is equal to 5.

00:09:42.070 --> 00:09:49.180
Or 7x minus 15/4
is equal to 5.

00:09:49.180 --> 00:09:54.600
Let's add 15/4-- Oh, sorry,
I didn't do that right.

00:09:54.600 --> 00:10:00.780
This would be 7x minus 3 times
4-- Oh, sorry, that was right.

00:10:00.780 --> 00:10:01.290
What am I doing?

00:10:01.290 --> 00:10:05.520
3 times is 15/4.

00:10:05.520 --> 00:10:07.290
Is equal to 5.

00:10:07.290 --> 00:10:16.960
Let's add 15/4 to both sides.

00:10:16.960 --> 00:10:18.290
And what do we get?

00:10:18.290 --> 00:10:20.940
The left-hand side just
becomes a 7x.

00:10:20.940 --> 00:10:22.630
These guys cancel out.

00:10:22.630 --> 00:10:24.810
And that's going to be
equal to 5, is the

00:10:24.810 --> 00:10:27.000
same thing as 20/4.

00:10:27.000 --> 00:10:31.740
20/4 plus 15/4.

00:10:31.740 --> 00:10:37.610
Or we get that-- let me scroll
down a little bit-- 7x is

00:10:37.610 --> 00:10:41.820
equal to 35/4.

00:10:41.820 --> 00:10:45.750
We can multiply both sides by
1/7, or we could divide both

00:10:45.750 --> 00:10:47.240
sides by 7, same thing.

00:10:47.240 --> 00:10:50.530
Let's multiply both
sides by 1/7.

00:10:50.530 --> 00:10:53.380
The same thing as
dividing by 7.

00:10:53.380 --> 00:10:56.520
So these cancel out and you're
left with x is equal to--

00:10:56.520 --> 00:10:59.190
Here, if you divide 35
by 7, you get 5.

00:10:59.190 --> 00:11:01.340
You divide 7 by 7, you get 1.

00:11:01.340 --> 00:11:05.100
So x is equal to 5/4 as well.

00:11:05.100 --> 00:11:09.960
So the point of intersection of
this right here is both x

00:11:09.960 --> 00:11:14.170
and y are going to
be equal to 5/4.

00:11:14.170 --> 00:11:18.900
So if you looked at it as a
graph, it'd be 5/4 comma 5/4.

00:11:18.900 --> 00:11:21.170
And let's verify that this
satisfies the top equation.

00:11:21.170 --> 00:11:29.480
And if you take 5 times
5/4, plus 7 times

00:11:29.480 --> 00:11:31.530
5/4, what do you get?

00:11:31.530 --> 00:11:33.330
It should be equal to 15.

00:11:33.330 --> 00:11:39.580
So this is equal to 25/4,
plus-- what is this?

00:11:39.580 --> 00:11:43.560
This is plus 35/4.

00:11:43.560 --> 00:11:48.020
Which is equal to 60/4, which
is indeed equal to 15.

00:11:48.020 --> 00:11:50.220
So it does definitely satisfy
that top equation.

00:11:50.220 --> 00:11:52.750
And you could check out this
bottom equation for yourself,

00:11:52.750 --> 00:11:54.630
but it should, because we
actually used this bottom

00:11:54.630 --> 00:11:57.750
equation to figure out that
x is equal to 5/4.