Graphing Using Intercepts

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.390
What we're going to do in this
video is graph some lines

00:00:04.390 --> 00:00:08.330
using the x- and y-intercepts
of the line.

00:00:08.330 --> 00:00:13.060
So first of all, a line is
defined by two points.

00:00:13.060 --> 00:00:15.760
If you find two points-- if
you're able to plot two points

00:00:15.760 --> 00:00:18.020
on a line, then you can connect
them and keep going in

00:00:18.020 --> 00:00:19.050
both directions.

00:00:19.050 --> 00:00:21.410
And you will have
drawn the line.

00:00:21.410 --> 00:00:23.750
What I'm going to do in this
video-- the two points I'm

00:00:23.750 --> 00:00:30.620
going to pick are the intercepts
of the lines.

00:00:30.620 --> 00:00:31.380
There's two of them.

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There's the x-intercept and then
there's the y-intercept.

00:00:36.040 --> 00:00:38.510
These are the points
where the lines

00:00:38.510 --> 00:00:40.860
intersect the x- and y-axes.

00:00:40.860 --> 00:00:45.240
And just to make it clear, this
right here is the x-axis

00:00:45.240 --> 00:00:47.540
on my graph paper.

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That right there is my x-axis.

00:00:50.090 --> 00:00:52.340
It just keeps going.

00:00:52.340 --> 00:00:54.030
That's my x-axis.

00:00:54.030 --> 00:00:56.770
This right here, in the
middle, going up

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and down, is my y-axis.

00:00:59.880 --> 00:01:03.845
That right there is my y-axis.

00:01:03.845 --> 00:01:05.250
I don't think you can see
what I just wrote.

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That is the y-axis
right there.

00:01:07.690 --> 00:01:10.780
So let me show you what I mean
by x- and y-intercepts.

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So let's do part (a).

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I'll do it in a different color,
not the same color as

00:01:18.240 --> 00:01:19.510
my intercepts.

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Part (a), we have the equation
y is equal to 2x plus 3.

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So first of all, let's think
about what happens when x is

00:01:26.550 --> 00:01:27.800
equal to 0.

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When x is equal to 0,
what is y equal to?

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y is going to be equal to
two times 0 plus 3.

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Well, 2 times 0,
that's just 0.

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So y is equal to 0 plus
3 or y is equal to 3.

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So the point x is equal to
0, y is equal to 3, that

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satisfies this equation.

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Or it's on the line defined
by this equation.

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So the point 0, 3.

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Let me plot that.

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The point 0, 3.

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Now notice, this point-- this is
when x is equal to 0, this

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sits on our y-axis.

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We call this right here
a y-intercept.

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It is when x is equal to 0.

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But it's called a y-intercept
because it sits on the y-axis.

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This is where the line
intercepts the y-axis.

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Now let's do the same
for the x-axis.

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Let's set y equal to 0.

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So if y is equal to 0, we have
0 is equal to 2x plus 3.

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We could subtract 3 from both
sides of that equation.

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And you get negative
3 is equal to 2x.

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We can divide both sides by 2.

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And you get negative
3/2 is equal to x.

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And negative 3/2, that's the
same thing as negative 1 1/2.

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So we know that the point
negative 3/2, 0, right?

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The second scenario,
y is equal to 0.

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We know that this set of values
of x and y also satisfy

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this equation.

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Or that this point also
sits on the line.

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So negative 3/2-- remember
that's the same thing as

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negative 1 1/2.

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That's right there. y is 0.

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So that's that second point
we just figured out.

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When y is 0, we're dealing
with our x-intercept.

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We're intercepting the x-axis.

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That is the x-intercept
right there.

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If I wanted to draw the line,
it'll look like this.

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Very roughly, I'll connect those
points and then just

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keep going.

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Keep going in both directions.

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Keep going in both directions
forever and then I will have

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drawn my line.

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Let's do one more of these.

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I don't want to make it
too messy over here.

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Let me clear all this stuff
out of the way.

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Let me clear this out
of the way as well.

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You could always pause it, if
you want to gaze at it-- oh I

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shouldn't have done that, I got
rid of my-- well you can

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see still the axes.

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I'll redraw the axes.

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I'll redraw it right there.

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Good enough.

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That's one, my x-axis.

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That is my y-axis right
over there.

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Let me do part (b).
(b) looks like an

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interesting one right there.

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(b) has 6 times x minus 1 is
equal to 2 times y plus 3.

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So let's look at the situation
when x is equal to zero.

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When x is equal to 0, that
equation up there becomes 6

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times negative 1.

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Right?

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Because this is 0.

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So 6 times negative 1 is
negative 6 is equal to 2-- let

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me distribute this--
2y plus 6.

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I multiplied 2 times the
y and times the 3.

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And then we can subtract 6 from
both sides, so that you

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get negative 12.

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Negative 12 is equal to 2y.

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I subtracted 6 from both sides
to essentially move this 6 on

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to the left-hand side.

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You subtract 6 here,
this disappears.

00:05:04.450 --> 00:05:06.910
You subtract 6 from here,
you get negative 12.

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Divide both sides by 2, you get
negative 6 is equal to y.

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So our first point that we know
is on the line is a point

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x is equal to 0, y is
equal to negative 6.

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That is our y-intercept.

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Now let's see what happens
when y is equal to 0.

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Go back to the original
equation.

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We have-- let me write that a
little bit different-- when y

00:05:40.330 --> 00:05:44.260
is equal to 0, we have
6 times x minus 1.

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Let me just distribute that.

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So 6x minus 6 is equal
to 2 times y.

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Well, y is just going to be 0.

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So it's going to be 2 times 3.

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This whole thing is
going to be 3.

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It's going to be equal to 6.

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So let's subtract 6 from both
sides-- or let's add 6 to both

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sides of this equation.

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I want to get rid of
that right there.

00:06:02.800 --> 00:06:05.380
So I need to add 6
to make that 0.

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So we get 6x is equal to-- you
add 6 to the right-hand side,

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you get 12.

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And you get x-- divide both
sides by 6, you get

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x is equal to 2.

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So our other intercept is x is
equal to 2, y is equal to 0.

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So we have x is equal to
2, y is equal to 0.

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Right there.

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That is our x-intercept.

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And then our actual line, we
just connect the dots.

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It will look something
like that.

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Obviously, you have trouble
drawing a straight line, but I

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think you got the idea.

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Two points define a line.

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Let me do one more of these.

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I could probably do it
up here, in this

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real estate up here.

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Let's do part d.

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You can do part c if you
want extra practice.

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So part d, we have x plus
y is equal to 8.

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This is actually very,
very straightforward.

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When x is equal to 0,
what is y equal to?

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If this is 0, all we have
left is y is equal to 8.

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And then when y is
0, what is x?

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Well, you can just almost
cover this up.

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Then you just get x plus
0 is equal to 8.

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x is equal to 8.

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So that was pretty
straightforward right there.

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So we have one point, 0, 8.

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So 0, 8.

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That is the y-intercept.

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And then we have
the point 8, 0.

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8, 0.

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That is the x-intercept.

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And then we can connect
the dots.

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It looks something like that.

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Now let's do this problem
down here.

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At a local grocery store,
strawberries cost $3.00 per

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pound and bananas cost
$1.00 per pound.

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If I have $10 to spend between
strawberries and bananas, draw

00:07:59.555 --> 00:08:03.140
a graph to show the combination
of each I can buy

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and spend exactly $10.

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Let me draw my axes.

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Make sure that these are
nice and filled in.

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So that remember is my
horizontal access.

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But just for fun, instead of
calling it the x-axis, I'm

00:08:22.070 --> 00:08:24.235
going to call it the
strawberry-axis.

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Actually let me call it s, the
s-axis, where s is for

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strawberries.

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Let s equal the number
of strawberries.

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I think you know where
I'm going.

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b will be number of bananas.

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Let me do my b-axis.

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I'm going to plot the number
bananas on the vertical axis.

00:08:55.340 --> 00:08:58.760
So this right there,
that is the b-axis.

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b for bananas.

00:09:06.070 --> 00:09:07.860
Strawberries cost $3.00.

00:09:07.860 --> 00:09:09.380
OK, this isn't the number
of strawberries.

00:09:09.380 --> 00:09:10.790
This is the number of pounds.

00:09:10.790 --> 00:09:12.040
Let me clear that up.

00:09:12.040 --> 00:09:13.430
This is pounds of
strawberries.

00:09:13.430 --> 00:09:14.990
We're not going by the number
of strawberries.

00:09:14.990 --> 00:09:16.920
We're going by pounds
of strawberries.

00:09:16.920 --> 00:09:21.660
This is pounds of bananas.

00:09:21.660 --> 00:09:22.680
All right.

00:09:22.680 --> 00:09:25.480
Strawberries cost $3.00
per pound, bananas

00:09:25.480 --> 00:09:27.080
cost $1.00 per pound.

00:09:27.080 --> 00:09:28.800
So how much am I
going to spend?

00:09:28.800 --> 00:09:33.690
I'm going to spend $3.00 times
the number of strawberries

00:09:33.690 --> 00:09:35.860
because they're $3.00.

00:09:35.860 --> 00:09:38.230
All right, I'm going to spend
$3.00 times the number pounds

00:09:38.230 --> 00:09:42.750
of strawberries because it's
$3.00 per pound plus $1.00

00:09:42.750 --> 00:09:45.810
times the number of
pounds of bananas.

00:09:45.810 --> 00:09:49.330
And they say I have $10.00
to spend on both.

00:09:49.330 --> 00:09:52.040
So that's going to
be equal to $10.

00:09:52.040 --> 00:09:56.010
So 3 times the pounds of
strawberries plus the pounds

00:09:56.010 --> 00:09:57.650
of bananas are going
to be equal to 10.

00:09:57.650 --> 00:10:01.710
1b, that's the same thing as b,
so I can rewrite this as 3s

00:10:01.710 --> 00:10:04.160
plus b is equal to 10.

00:10:04.160 --> 00:10:06.610
Now let's plot this.

00:10:06.610 --> 00:10:09.235
Let's look at the situation
where I get no strawberries.

00:10:13.030 --> 00:10:16.120
So my pounds of strawberries
are 0.

00:10:16.120 --> 00:10:18.280
What does this equation
become?

00:10:18.280 --> 00:10:22.860
It becomes 3 times 0 plus
b is equal to 10.

00:10:22.860 --> 00:10:24.700
That's just 0.

00:10:24.700 --> 00:10:26.550
So b will be equal to 10.

00:10:26.550 --> 00:10:29.300
So I would have the point 0
pounds of strawberries, I

00:10:29.300 --> 00:10:31.380
could get 10 pounds
of bananas.

00:10:31.380 --> 00:10:34.650
So if I get 0 pounds of
strawberries, I can get 10

00:10:34.650 --> 00:10:38.710
pounds, right there,
of bananas.

00:10:38.710 --> 00:10:41.510
Now what about the
other scenario?

00:10:41.510 --> 00:10:44.330
What about the scenario where
I get no bananas?

00:10:44.330 --> 00:10:46.400
I get 0 pounds of bananas.

00:10:46.400 --> 00:10:47.610
Now let's substitute
back here.

00:10:47.610 --> 00:10:52.120
We have 3 times my pounds of
strawberries plus 0 pounds of

00:10:52.120 --> 00:10:53.780
bananas will equal 10.

00:10:53.780 --> 00:10:55.410
That's just a 0.

00:10:55.410 --> 00:10:57.470
Divide both sides by 3.

00:10:57.470 --> 00:11:00.960
I could-- if I get no bananas--
I can get 10/3

00:11:00.960 --> 00:11:02.400
pounds of strawberries.

00:11:02.400 --> 00:11:05.480
Or this is equal to 3 1/3.

00:11:05.480 --> 00:11:11.170
So it would have the point 3 1/3
pounds of strawberries, 0

00:11:11.170 --> 00:11:12.890
pounds of bananas.

00:11:12.890 --> 00:11:15.240
So 3 1/3 pounds of

00:11:15.240 --> 00:11:18.290
strawberries, 0 pounds of bananas.

00:11:18.290 --> 00:11:21.110
And then we can connect
the line.

00:11:21.110 --> 00:11:23.010
This whole line I'm going to
draw, I am just going to draw

00:11:23.010 --> 00:11:24.090
it in the first quadrant.

00:11:24.090 --> 00:11:25.880
Because I can't have
negative bananas.

00:11:25.880 --> 00:11:27.760
I'm not going to sell
pounds of bananas.

00:11:27.760 --> 00:11:29.210
And I can't have negative
strawberries.

00:11:29.210 --> 00:11:30.620
We're not talking about
selling strawberries.

00:11:30.620 --> 00:11:32.810
So I can only buy
these things.

00:11:32.810 --> 00:11:34.150
This is the amount
that I'm buying.

00:11:34.150 --> 00:11:38.010
So let me to connect the
dots right there.

00:11:38.010 --> 00:11:41.700
This is neat because this line
shows all of the possible

00:11:41.700 --> 00:11:45.580
combinations of pounds of
strawberries and bananas.

00:11:45.580 --> 00:11:49.700
For example, this point right
here-- I don't know if I'm

00:11:49.700 --> 00:11:54.360
drawing it exactly-- it looks
like if I get about 5 pounds

00:11:54.360 --> 00:11:59.200
of bananas, I can get a little
under 2 pounds of

00:11:59.200 --> 00:12:00.350
strawberries.

00:12:00.350 --> 00:12:01.110
That's what that tells me.

00:12:01.110 --> 00:12:06.460
If I get 3 pounds of
strawberries, if I get exactly

00:12:06.460 --> 00:12:10.010
3 pounds of strawberries, I can
get 1 pound of bananas.

00:12:10.010 --> 00:12:12.720
Every point here represents
a combination that

00:12:12.720 --> 00:12:16.040
I can get for $10.


Coordinate plane examples

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.730
What we're going to do in this
video is, through a bunch of

00:00:03.730 --> 00:00:06.710
examples, familiarize
ourselves with the

00:00:06.710 --> 00:00:07.670
x,y-coordinate plane.

00:00:07.670 --> 00:00:10.810
And first we're going to just
look at some points that are

00:00:10.810 --> 00:00:13.500
already plotted and figure
out their coordinates.

00:00:13.500 --> 00:00:15.390
Then we're going to look at some
coordinates and figure

00:00:15.390 --> 00:00:16.290
out where those points are.

00:00:16.290 --> 00:00:17.395
Then we'll do one
more problem.

00:00:17.395 --> 00:00:19.700
So let's figure out
what are the

00:00:19.700 --> 00:00:21.890
coordinates of these points?

00:00:21.890 --> 00:00:24.550
So you have this point
right here, A.

00:00:24.550 --> 00:00:26.680
So its x-coordinate, you
can see it right there.

00:00:26.680 --> 00:00:27.600
You just drop down.

00:00:27.600 --> 00:00:29.800
Where does it intersect
the x-axis?

00:00:29.800 --> 00:00:31.346
x is equal to 5.

00:00:31.346 --> 00:00:36.960
So it's the point 5 comma and
y is going to be equal to 6.

00:00:36.960 --> 00:00:38.990
5 comma 6.

00:00:38.990 --> 00:00:41.920
Now this point B here, what's
the x-coordinate?

00:00:41.920 --> 00:00:43.920
It is 5 to the left.

00:00:43.920 --> 00:00:45.940
5 to the left of the x-axis.

00:00:45.940 --> 00:00:47.496
This is negative 5.

00:00:47.496 --> 00:00:50.590
Its x-coordinate
is negative 5.

00:00:50.590 --> 00:00:54.340
y-coordinate is, if you just
go straight to the right,

00:00:54.340 --> 00:00:57.210
you're going to hit
y is equal to 5.

00:00:57.210 --> 00:00:58.620
y is equal 5.

00:00:58.620 --> 00:01:00.190
Let me switch colors.

00:01:00.190 --> 00:01:01.050
C.

00:01:01.050 --> 00:01:02.750
I think you're getting
the hang of this.

00:01:02.750 --> 00:01:06.040
Let's do the y-coordinate first.
The y-coordinate is 3.

00:01:06.040 --> 00:01:07.760
You see that right there.

00:01:07.760 --> 00:01:10.440
And then the x-coordinate
is negative 2.

00:01:10.440 --> 00:01:11.130
Negative 2.

00:01:11.130 --> 00:01:12.850
You always put the x-coordinate
first. That's

00:01:12.850 --> 00:01:14.800
just the convention we use.

00:01:14.800 --> 00:01:18.660
D, x-coordinate negative 2.

00:01:18.660 --> 00:01:20.210
You see that right there.

00:01:20.210 --> 00:01:23.750
y-coordinate negative
2, as well.

00:01:23.750 --> 00:01:25.040
Let me get another color.

00:01:25.040 --> 00:01:28.170
E, let's do the y-coordinate.

00:01:28.170 --> 00:01:29.650
We'll figure it out first,
but you always

00:01:29.650 --> 00:01:31.450
have to write it second.

00:01:31.450 --> 00:01:32.780
It's negative 4.

00:01:32.780 --> 00:01:34.510
You see that right there,
the y-coordinate.

00:01:34.510 --> 00:01:37.420
The x-coordinate is 3.

00:01:37.420 --> 00:01:39.820
And then finally, F.

00:01:39.820 --> 00:01:42.046
The x-coordinate is 2.

00:01:42.046 --> 00:01:46.000
And the y-coordinate
is negative 6.

00:01:46.000 --> 00:01:48.550
Hopefully that gives you a sense
of at least figuring out

00:01:48.550 --> 00:01:49.570
the coordinates.

00:01:49.570 --> 00:01:50.550
Now let's go the other way.

00:01:50.550 --> 00:01:53.190
Let's start with coordinates
and figure out where those

00:01:53.190 --> 00:01:54.600
points are.

00:01:54.600 --> 00:01:55.900
So you have this first one.

00:01:55.900 --> 00:01:58.560
I'll do it lowercase case a in
parentheses to differentiate

00:01:58.560 --> 00:02:00.200
it from this uppercase A.

00:02:00.200 --> 00:02:03.190
So it's at 4 comma 2.
x is equal to 4.

00:02:03.190 --> 00:02:04.500
y is equal to 2.

00:02:08.410 --> 00:02:11.360
So that's that point
right there.

00:02:11.360 --> 00:02:12.855
Let's do the next one.

00:02:12.855 --> 00:02:15.850
Let me do it in a color that
you'll be able to read.

00:02:15.850 --> 00:02:17.150
b.

00:02:17.150 --> 00:02:20.680
x is equal to negative 3.

00:02:20.680 --> 00:02:23.190
y is equal to 5.5.

00:02:23.190 --> 00:02:27.480
So you go all the
way up to 5.5.

00:02:27.480 --> 00:02:29.440
y is equal to 5.5.

00:02:29.440 --> 00:02:34.100
So that is the point lowercase
b with parentheses around it.

00:02:34.100 --> 00:02:39.050
Then c, 4 negative 4.
x is equal to 4.

00:02:39.050 --> 00:02:41.300
y is equal to negative 4.

00:02:41.300 --> 00:02:44.190
Right over there.

00:02:44.190 --> 00:02:45.610
And then one last one.

00:02:45.610 --> 00:02:47.230
I'll do it in orange.

00:02:47.230 --> 00:02:50.915
d, x is negative 2,
y is negative 3.

00:02:54.760 --> 00:02:56.140
Right there.

00:02:56.140 --> 00:02:58.180
That's the d with parentheses.

00:02:58.180 --> 00:02:58.980
And you could have gone
the other way.

00:02:58.980 --> 00:03:01.790
You could have said, hey, y
is equal to negative 3.

00:03:01.790 --> 00:03:02.690
x is equal to negative 2.

00:03:02.690 --> 00:03:04.360
So you could go to the
left and down.

00:03:04.360 --> 00:03:06.070
Or you could go down
and to the left.

00:03:06.070 --> 00:03:09.610
And you're still going to
get to the same point.

00:03:09.610 --> 00:03:12.400
So hopefully that gives you a
good sense of how to figure

00:03:12.400 --> 00:03:13.160
out coordinates.

00:03:13.160 --> 00:03:15.160
Or if you're given coordinates,
how to figure out

00:03:15.160 --> 00:03:17.550
where to plot something on
the x,y-coordinate plane.

00:03:17.550 --> 00:03:20.480
Now let's do a slightly
more involved problem.

00:03:20.480 --> 00:03:24.100
So it says the following 3
points are 3 vertices of

00:03:24.100 --> 00:03:26.000
square A, B, C, D.

00:03:26.000 --> 00:03:27.440
Plot them on a graph.

00:03:27.440 --> 00:03:29.180
Then determine what the
coordinates of the fourth

00:03:29.180 --> 00:03:31.030
point, D, would be.

00:03:31.030 --> 00:03:33.210
All right, let's plot
these on a graph, as

00:03:33.210 --> 00:03:35.230
they tell us to do.

00:03:35.230 --> 00:03:35.860
All right.

00:03:35.860 --> 00:03:38.890
That'll be my y-axis.

00:03:38.890 --> 00:03:41.410
That's my y-axis.

00:03:41.410 --> 00:03:42.910
The vertical axis.

00:03:42.910 --> 00:03:44.160
That'll be my x-axis.

00:03:46.620 --> 00:03:48.750
And let me put some--
let me mark it.

00:03:48.750 --> 00:03:52.800
So that's x equals 1, 2, 3, 4.

00:03:52.800 --> 00:03:56.700
This is x is equal to negative
1, negative 2, negative 3,

00:03:56.700 --> 00:03:57.460
negative 4.

00:03:57.460 --> 00:04:01.000
That's y is equal
to 1, 2, 3, 4.

00:04:01.000 --> 00:04:04.270
This is y is equal to negative
1, negative 2, negative 3,

00:04:04.270 --> 00:04:05.590
negative 4.

00:04:05.590 --> 00:04:07.350
I could write that
this y equals 4.

00:04:07.350 --> 00:04:10.530
This y equals negative
4. x is equal to 4.

00:04:10.530 --> 00:04:13.950
x is equal to negative 4.

00:04:13.950 --> 00:04:15.180
And let's see.

00:04:15.180 --> 00:04:16.649
Let's plot these points.

00:04:16.649 --> 00:04:18.970
So first, we have the point
A is equal to negative 4,

00:04:18.970 --> 00:04:20.730
negative 4.

00:04:20.730 --> 00:04:23.230
So we go x is negative 4.

00:04:23.230 --> 00:04:24.810
And then y is negative 4.

00:04:24.810 --> 00:04:27.040
So we drop down 4 right there.

00:04:27.040 --> 00:04:29.790
And that is our point A.

00:04:29.790 --> 00:04:32.080
Negative 4, negative 4.

00:04:32.080 --> 00:04:34.350
And just to familiarize yourself
with a labeling

00:04:34.350 --> 00:04:36.790
scheme that you may or may not
have seen before, is that

00:04:36.790 --> 00:04:39.600
people label these sections
of the coordinate plane.

00:04:39.600 --> 00:04:41.440
They call this the
first quadrant.

00:04:41.440 --> 00:04:43.510
They call this the
second quadrant.

00:04:43.510 --> 00:04:45.850
They call this the
third quadrant.

00:04:45.850 --> 00:04:47.950
And they call this the
fourth quadrant.

00:04:47.950 --> 00:04:51.290
And these are just the Roman
numerals for I, II, III, and

00:04:51.290 --> 00:04:53.570
IV, So this point is in
the third quadrant.

00:04:53.570 --> 00:04:56.480
When we looked up at this stuff
over here, these points

00:04:56.480 --> 00:04:57.840
are in the fourth quadrant.

00:04:57.840 --> 00:05:01.160
These are in the third,
second, first. Just an

00:05:01.160 --> 00:05:02.430
interesting thing to know.

00:05:02.430 --> 00:05:03.920
Sometimes someone might
ask you, what quadrant

00:05:03.920 --> 00:05:04.540
is that point in?

00:05:04.540 --> 00:05:05.990
And you just say, OK, I see.

00:05:05.990 --> 00:05:08.850
If they're both negative,
they're going to be in the

00:05:08.850 --> 00:05:10.000
third quadrant.

00:05:10.000 --> 00:05:13.100
If just the y is negative, but
the x is positive, you're

00:05:13.100 --> 00:05:14.640
going to be in the fourth.

00:05:14.640 --> 00:05:17.560
If they're both positive, you're
in the first. If y is

00:05:17.560 --> 00:05:19.760
positive, but x is negative,
you're in the second.

00:05:19.760 --> 00:05:21.150
And we'll talk a little
bit about that as

00:05:21.150 --> 00:05:22.480
we plot these points.

00:05:22.480 --> 00:05:24.880
So point B, x is positive.

00:05:24.880 --> 00:05:26.730
It's 1, 2, 3.

00:05:26.730 --> 00:05:28.430
And y is negative 4.

00:05:28.430 --> 00:05:31.710
So we drop down here into
the fourth quadrant.

00:05:31.710 --> 00:05:33.580
That is the point B.

00:05:33.580 --> 00:05:35.620
It's 3, negative 4.

00:05:35.620 --> 00:05:38.280
So we can already see the bottom
of our rectangle that

00:05:38.280 --> 00:05:39.920
they're talking about,
right there.

00:05:39.920 --> 00:05:43.700
And notice, both of these
have the exact same y.

00:05:43.700 --> 00:05:48.080
They're both at the same
level below the x-axis.

00:05:48.080 --> 00:05:49.600
And then what's the
next point?

00:05:49.600 --> 00:05:51.880
Point C is 3 comma 3.

00:05:51.880 --> 00:05:53.880
So 3 comma 3.

00:05:53.880 --> 00:05:55.100
It's in the first quadrant.

00:05:55.100 --> 00:05:57.600
Both of its coordinates
are positive.

00:05:57.600 --> 00:05:58.490
3 comma 3.

00:05:58.490 --> 00:05:59.910
Both x and y are positive.

00:05:59.910 --> 00:06:02.620
And notice, it's on the
same vertical as B.

00:06:02.620 --> 00:06:04.690
It has the same x value.

00:06:04.690 --> 00:06:06.460
They both have an
x value of 3.

00:06:06.460 --> 00:06:08.220
So it's right above it.

00:06:08.220 --> 00:06:08.910
Right above it.

00:06:08.910 --> 00:06:12.210
Now we have to figure out
the last point here.

00:06:12.210 --> 00:06:14.740
Well, the point is going to
have to be on the same

00:06:14.740 --> 00:06:15.760
vertical as this point.

00:06:15.760 --> 00:06:17.690
It's going to have to be on
the same vertical as this

00:06:17.690 --> 00:06:20.290
point, which means it's going
to have the same x value as

00:06:20.290 --> 00:06:21.250
this point.

00:06:21.250 --> 00:06:24.320
So its x value is going
to be negative 4.

00:06:24.320 --> 00:06:26.360
And then it's going to have to
be on the same horizontal as

00:06:26.360 --> 00:06:27.660
this point.

00:06:27.660 --> 00:06:28.680
It's going to have
to be on the same

00:06:28.680 --> 00:06:30.020
horizontal as that point.

00:06:30.020 --> 00:06:32.560
So it's going to have to have
the same y value at the same

00:06:32.560 --> 00:06:34.320
height above the x-axis.

00:06:34.320 --> 00:06:36.270
So it's going to have to be 3.

00:06:36.270 --> 00:06:38.570
So that is our point D.

00:06:38.570 --> 00:06:41.850
Notice it's at negative
4, right above A.

00:06:41.850 --> 00:06:44.230
And it's at y is equal to 3.

00:06:44.230 --> 00:06:47.210
Right to the left of point C.

00:06:47.210 --> 00:06:48.980
And we are done.


Problem Solving Word Problems 2

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:03.500
Let's do a couple of word
problems. This tells us that

00:00:03.500 --> 00:00:07.770
Nadia's at home and Peter is
at school, which is 6 miles

00:00:07.770 --> 00:00:08.660
away from home.

00:00:08.660 --> 00:00:10.820
So let me draw a little
diagram here.

00:00:10.820 --> 00:00:13.230
So Nadia is there.

00:00:13.230 --> 00:00:17.950
Peter is over there, and
they're 6 miles apart.

00:00:17.950 --> 00:00:20.470
So this is the distance between
home and school.

00:00:20.470 --> 00:00:22.115
That is 6 miles.

00:00:25.630 --> 00:00:28.440
They start traveling to each
other at the same time.

00:00:28.440 --> 00:00:31.360
So they both leave at the
same time, traveling

00:00:31.360 --> 00:00:32.680
towards each other.

00:00:32.680 --> 00:00:35.890
Nadia is walking at 3 and
1/2 miles per hour.

00:00:35.890 --> 00:00:44.460
So she's travelling at
3 and 1/2 miles per

00:00:44.460 --> 00:00:46.990
hour in that direction.

00:00:46.990 --> 00:00:49.150
And Peter is skateboarding
at 6 miles per

00:00:49.150 --> 00:00:50.060
hour in that direction.

00:00:50.060 --> 00:00:54.160
So Peter is going in that
direction at 6 miles per hour.

00:00:54.160 --> 00:00:57.410
When will they meet and
how far from home is

00:00:57.410 --> 00:00:58.980
their meeting place?

00:00:58.980 --> 00:01:02.200
So let's say that their meeting
place is right-- I

00:01:02.200 --> 00:01:02.390
don't know.

00:01:02.390 --> 00:01:04.120
It's going to be closer to
home, because this guy is

00:01:04.120 --> 00:01:05.000
going faster.

00:01:05.000 --> 00:01:06.630
Peter is going faster.

00:01:06.630 --> 00:01:12.020
So let's say, let x be
equal to the distance

00:01:12.020 --> 00:01:13.270
from home they meet.

00:01:19.340 --> 00:01:23.120
So this distance right here
is going to be x.

00:01:23.120 --> 00:01:24.750
And then what's this distance
going to be?

00:01:24.750 --> 00:01:28.300
Well, if that's x, and this
whole thing is 6 miles, then

00:01:28.300 --> 00:01:32.460
this distance right there is
going to be 6 minus x.

00:01:32.460 --> 00:01:35.760
So Nadia will travel x
miles and Peter will

00:01:35.760 --> 00:01:39.400
travel 6 minus x miles.

00:01:39.400 --> 00:01:41.340
And they're going to travel
the same amount of time.

00:01:41.340 --> 00:01:45.730
Remember, they both leave at
time 0, and after time t, they

00:01:45.730 --> 00:01:48.050
will both meet right there.

00:01:48.050 --> 00:01:50.050
So they're both going to take
the same amount of time.

00:01:50.050 --> 00:01:54.180
So Nadia's equation, if we just
remember that distance is

00:01:54.180 --> 00:01:57.510
equal to rate times time, we
could use that for both of

00:01:57.510 --> 00:02:02.950
them, in Nadia's case, the
distance she travels is x.

00:02:02.950 --> 00:02:08.280
x is going to be equal to her
rate, which is 3.5 miles per

00:02:08.280 --> 00:02:12.410
hour, times t, which is the
time that she travels.

00:02:12.410 --> 00:02:15.660
And then Peter's equation is
going to be his distance.

00:02:15.660 --> 00:02:20.880
Well, he travels 6 minus x
miles, so it's 6 minus x is

00:02:20.880 --> 00:02:25.190
going to be equal to his rate,
which is 6 miles per hour

00:02:25.190 --> 00:02:26.600
times the time he travels.

00:02:26.600 --> 00:02:31.960
Well, he also travels
t hours, so times t.

00:02:31.960 --> 00:02:34.660
So we have two equations
and two unknowns, so we

00:02:34.660 --> 00:02:36.420
can solve for them.

00:02:36.420 --> 00:02:39.840
We've already solved
for x here.

00:02:39.840 --> 00:02:44.660
So let's substitute this
for x right there.

00:02:44.660 --> 00:02:49.510
So we can rewrite this equation
as 6 minus, but

00:02:49.510 --> 00:02:55.390
instead of an x, we can put
3.5t, 6 minus 3.5t, because we

00:02:55.390 --> 00:03:02.160
know that x is equal to
3.5t, is equal to 6t.

00:03:02.160 --> 00:03:07.270
And then let's see, if we add
3.5t to both sides of this

00:03:07.270 --> 00:03:17.130
equation, we get 6 is equal to
3.5t plus 6t is 9.5t, or if we

00:03:17.130 --> 00:03:27.880
divide both sides by 9.5, you
get t is equal to 6 over 9.5

00:03:27.880 --> 00:03:30.110
hours, which is a bit
of a bizarre number.

00:03:30.110 --> 00:03:32.520
But 9.5, we can rewrite that.

00:03:32.520 --> 00:03:36.670
That is the same thing as 6
over-- see, 9.5, that's the

00:03:36.670 --> 00:03:40.020
same thing as 19/2, right?

00:03:40.020 --> 00:03:44.530
19/2, you divide that and
you would get 9.5.

00:03:44.530 --> 00:03:48.620
So this is equal to
6 times 2 over 19.

00:03:48.620 --> 00:03:53.860
So this is equal to
12/19 of an hour.

00:03:53.860 --> 00:03:56.290
That's how long it'll
take them to meet.

00:03:56.290 --> 00:03:59.210
And then if you ask the question
how far from home

00:03:59.210 --> 00:04:00.170
will they meet?

00:04:00.170 --> 00:04:03.310
If you want to figure out the
x value, x is going to be

00:04:03.310 --> 00:04:09.390
equal to 3.5-- actually, instead
of writing 3.5, I can

00:04:09.390 --> 00:04:16.779
write 3.5 as 7/2; that's the
same thing as 3.5-- times our

00:04:16.779 --> 00:04:21.672
time, times 12/19.

00:04:21.672 --> 00:04:25.740
So you divide the numerator and
the denominator by 2, and

00:04:25.740 --> 00:04:33.360
we get 7 times 6 is
42 over 19 miles.

00:04:33.360 --> 00:04:36.230
And this right here,
that is in hours.

00:04:36.230 --> 00:04:40.110
So in 12/19 of an hour-- it's
a weird fraction-- they will

00:04:40.110 --> 00:04:43.560
meet exactly 42/19
miles from home.

00:04:43.560 --> 00:04:47.580
So it's a little over
2 miles from home.

00:04:47.580 --> 00:04:50.150
Next problem.

00:04:50.150 --> 00:04:55.760
Peter bought several notebooks
at Staples for $2.25.

00:04:55.760 --> 00:04:58.650
And he bought a few
more notebooks at

00:04:58.650 --> 00:05:00.650
Rite Aid for $2 each.

00:05:00.650 --> 00:05:03.900
He spent the same amount of
money in both places and he

00:05:03.900 --> 00:05:06.620
bought 17 notebooks in total.

00:05:06.620 --> 00:05:09.530
How many notebooks did
he buy in each store?

00:05:09.530 --> 00:05:12.320
So let's define some
variables.

00:05:12.320 --> 00:05:22.120
Let's let S equal number
bought at Staples.

00:05:22.120 --> 00:05:28.990
And then we could say that R is
equal to the number bought

00:05:28.990 --> 00:05:32.910
at Rite Aid.

00:05:32.910 --> 00:05:34.870
So what do we know?

00:05:37.430 --> 00:05:41.630
So he bought a total
of 17 notebooks.

00:05:41.630 --> 00:05:42.800
Let me do that in a
different color.

00:05:42.800 --> 00:05:47.550
He bought a total
of 17 notebooks.

00:05:47.550 --> 00:05:53.330
So that tells us that S
plus R is equal to 17.

00:05:53.330 --> 00:05:58.600
And we also know that he spent
the same amount of money in

00:05:58.600 --> 00:06:00.320
both places.

00:06:00.320 --> 00:06:02.110
So how much did he spend
at Rite Aid?

00:06:02.110 --> 00:06:02.990
Let me do this in orange.

00:06:02.990 --> 00:06:07.190
Spent the same amount of
money in both places.

00:06:07.190 --> 00:06:10.210
So at Staples, how much
did he spend?

00:06:10.210 --> 00:06:21.940
He bought S notebooks for $2.25
each, so 2.25S, that's

00:06:21.940 --> 00:06:23.345
how much he spent at Staples.

00:06:29.290 --> 00:06:31.030
That's going to be equal
to the amount he

00:06:31.030 --> 00:06:31.930
spent at Rite Aid.

00:06:31.930 --> 00:06:34.340
He spent $2 per notebook
at Rite Aid.

00:06:37.540 --> 00:06:41.310
So $2 times the number of
notebooks from Rite Aid, so

00:06:41.310 --> 00:06:47.680
this is spent at Rite Aid.

00:06:47.680 --> 00:06:49.890
So once again, two equations
with two unknowns.

00:06:49.890 --> 00:06:53.710
We can do a little bit of
substitution maybe.

00:06:53.710 --> 00:06:55.900
So what's the best way
to substitute here?

00:06:55.900 --> 00:06:57.870
Well, let's divide both
sides of this

00:06:57.870 --> 00:07:01.070
equation by 2 right here.

00:07:01.070 --> 00:07:05.350
So you have-- I'm going to
rewrite 2.2-- well, I'll just

00:07:05.350 --> 00:07:05.950
do it like this.

00:07:05.950 --> 00:07:07.590
So you have R.

00:07:07.590 --> 00:07:12.570
If you divide both sides of this
by 2, you have 2.25 over

00:07:12.570 --> 00:07:16.360
2S is equal to R, right?

00:07:16.360 --> 00:07:19.750
I just divided both sides
of this by 2.

00:07:19.750 --> 00:07:27.080
So if you take this value and
you substitute it in for R

00:07:27.080 --> 00:07:34.540
right there, this equation
becomes S plus-- instead of an

00:07:34.540 --> 00:07:44.930
R we have 2.25 over
2S is equal to 17.

00:07:44.930 --> 00:07:47.610
And let's just simplify.

00:07:47.610 --> 00:07:51.200
So this is 1-- we could view
this as 2 over 2S, right?

00:07:51.200 --> 00:07:53.200
The coefficient there
is just 1.

00:07:53.200 --> 00:07:56.310
So we can view this as we have
a common denominator.

00:07:56.310 --> 00:08:02.550
This is the same thing is 4.25
over 2S is equal to 17.

00:08:02.550 --> 00:08:05.710
I'm avoiding doing any
hard math just yet.

00:08:05.710 --> 00:08:09.280
Let me multiply both sides
by the inverse.

00:08:09.280 --> 00:08:11.390
2/4.25.

00:08:11.390 --> 00:08:14.490
It's a little bizarre to have
an expression with both a

00:08:14.490 --> 00:08:17.045
fraction and a decimal,
but it's not illegal.

00:08:17.045 --> 00:08:21.680
Let's multiply both
times 2/4.25.

00:08:21.680 --> 00:08:27.720
These cancel out, and so you get
S is equal to 17 times 2

00:08:27.720 --> 00:08:33.360
is 34 over 4.25.

00:08:33.360 --> 00:08:34.850
And actually, I can
eyeball that.

00:08:34.850 --> 00:08:40.169
That looks like that should
be equal to 8, right?

00:08:40.169 --> 00:08:48.300
4.25 is 4 and 1/4, which is the
same thing as 17/4 over 4.

00:08:48.300 --> 00:08:53.530
So this is the same thing as
34 over 17/4, which is the

00:08:53.530 --> 00:08:58.750
same thing as 34 times 4/17.

00:08:58.750 --> 00:09:02.690
Put a 1 there, divide by 17, you
get a 2, divide by 17, you

00:09:02.690 --> 00:09:05.210
get a 1, 2 times 4 is 8.

00:09:05.210 --> 00:09:10.530
So he bought 8 notebooks at
Staples, this is 8, and then

00:09:10.530 --> 00:09:15.940
at Rite Aid, he bought-- well,
8 plus R is equal to 17.

00:09:15.940 --> 00:09:17.820
Subtract 8 from both sides.

00:09:17.820 --> 00:09:22.920
He must have bought 9 notebooks
at Rite Aid.

00:09:22.920 --> 00:09:24.540
Let's do one more.

00:09:24.540 --> 00:09:27.400
This one is especially
fun-looking.

00:09:27.400 --> 00:09:29.980
Peter is outside, looking
at the pigs and

00:09:29.980 --> 00:09:31.430
chickens in the yard.

00:09:31.430 --> 00:09:34.310
Nadia is indoors and cannot
see the animals.

00:09:34.310 --> 00:09:35.590
Peter gives her a puzzle.

00:09:35.590 --> 00:09:40.200
He tells her that he counts 13
heads and 36 feet and asks her

00:09:40.200 --> 00:09:43.540
how many pigs and how many
chickens are in the yard.

00:09:43.540 --> 00:09:45.520
So let's once again define
our variables.

00:09:45.520 --> 00:09:50.160
P is equal to the
number of pigs.

00:09:50.160 --> 00:09:56.510
And let C is equal to the
number of chickens.

00:09:56.510 --> 00:09:59.550
So the number of heads will
essentially be the number of

00:09:59.550 --> 00:10:02.660
pigs and chickens, assuming
they each have one head.

00:10:02.660 --> 00:10:04.920
So the number of pigs plus the
number of chickens will be

00:10:04.920 --> 00:10:08.190
equal to the number of heads,
so that is right here.

00:10:08.190 --> 00:10:12.490
That is 13 heads.

00:10:12.490 --> 00:10:16.160
And then 4 times the number
of pigs, right?

00:10:16.160 --> 00:10:20.460
Each pig has 4 legs, so 4 times
the number of pigs plus

00:10:20.460 --> 00:10:22.410
2 times the number of chickens,
assuming we're

00:10:22.410 --> 00:10:25.130
dealing with two-legged
chickens, is going to be equal

00:10:25.130 --> 00:10:30.100
to the number of feet,
is equal to 36.

00:10:30.100 --> 00:10:33.400
So once again, we have two
equations with two unknowns.

00:10:33.400 --> 00:10:35.750
Instead of solving it with
substitution, I'm going to do

00:10:35.750 --> 00:10:38.230
it by adding and subtracting
the two equations.

00:10:38.230 --> 00:10:40.990
So what we can do, this
equation, we can

00:10:40.990 --> 00:10:43.740
multiply it times 2.

00:10:43.740 --> 00:10:46.240
And when I say multiply it times
2, we have to multiply

00:10:46.240 --> 00:10:49.720
the entire equation times 2.

00:10:49.720 --> 00:10:51.970
And so it's still true.

00:10:51.970 --> 00:10:53.590
So we multiply the entire
equation times 2.

00:10:53.590 --> 00:11:00.150
This becomes 2P plus
2C is equal to 26.

00:11:00.150 --> 00:11:04.250
And now what I'm going to do is
I'm going to subtract this

00:11:04.250 --> 00:11:06.790
equation from that equation,
which you can do.

00:11:06.790 --> 00:11:09.660
Because that is equal to that,
that is equal to that, and so

00:11:09.660 --> 00:11:12.270
we're not doing anything that
violates the laws of algebra.

00:11:12.270 --> 00:11:14.720
So if you subtract the bottom
equation from the top

00:11:14.720 --> 00:11:17.840
equation, or I could just put
a negative sign everywhere

00:11:17.840 --> 00:11:19.760
here, just so you know
we're subtracting,

00:11:19.760 --> 00:11:22.650
4P minus 2P is 2P.

00:11:22.650 --> 00:11:24.530
2C minus 2C is 0.

00:11:24.530 --> 00:11:27.440
That's why I did this, so that
these would cancel out.

00:11:27.440 --> 00:11:30.750
And then 36 minus 26
is equal to 10.

00:11:30.750 --> 00:11:32.670
So 2P is equal to 10.

00:11:32.670 --> 00:11:34.830
Divide both sides by 2.

00:11:34.830 --> 00:11:36.910
We have 5 pigs.

00:11:36.910 --> 00:11:42.440
And then 5 plus the number of
chickens is equal to 13, so we

00:11:42.440 --> 00:11:46.630
must have-- subtract 5 from
both sides-- 8 chickens.

00:11:46.630 --> 00:11:48.420
Now if this confuses you, you
might want to try it with

00:11:48.420 --> 00:11:51.750
substitution, but I have many
videos on solving systems of

00:11:51.750 --> 00:11:54.430
equations where I go a little
bit slower and explain a

00:11:54.430 --> 00:11:55.710
little bit more of
the logic of it.

00:11:55.710 --> 00:11:57.280
So whatever floats your boat.

00:11:57.280 --> 00:12:00.500
But either way, Nadia should
hopefully guess or hopefully

00:12:00.500 --> 00:12:05.150
solve for 5 pigs
and 8 chickens.

00:12:05.150 --> 00:12:06.160
And it should work out, right?

00:12:06.160 --> 00:12:07.280
You have 13 animals.

00:12:07.280 --> 00:12:08.620
You have 13 heads.

00:12:08.620 --> 00:12:15.480
And if you multiply 5 times 4,
that's 20 pig feet plus 16

00:12:15.480 --> 00:12:16.190
chicken feet.

00:12:16.190 --> 00:12:19.400
20 plus 16 is 36 total feet.


Square roots and real numbers

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.480
I have here a bunch of radical
expressions, or square root

00:00:04.480 --> 00:00:05.110
expressions.

00:00:05.110 --> 00:00:07.600
And what I'm going to do is
go through all of them and

00:00:07.600 --> 00:00:08.500
simplify them.

00:00:08.500 --> 00:00:11.240
And we'll talk about whether
these are rational or

00:00:11.240 --> 00:00:13.390
irrational numbers.

00:00:13.390 --> 00:00:15.710
So let's start with A.

00:00:15.710 --> 00:00:20.440
A is equal to the square
root of 25.

00:00:20.440 --> 00:00:26.560
Well that's the same thing as
the square root of 5 times 5,

00:00:26.560 --> 00:00:31.000
which is a clearly
going to be 5.

00:00:31.000 --> 00:00:34.440
We're focusing on the positive
square root here.

00:00:34.440 --> 00:00:37.060
Now let's do B.

00:00:37.060 --> 00:00:39.920
B I'll do in a different color,
for the principal root,

00:00:39.920 --> 00:00:42.250
when we say positive
square root.

00:00:42.250 --> 00:00:46.200
B, we have the square
root of 24.

00:00:46.200 --> 00:00:47.960
So what you want to do, is
you want to get the prime

00:00:47.960 --> 00:00:50.530
factorization of this
number right here.

00:00:50.530 --> 00:00:53.560
So 24, let's do its prime
factorization.

00:00:53.560 --> 00:00:56.250
This is 2 times 12.

00:00:56.250 --> 00:00:59.720
12 is 2 times 6.

00:00:59.720 --> 00:01:03.430
6 is 2 times 3.

00:01:03.430 --> 00:01:07.220
So the square root of 24, this
is the same thing as the

00:01:07.220 --> 00:01:15.320
square root of 2 times
2 times 2 times 3.

00:01:15.320 --> 00:01:18.080
That's the same thing as 24.

00:01:18.080 --> 00:01:22.530
Well, we see here, we have one
perfect square right there.

00:01:22.530 --> 00:01:23.870
So we could rewrite this.

00:01:23.870 --> 00:01:30.330
This is the same thing as the
square root of 2 times 2 times

00:01:30.330 --> 00:01:34.030
the square root of 2 times 3.

00:01:34.030 --> 00:01:35.890
Now this is clearly 2.

00:01:35.890 --> 00:01:37.010
This is the square root of 4.

00:01:37.010 --> 00:01:38.920
The square root of 4 is 2.

00:01:38.920 --> 00:01:40.710
And then this we can't
simplify anymore.

00:01:40.710 --> 00:01:44.520
We don't see two numbers
multiplied by itself here.

00:01:44.520 --> 00:01:47.940
So this is going to be times
the square root of 6.

00:01:47.940 --> 00:01:50.110
Or we could even right this as
the square root of 2 times the

00:01:50.110 --> 00:01:51.540
square root of 3.

00:01:51.540 --> 00:01:53.210
Now I said I would talk
about whether things

00:01:53.210 --> 00:01:54.550
are rational or not.

00:01:54.550 --> 00:01:56.460
This is rational.

00:01:56.460 --> 00:02:03.630
This part A can be expressed
as the ratio of 2 integers.

00:02:03.630 --> 00:02:05.920
Namely 5/1.

00:02:05.920 --> 00:02:07.340
This is rational.

00:02:07.340 --> 00:02:08.590
This is irrational.

00:02:11.840 --> 00:02:14.060
I'm not going to prove
it in this video.

00:02:14.060 --> 00:02:18.770
But anything that is the product
of irrational numbers.

00:02:18.770 --> 00:02:24.920
And the square root of any prime
number is irrational.

00:02:24.920 --> 00:02:25.790
I'm not proving it here.

00:02:25.790 --> 00:02:29.060
This is the square root of 2
times the square root of 3.

00:02:29.060 --> 00:02:30.365
That's what the square
root of 6 is.

00:02:30.365 --> 00:02:32.280
And that's what makes
this irrational.

00:02:32.280 --> 00:02:35.910
I cannot express this as
any type of fraction.

00:02:35.910 --> 00:02:40.830
I can't express this as some
integer over some other

00:02:40.830 --> 00:02:42.280
integer like I did there.

00:02:42.280 --> 00:02:43.250
And I'm not proving it here.

00:02:43.250 --> 00:02:45.910
I'm just giving you a little
bit of practice.

00:02:45.910 --> 00:02:47.010
And a quicker way to do this.

00:02:47.010 --> 00:02:48.300
You could say, hey,
4 goes into this.

00:02:48.300 --> 00:02:49.770
4 is a perfect square.

00:02:49.770 --> 00:02:50.830
Let me take a 4 out.

00:02:50.830 --> 00:02:52.120
This is 4 times 6.

00:02:52.120 --> 00:02:54.770
The square root of 4 is 2, leave
the 6 in, and you would

00:02:54.770 --> 00:02:56.160
have gotten the 2 square
roots of 6.

00:02:56.160 --> 00:02:58.990
Which you will get the hang of
it eventually, but I want to

00:02:58.990 --> 00:03:01.590
do it systematically first.

00:03:01.590 --> 00:03:03.820
Let's do part C.

00:03:03.820 --> 00:03:06.610
Square root of 20.

00:03:06.610 --> 00:03:12.350
Once again, 20 is 2 times
10, which is 2 times 5.

00:03:12.350 --> 00:03:18.050
So this is the same thing as the
square root of 2 times 2,

00:03:18.050 --> 00:03:20.740
right, times 5.

00:03:20.740 --> 00:03:22.690
Now, the square root of 2 times
2, that's clearly just

00:03:22.690 --> 00:03:25.120
going to be 2.

00:03:25.120 --> 00:03:26.530
It's going to be the square
root of this times

00:03:26.530 --> 00:03:27.380
square root of that.

00:03:27.380 --> 00:03:29.400
2 times the square root of 5.

00:03:29.400 --> 00:03:31.090
And once again, you could
probably do that in your head

00:03:31.090 --> 00:03:31.910
with a little practice.

00:03:31.910 --> 00:03:34.920
The square root of the
20 is 4 times 5.

00:03:34.920 --> 00:03:36.550
The square root of 4 is 2.

00:03:36.550 --> 00:03:39.080
You leave the 5 in
the radical.

00:03:39.080 --> 00:03:43.200
So let's do part D.

00:03:43.200 --> 00:03:47.380
We have to do the square
root of 200.

00:03:47.380 --> 00:03:48.350
Same process.

00:03:48.350 --> 00:03:50.390
Let's take the prime
factors of it.

00:03:50.390 --> 00:03:56.310
So it's 2 times 100, which is
2 times 50, which is 2 times

00:03:56.310 --> 00:04:01.030
25, which is 5 times 5.

00:04:01.030 --> 00:04:03.640
So this right here,
we can rewrite it.

00:04:03.640 --> 00:04:05.800
Let me scroll to the
right a little bit.

00:04:05.800 --> 00:04:15.030
This is equal to the square
root of 2 times 2 times 2

00:04:15.030 --> 00:04:18.390
times 5 times 5.

00:04:18.390 --> 00:04:20.730
Well we have one perfect square
there, and we have

00:04:20.730 --> 00:04:23.350
another perfect square there.

00:04:23.350 --> 00:04:25.290
So if I just want to write out
all the steps, this would be

00:04:25.290 --> 00:04:31.170
the square root of 2 times 2
times the square root of 2

00:04:31.170 --> 00:04:35.120
times the square root
of 5 times 5.

00:04:35.120 --> 00:04:37.345
The square root of
2 times 2 is 2.

00:04:37.345 --> 00:04:40.245
The square root of 2 is just
the square root of 2.

00:04:40.245 --> 00:04:43.680
The square root of 5 times 5,
that's the square root of 25,

00:04:43.680 --> 00:04:45.430
that's just going to be 5.

00:04:45.430 --> 00:04:46.880
So you can rearrange these.

00:04:46.880 --> 00:04:48.830
2 times 5 is 10.

00:04:48.830 --> 00:04:50.730
10 square roots of 2.

00:04:50.730 --> 00:04:53.150
And once again, this
it is irrational.

00:04:53.150 --> 00:04:58.800
You can't express it as a
fraction with an integer and a

00:04:58.800 --> 00:05:00.850
numerator and the denominator.

00:05:00.850 --> 00:05:04.270
And if you were to actually try
to express this number, it

00:05:04.270 --> 00:05:08.610
will just keep going on and on
and on, and never repeating.

00:05:08.610 --> 00:05:10.790
Well let's do part E.

00:05:10.790 --> 00:05:13.720
The square root of 2000.

00:05:13.720 --> 00:05:15.660
I'll do it down here.

00:05:15.660 --> 00:05:20.620
Part E, the square
root of 2000.

00:05:20.620 --> 00:05:23.950
Same exact process that we've
been doing so far.

00:05:23.950 --> 00:05:25.820
Let's do the prime
factorization.

00:05:25.820 --> 00:05:35.680
That is 2 times 1000, which is
2 times 500, which is 2 times

00:05:35.680 --> 00:05:45.930
250, which is 2 times 125,
which is 5 times 25,

00:05:45.930 --> 00:05:49.580
which is 5 times 5.

00:05:49.580 --> 00:05:50.600
And we're done.

00:05:50.600 --> 00:05:56.180
So this is going to be equal to
the square root of 2 times

00:05:56.180 --> 00:05:59.630
2-- I'll put it in parentheses--
2 times 2, times

00:05:59.630 --> 00:06:06.350
2 times 2, times 2 times
2, times 5 times 5,

00:06:06.350 --> 00:06:08.840
times 5 times 5, right?

00:06:08.840 --> 00:06:15.390
We have 1, 2, 3, 4, 2's, and
then 3, 5's, times 5.

00:06:15.390 --> 00:06:18.000
Now what is this going
to be equal to?

00:06:18.000 --> 00:06:20.520
Well, one thing you might see
is, hey, I could write this

00:06:20.520 --> 00:06:25.140
as, this is a 4, this is a 4.

00:06:25.140 --> 00:06:27.510
So we're going to have
a 4 repeated.

00:06:27.510 --> 00:06:32.600
And so this the same thing as
the square root of 4 times 4

00:06:32.600 --> 00:06:37.330
times the square root of
5 times 5 times the

00:06:37.330 --> 00:06:39.480
square root of 5.

00:06:39.480 --> 00:06:42.310
So this right here
is obviously 4.

00:06:42.310 --> 00:06:44.570
This right here is 5.

00:06:44.570 --> 00:06:47.070
And then times the
square root of 5.

00:06:47.070 --> 00:06:52.070
So 4 times 5 is 20 square
roots of 5.

00:06:52.070 --> 00:06:54.290
And once again, this
is irrational.

00:06:58.290 --> 00:07:00.990
Well, let's do F.

00:07:00.990 --> 00:07:16.850
The square root of 1/4, which
we can view this is the same

00:07:16.850 --> 00:07:21.250
thing as the square root of 1
over the square root of 4,

00:07:21.250 --> 00:07:24.180
which is equal to 1/2.

00:07:24.180 --> 00:07:25.170
Which is clearly rational.

00:07:25.170 --> 00:07:27.400
It can be expressed
as a fraction.

00:07:27.400 --> 00:07:33.050
So that's clearly rational.

00:07:33.050 --> 00:07:39.380
Part G is the square
root of 9/4.

00:07:43.800 --> 00:07:44.600
Same logic.

00:07:44.600 --> 00:07:48.160
This is equal to the square root
of 9 over the square root

00:07:48.160 --> 00:07:52.910
of 4, which is equal to 3/2.

00:07:52.910 --> 00:07:56.960
Let's do part H.

00:07:56.960 --> 00:08:02.720
The square root of 0.16.

00:08:02.720 --> 00:08:05.250
Now you could do this in your
head if you immediately

00:08:05.250 --> 00:08:07.670
recognize that, gee, if
I multiply 0.4 times

00:08:07.670 --> 00:08:10.170
0.4, I'll get this.

00:08:10.170 --> 00:08:14.190
But I'll show you a more
systematic way of doing it, if

00:08:14.190 --> 00:08:16.040
that wasn't obvious to you.

00:08:16.040 --> 00:08:18.330
So this is the same thing
as the square

00:08:18.330 --> 00:08:22.730
root of 16/100, right?

00:08:22.730 --> 00:08:24.840
That's what 0.16 is.

00:08:24.840 --> 00:08:28.740
So this is equal to the square
root of 16 over the square

00:08:28.740 --> 00:08:37.010
root of 100, which is equal to
4/10, which is equal to 0.4.

00:08:37.010 --> 00:08:39.260
Let's do a couple
more like that.

00:08:39.260 --> 00:08:39.429
OK.

00:08:39.429 --> 00:08:46.180
Part I was the square root of
0.1, which is equal to the

00:08:46.180 --> 00:08:50.840
square root of 1/10, which is
equal to the square root of 1

00:08:50.840 --> 00:08:55.980
over the square root of 10,
which is equal to 1 over--

00:08:55.980 --> 00:08:59.890
now, the square root of 10--
10 is just 2 times 5.

00:08:59.890 --> 00:09:01.380
So that doesn't really
help us much.

00:09:01.380 --> 00:09:04.920
So that's just the square
root of 10 like that.

00:09:04.920 --> 00:09:08.130
A lot of math teachers don't
like you leaving that radical

00:09:08.130 --> 00:09:08.870
in the denominator.

00:09:08.870 --> 00:09:10.330
But I can already tell you
that this is irrational.

00:09:13.940 --> 00:09:15.650
You'll just keep getting
numbers.

00:09:15.650 --> 00:09:16.850
You can try it on your
calculator, and

00:09:16.850 --> 00:09:17.530
it will never repeat.

00:09:17.530 --> 00:09:19.430
Your calculator will just give
you an approximation.

00:09:19.430 --> 00:09:21.100
Because in order to give the
exact value, you'd have to

00:09:21.100 --> 00:09:23.560
have an infinite number
of digits.

00:09:23.560 --> 00:09:25.770
But if you wanted to
rationalize this,

00:09:25.770 --> 00:09:26.820
just to show you.

00:09:26.820 --> 00:09:28.620
If you want to get rid of the
radical in the denominator,

00:09:28.620 --> 00:09:32.090
you can multiply this times the
square root of 10 over the

00:09:32.090 --> 00:09:33.520
square root of 10, right?

00:09:33.520 --> 00:09:34.910
This is just 1.

00:09:34.910 --> 00:09:38.130
So you get the square
root of 10/10.

00:09:38.130 --> 00:09:40.630
These are equivalent statements,
but both of them

00:09:40.630 --> 00:09:41.540
are irrational.

00:09:41.540 --> 00:09:43.870
You take an irrational number,
divide it by 10, you still

00:09:43.870 --> 00:09:45.660
have an irrational number.

00:09:45.660 --> 00:09:46.930
Let's do J.

00:09:49.520 --> 00:09:53.820
We have the square
root of 0.01.

00:09:53.820 --> 00:09:57.570
This is the same thing as the
square root of 1/100.

00:09:57.570 --> 00:10:00.680
Which is equal to the square
root of 1 over the square root

00:10:00.680 --> 00:10:07.050
of 100, which is equal
to 1/10, or 0.1.

00:10:07.050 --> 00:10:10.030
Clearly once again
this is rational.

00:10:10.030 --> 00:10:12.880
It's being written
as a fraction.

00:10:12.880 --> 00:10:14.185
This one up here was
also rational.

00:10:14.185 --> 00:10:16.030
It can be written expressed
as a fraction.


Division of Rational Numbers

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.890
Let's do a little bit of
practice dividing rational

00:00:03.890 --> 00:00:05.105
numbers, or another
way to think about

00:00:05.105 --> 00:00:06.860
it, dividing fractions.

00:00:06.860 --> 00:00:08.039
Same thing.

00:00:08.039 --> 00:00:09.210
So we have part A.

00:00:09.210 --> 00:00:10.160
Well, number 1.

00:00:10.160 --> 00:00:12.560
Find the multiplicative inverse

00:00:12.560 --> 00:00:13.840
of each of the following.

00:00:13.840 --> 00:00:18.290
Now, all that means is if I
have some number a, its

00:00:18.290 --> 00:00:21.370
inverse is going to be 1/a.

00:00:21.370 --> 00:00:23.580
When I take a number and
I multiply it by its

00:00:23.580 --> 00:00:28.940
multiplicative inverse, I'm
going to get a/a, which is

00:00:28.940 --> 00:00:30.120
equal to 1.

00:00:30.120 --> 00:00:34.410
So a times its multiplicative
inverse is going to be 1.

00:00:34.410 --> 00:00:35.390
So let's do these

00:00:35.390 --> 00:00:38.160
problems. So part A.

00:00:38.160 --> 00:00:39.600
We have 100.

00:00:39.600 --> 00:00:41.460
So the multiplicative
inverse of that is

00:00:41.460 --> 00:00:44.130
just going to be 1/100.

00:00:44.130 --> 00:00:47.810
And if you multiplied that
times that, you'd get 1.

00:00:47.810 --> 00:00:49.730
Part B.

00:00:49.730 --> 00:00:51.040
I'll do all of these.

00:00:51.040 --> 00:00:51.950
We have 2/8.

00:00:51.950 --> 00:00:57.530
The inverse of 2/8 is 1 over
2/8, which is the same thing--

00:00:57.530 --> 00:01:03.320
this is equal to 1 times
the inverse of this.

00:01:03.320 --> 00:01:08.960
Times 8/2, which is the
same thing as 4.

00:01:08.960 --> 00:01:11.380
Now immediately, you might
see something.

00:01:11.380 --> 00:01:12.760
I guess, you see
a pattern here.

00:01:12.760 --> 00:01:15.430
If I take 1 over a fraction,
the result is

00:01:15.430 --> 00:01:16.930
the fraction swapped.

00:01:16.930 --> 00:01:19.950
So the inverse of 2/8 is 8/2.

00:01:19.950 --> 00:01:22.640
You just swap the numerator
and the denominator.

00:01:22.640 --> 00:01:26.150
So let's apply that to part C.

00:01:26.150 --> 00:01:32.840
If I have negative 19/21, its
inverse is just going to be

00:01:32.840 --> 00:01:34.570
you swap the numerator
and the denominator.

00:01:34.570 --> 00:01:38.720
Negative 21/19.

00:01:38.720 --> 00:01:40.690
And if you were to multiply
these two numbers,

00:01:40.690 --> 00:01:42.270
you would get 1.

00:01:42.270 --> 00:01:43.470
The negatives would
cancel out.

00:01:43.470 --> 00:01:47.100
The 21 and the 21 would cancel
out, the 19 and the 19 would

00:01:47.100 --> 00:01:48.380
cancel out.

00:01:48.380 --> 00:01:52.600
Part D.

00:01:52.600 --> 00:01:57.980
The inverse of 7,

00:01:57.980 --> 00:02:01.610
multiplicative inverse, is 1/7.

00:02:01.610 --> 00:02:04.060
And then finally, let's do E.

00:02:04.060 --> 00:02:08.000
So the inverse of-- so they give
us z to the third over--

00:02:08.000 --> 00:02:12.880
so it's negative z to the third
over 2xy to the third.

00:02:12.880 --> 00:02:16.150
So the multiplicative inverse
of that-- one of the harder

00:02:16.150 --> 00:02:19.630
words for me to say-- is just
going to be equal to-- so

00:02:19.630 --> 00:02:22.700
that's the inverse-- is going
to be 1 over this, which is

00:02:22.700 --> 00:02:24.210
just going to be negative.

00:02:24.210 --> 00:02:26.180
The denominator becomes
the numerator.

00:02:26.180 --> 00:02:31.080
2y to the third over z cubed.

00:02:31.080 --> 00:02:34.190
Now let's go to this
next section.

00:02:34.190 --> 00:02:36.610
Divide the following
rational numbers.

00:02:36.610 --> 00:02:39.400
Be sure that your answer
is in simplest form.

00:02:39.400 --> 00:02:41.650
So I'm just going to do every
other one of these problems,

00:02:41.650 --> 00:02:44.010
just so that I don't use
up all of your time.

00:02:44.010 --> 00:02:46.195
So let's do this first
one: part A.

00:02:46.195 --> 00:02:47.890
Well, I'll do it over here.

00:02:47.890 --> 00:02:49.170
Part A.

00:02:49.170 --> 00:02:54.000
5/2 divided by 1/4.

00:02:54.000 --> 00:03:01.010
This is the same thing
as 5/2 times 4/1.

00:03:05.010 --> 00:03:08.920
You divide by something is the
same thing as multiplying by

00:03:08.920 --> 00:03:10.310
its inverse.

00:03:10.310 --> 00:03:12.580
So I'm multiplying by
the inverse of 1/4.

00:03:12.580 --> 00:03:15.670
So this is going to be equal
to-- well, we could divide the

00:03:15.670 --> 00:03:19.160
numerator and denominator by 2,
so the 4 becomes a 2, the 2

00:03:19.160 --> 00:03:20.360
becomes a 1.

00:03:20.360 --> 00:03:26.130
5 times 2 is 10/1,
So it's just 10.

00:03:26.130 --> 00:03:27.050
Part B.

00:03:27.050 --> 00:03:28.440
Or let me do every other one.

00:03:28.440 --> 00:03:30.500
C.

00:03:30.500 --> 00:03:38.300
5/11 divided by 6/7.

00:03:38.300 --> 00:03:43.580
Once again, this is the same
thing as 5/11 times the

00:03:43.580 --> 00:03:47.235
inverse of 6/7, so times 7/6.

00:03:50.730 --> 00:03:57.720
And so we get, this is 5
times 7 is 35 over 66.

00:03:57.720 --> 00:04:02.270
And that's in lowest common
form or simplified form.

00:04:02.270 --> 00:04:03.040
So E.

00:04:03.040 --> 00:04:05.030
Let's do E.

00:04:05.030 --> 00:04:12.440
E, we have negative x over
2 divided by 5 over 7.

00:04:12.440 --> 00:04:15.770
Once again, this is the same
thing as negative x over 2

00:04:15.770 --> 00:04:26.470
times 7/5, which is equal
to negative 7x over 10.

00:04:26.470 --> 00:04:28.390
Think you're getting
the hang of it.

00:04:28.390 --> 00:04:33.050
Let's do part G here.

00:04:33.050 --> 00:04:41.410
We have a negative 1/3 divided
by negative 3/5.

00:04:41.410 --> 00:04:46.100
Well, this is going to be the
same thing as negative 1/3

00:04:46.100 --> 00:04:50.650
times the inverse of this,
times negative 5/3.

00:04:50.650 --> 00:04:53.600
I just swapped the numerator
and the denominator.

00:04:53.600 --> 00:04:56.030
So this is going to be equal
to-- the negatives cancel out.

00:04:56.030 --> 00:04:58.160
A negative times a negative
is a positive.

00:04:58.160 --> 00:05:00.860
1 times 5 is 5.

00:05:00.860 --> 00:05:03.540
3 times 3 is 9.

00:05:03.540 --> 00:05:04.490
Let's do one more.

00:05:04.490 --> 00:05:06.550
Let's do I.

00:05:06.550 --> 00:05:07.740
Part I.

00:05:07.740 --> 00:05:12.790
11 divided by negative x/4.

00:05:12.790 --> 00:05:18.050
Once again, this is the
same thing as 11 times

00:05:18.050 --> 00:05:19.010
the inverse of this.

00:05:19.010 --> 00:05:24.600
The multiplicative inverse of
it, times negative 4/x.

00:05:24.600 --> 00:05:26.860
And if this confuses you--
actually, I shouldn't write

00:05:26.860 --> 00:05:27.690
multiplying.

00:05:27.690 --> 00:05:28.665
That looks just like an x.

00:05:28.665 --> 00:05:30.590
I should say this the
same thing as 11

00:05:30.590 --> 00:05:34.810
times negative 4/x.

00:05:34.810 --> 00:05:39.180
Or you could view this as 11
over 1 times-- you could even

00:05:39.180 --> 00:05:44.750
view it as negative 4/x, which
is equal to minus 44/x, or

00:05:44.750 --> 00:05:48.360
negative 44/x.

00:05:48.360 --> 00:05:51.340
Let's do this last problem
right here.

00:05:51.340 --> 00:05:57.850
The world's largest trench
digger, Bagger 288, moves at

00:05:57.850 --> 00:06:00.360
3/8 miles per hour.

00:06:00.360 --> 00:06:07.590
So its speed, or its velocity,
or its rate is equal to 3/8

00:06:07.590 --> 00:06:09.620
miles per hour.

00:06:09.620 --> 00:06:10.870
We could write it like this.

00:06:16.270 --> 00:06:20.010
How long will it take to dig
a trench 2/3 miles long?

00:06:20.010 --> 00:06:25.180
So we want to go a distance
that's equal to 2/3 miles.

00:06:25.180 --> 00:06:29.590
So we just remember that
distance is always going to be

00:06:29.590 --> 00:06:33.710
equal to rate times time.

00:06:33.710 --> 00:06:35.760
So our distance is 2/3 miles.

00:06:39.730 --> 00:06:48.260
That's going to be equal to our
rate, 3/8 miles per hour,

00:06:48.260 --> 00:06:51.800
times our time.

00:06:51.800 --> 00:06:57.200
Or we could divide both sides
by 3/8 miles per hour, so if

00:06:57.200 --> 00:07:00.030
we divide both sides by
3/8 miles per hour,

00:07:00.030 --> 00:07:01.850
we'll get our time.

00:07:01.850 --> 00:07:07.661
So divide by 3/8
miles per hour.

00:07:07.661 --> 00:07:14.100
Divide by 3/8 miles per hour.

00:07:14.100 --> 00:07:15.630
This cancels out.

00:07:15.630 --> 00:07:19.510
So time, we're going to get
time is equal to this.

00:07:19.510 --> 00:07:21.103
So we have 2/3 miles.

00:07:24.480 --> 00:07:26.430
Let me write it this way.

00:07:26.430 --> 00:07:32.150
2/3 miles divided by
3/8 miles per hour.

00:07:32.150 --> 00:07:35.580
Or another way to think of it,
that's the same thing as times

00:07:35.580 --> 00:07:45.270
the inverse, times 8/3
hours per mile.

00:07:50.120 --> 00:07:51.400
I just took the inverse
of this.

00:07:51.400 --> 00:07:52.450
We divided by this.

00:07:52.450 --> 00:07:54.920
That's the same thing as
multiplying by its inverse.

00:07:54.920 --> 00:07:57.310
And we inverted the
units as well.

00:07:57.310 --> 00:07:59.350
And so we'll see that the
units cancel out.

00:07:59.350 --> 00:08:01.920
Mile in the numerator, mile
in the denominator.

00:08:01.920 --> 00:08:03.360
We're left with hours.

00:08:03.360 --> 00:08:09.000
And so this is equal to 2 times
8 is 16 over 3 times 3,

00:08:09.000 --> 00:08:12.250
9, 16/9 hours.

00:08:12.250 --> 00:08:16.810
So like an hour and 7/9, or
1 and 7/9 hours to dig

00:08:16.810 --> 00:08:19.020
the 2/3 of a mile.


Distributive property algebraic expressions

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.820
Let's do some problems with
the distributive property.

00:00:03.820 --> 00:00:06.750
And the distributive property
just essentially reminds us

00:00:06.750 --> 00:00:11.620
that if we have, let's say, a
times b plus c, and then we

00:00:11.620 --> 00:00:14.570
need to multiply a times this,
we have to multiply a times

00:00:14.570 --> 00:00:15.870
both of these numbers.

00:00:15.870 --> 00:00:21.300
So this is going to be equal to
a times b plus a times c.

00:00:21.300 --> 00:00:25.500
It will not be just a
times b then plus c.

00:00:25.500 --> 00:00:27.690
And that makes complete sense.

00:00:27.690 --> 00:00:28.490
Let me give you an example.

00:00:28.490 --> 00:00:33.450
If I had said 5 times 3 plus
7, now, if you were to work

00:00:33.450 --> 00:00:35.380
this out using order of
operations, you'd say, this is

00:00:35.380 --> 00:00:37.252
5 times 10.

00:00:37.252 --> 00:00:42.840
So you'd say, this is 5 times
10, which is equal to 50.

00:00:42.840 --> 00:00:44.470
And we know that that's
the right answer.

00:00:44.470 --> 00:00:46.870
Now, use the distributive
property, that tells us that

00:00:46.870 --> 00:00:52.590
this is going to be equal to 5
times 3, which is 15, plus 5

00:00:52.590 --> 00:00:55.680
times 7, which is 35.

00:00:55.680 --> 00:00:59.370
And 15 plus 35 is
definitely 50.

00:00:59.370 --> 00:01:02.950
If you only multiplied the 5
times the 3, you'd have 15,

00:01:02.950 --> 00:01:05.370
and then plus the seven, you'd
get the wrong answer.

00:01:05.370 --> 00:01:07.320
You're multiplying 5 times
these things, you have to

00:01:07.320 --> 00:01:09.410
multiply 5 times both
of these things.

00:01:09.410 --> 00:01:11.610
Because you're multiplying
the sum of these guys.

00:01:11.610 --> 00:01:12.370
Anyway.

00:01:12.370 --> 00:01:16.260
Let's just apply that to a
sampling of these problems.

00:01:16.260 --> 00:01:18.040
Let's do A.

00:01:18.040 --> 00:01:23.050
So we have 1/2 times
x minus y minus 4.

00:01:23.050 --> 00:01:25.270
Well, we multiply 1/2
times both of these.

00:01:25.270 --> 00:01:30.480
So it's going to be 1/2
x minus 1/2 y minus

00:01:30.480 --> 00:01:32.240
4, and we're done.

00:01:32.240 --> 00:01:35.540
Let's do C.

00:01:35.540 --> 00:01:41.330
We have 6 plus x
minus 5 plus 7.

00:01:41.330 --> 00:01:42.850
Well, here there's actually
no distributive

00:01:42.850 --> 00:01:43.940
property to even do.

00:01:43.940 --> 00:01:45.800
We can actually just remove
the parentheses.

00:01:45.800 --> 00:01:51.010
6 plus this thing, that's the
same thing as 6 plus x plus

00:01:51.010 --> 00:01:54.610
negative 5 plus 7.

00:01:54.610 --> 00:01:56.610
Or you could view this
as 6 plus-- So this

00:01:56.610 --> 00:01:58.360
right here is 2, right?

00:01:58.360 --> 00:02:02.190
Negative 5 plus 7 is 2,
2 plus 6 is 8, so it

00:02:02.190 --> 00:02:04.730
becomes 8 plus x.

00:02:04.730 --> 00:02:05.450
All right.

00:02:05.450 --> 00:02:07.010
Not too bad.

00:02:07.010 --> 00:02:07.760
That was C.

00:02:07.760 --> 00:02:10.970
Let's do E.

00:02:10.970 --> 00:02:21.140
We have 4 times m plus 7 minus
6 times 4 minus m.

00:02:21.140 --> 00:02:22.360
Let's do the distributive
property.

00:02:22.360 --> 00:02:28.200
4 times m is 4m plus
4 times 7 is 28.

00:02:28.200 --> 00:02:31.330
And then we could
do it two ways.

00:02:31.330 --> 00:02:35.850
Let's do it this way first.
So we could have minus

00:02:35.850 --> 00:02:38.580
6 times 4 is 24.

00:02:38.580 --> 00:02:43.030
6 times negative
m is minus 6m.

00:02:43.030 --> 00:02:45.930
And notice, I could have just
said, times negative 6, and

00:02:45.930 --> 00:02:47.550
have a plus here, but I'm
doing it in two steps.

00:02:47.550 --> 00:02:51.350
I'm doing the 6 first, and then
I'll do the negative 1.

00:02:51.350 --> 00:02:55.520
And so this is going to be
4m plus 28, and then you

00:02:55.520 --> 00:02:56.760
distribute the negative sign.

00:02:56.760 --> 00:02:59.600
You can view this as a negative
1 times all of this.

00:02:59.600 --> 00:03:02.630
So negative 1 times
24 is minus 24.

00:03:02.630 --> 00:03:06.620
Negative 1 times minus
6m is plus 6m.

00:03:06.620 --> 00:03:12.920
Now you add the m terms.
4m plus 6m is 10m.

00:03:12.920 --> 00:03:17.200
And then add the constant terms.
28 minus 24, that is

00:03:17.200 --> 00:03:21.580
equal to plus 4.

00:03:21.580 --> 00:03:23.200
Let's go down here.

00:03:23.200 --> 00:03:25.610
Use the distributive property
to simplify

00:03:25.610 --> 00:03:26.730
the following fractions.

00:03:26.730 --> 00:03:28.400
So I'll do every other
one again.

00:03:28.400 --> 00:03:36.520
So the first one is, a
is 8x plus 12 over 4.

00:03:36.520 --> 00:03:37.870
So the reason why they're
saying the distributive

00:03:37.870 --> 00:03:40.150
property, you're essentially
saying, let's divide this

00:03:40.150 --> 00:03:41.980
whole thing by 4.

00:03:41.980 --> 00:03:44.590
And to divide the whole thing by
4, you have to divide each

00:03:44.590 --> 00:03:45.440
of the things by 4.

00:03:45.440 --> 00:03:47.790
You could even view this as,
this is the same thing as

00:03:47.790 --> 00:03:52.440
multiplying 1/4 times
8x plus 12.

00:03:52.440 --> 00:03:53.620
These two things
are equivalent.

00:03:53.620 --> 00:03:55.680
Here you're dividing each
by 4, here you're

00:03:55.680 --> 00:03:57.340
multiplying each by 4.

00:03:57.340 --> 00:04:02.060
If you did it this way, this is
the same thing as 8x over 4

00:04:02.060 --> 00:04:03.810
plus 12 over 4.

00:04:03.810 --> 00:04:07.130
You're kind of doing a adding
fractions problem in reverse.

00:04:07.130 --> 00:04:10.680
And then this 8 divided
by 4 is going to be,

00:04:10.680 --> 00:04:13.360
this'll be 2x plus 3.

00:04:13.360 --> 00:04:14.600
That's one way to do it.

00:04:14.600 --> 00:04:15.580
Or you could do it this way.

00:04:15.580 --> 00:04:22.810
1/4 times 8x is 2x, plus
1/4 times 12 is 3.

00:04:22.810 --> 00:04:26.960
Either way, we got
the same answer.

00:04:26.960 --> 00:04:29.050
C.

00:04:29.050 --> 00:04:34.300
We have 11x plus 12 over 2.

00:04:34.300 --> 00:04:35.050
Just like here.

00:04:35.050 --> 00:04:37.895
We could say, this is the same
thing as 11-- We could write

00:04:37.895 --> 00:04:40.480
it as 11 over 2x, if we like.

00:04:40.480 --> 00:04:42.950
Or 11x over 2, either way.

00:04:42.950 --> 00:04:47.620
Plus 12 over 2 plus 6.

00:04:47.620 --> 00:04:50.360
And let's just do one more.

00:04:50.360 --> 00:04:52.140
E.

00:04:52.140 --> 00:04:52.810
This looks interesting.

00:04:52.810 --> 00:04:56.570
We have a negative out in front,
and then we have a 6z

00:04:56.570 --> 00:04:59.650
minus 2 over 3.

00:04:59.650 --> 00:05:03.140
So one way we can view this,
this is the same thing, this

00:05:03.140 --> 00:05:09.190
is equal to, negative 1/3
times 6z minus 2.

00:05:09.190 --> 00:05:13.070
These two things
are equivalent.

00:05:13.070 --> 00:05:13.340
Right?

00:05:13.340 --> 00:05:14.550
This is a negative 1/3.

00:05:14.550 --> 00:05:16.650
You could imagine a
1 right out here.

00:05:16.650 --> 00:05:16.830
Right?

00:05:16.830 --> 00:05:20.610
Negative 1/3 times 6z minus 2.

00:05:20.610 --> 00:05:22.210
And then you just do the
distributive property.

00:05:22.210 --> 00:05:28.280
Negative 1/3 times 6z is
going to be minus 2z.

00:05:28.280 --> 00:05:32.090
And then negative 1/3 times
negative 2, negatives cancel

00:05:32.090 --> 00:05:35.530
out, you get plus 2/3.

00:05:35.530 --> 00:05:38.180
And you are done.


Multiplication of Rational Numbers

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.800
I'm going to warm-up with
a couple of straight-up

00:00:02.800 --> 00:00:06.280
multiplying fractions or
multiplying rational number

00:00:06.280 --> 00:00:08.670
problems-- rational number is
just another way to say

00:00:08.670 --> 00:00:11.360
fractions, it's just any number
that can be expressed

00:00:11.360 --> 00:00:13.790
as a fraction.

00:00:13.790 --> 00:00:16.190
And then we'll actually do a
word problem that deals with

00:00:16.190 --> 00:00:17.750
the multiplication
of fractions.

00:00:17.750 --> 00:00:19.280
Let's do part a.

00:00:19.280 --> 00:00:24.420
5 over 12 times 9 over 10.

00:00:24.420 --> 00:00:26.100
And the fun thing about
multiplying and dividing

00:00:26.100 --> 00:00:28.750
fractions, at least in my mind,
it's easier than adding

00:00:28.750 --> 00:00:31.300
or subtracting, you literally
just multiply the numerators

00:00:31.300 --> 00:00:34.250
when you're multiplying and
multiply the denominators.

00:00:34.250 --> 00:00:38.740
So you could say that this is
the same thing as 5 times 9

00:00:38.740 --> 00:00:41.450
over 12 times 10.

00:00:41.450 --> 00:00:44.730
And we could just multiply
these out, 45 over 120.

00:00:44.730 --> 00:00:46.170
But that won't be in
simplest form.

00:00:46.170 --> 00:00:47.970
We'd be multiplying
these big numbers.

00:00:47.970 --> 00:00:50.830
And before we do that we can
divide the numerator and

00:00:50.830 --> 00:00:53.020
denominator by numbers that will
simplify it a little bit.

00:00:53.020 --> 00:00:54.540
So let's do that.

00:00:54.540 --> 00:00:58.330
So if we divide the numerator
and the denominator by 5, then

00:00:58.330 --> 00:01:01.790
this will become a 1, this
will become a 2.

00:01:01.790 --> 00:01:05.090
If we divide the numerator and
the denominator by 3, this

00:01:05.090 --> 00:01:08.430
will become a 3, and this
will become a 4.

00:01:08.430 --> 00:01:10.850
So the whole thing is simplified
to 1 times 3, which

00:01:10.850 --> 00:01:15.250
is 3 over 4 times 2, 3/8,
which is a lot easier

00:01:15.250 --> 00:01:18.020
than 45 over 120.

00:01:18.020 --> 00:01:20.600
All right.

00:01:20.600 --> 00:01:23.210
Let's just do part c here.

00:01:23.210 --> 00:01:27.690
3/4 times 1/3.

00:01:27.690 --> 00:01:29.880
We could do that exact
same principle.

00:01:29.880 --> 00:01:33.000
This is going to be 3 times
1 over 4 times 3.

00:01:33.000 --> 00:01:36.730
We can skip a step if we like
and cancel out the 3's in the

00:01:36.730 --> 00:01:38.110
numerator and the denominator.

00:01:38.110 --> 00:01:40.890
So you divide the numerator
and denominator by 3.

00:01:40.890 --> 00:01:44.270
You'd say that that's
a 1, and that's a 1.

00:01:44.270 --> 00:01:49.930
And you'd say OK, 1/4 times 1,
1 over 1, is equal to 1/4.

00:01:49.930 --> 00:01:53.000
And all I did there, I skipped a
step, but all I did, I said,

00:01:53.000 --> 00:01:55.950
OK, that's going to be the
same thing as 3 times 1

00:01:55.950 --> 00:01:57.970
over 4 times 3.

00:01:57.970 --> 00:02:00.030
Divide the numerator and
the denominator by

00:02:00.030 --> 00:02:02.730
3 and you get 1/4.

00:02:02.730 --> 00:02:04.420
These are the exact
same thing.

00:02:07.800 --> 00:02:11.410
I'll do e and f just to
finish up things.

00:02:11.410 --> 00:02:18.780
So e is 1/13 times 1 over 11.

00:02:18.780 --> 00:02:20.570
Well, we just multiply
the numerators.

00:02:20.570 --> 00:02:24.250
We get 1 over whatever
13 times 11 is.

00:02:24.250 --> 00:02:27.790
Let's see, 13 times 10 is 130,
and so we're going to have to

00:02:27.790 --> 00:02:29.690
get one more 13 in there.

00:02:29.690 --> 00:02:36.350
So it's going to be 143, if
I did that correctly.

00:02:36.350 --> 00:02:38.290
Yeah, 13 times 10 is 130.

00:02:38.290 --> 00:02:38.900
Yup.

00:02:38.900 --> 00:02:40.330
Looks good to me.

00:02:40.330 --> 00:02:43.230
All right, one more, get
us really warmed-up.

00:02:43.230 --> 00:02:44.570
All right.

00:02:44.570 --> 00:02:46.380
I'll do part f.

00:02:46.380 --> 00:02:51.660
7 over 27 times 9 over 14.

00:02:51.660 --> 00:02:53.890
We can do the same trick
we did over here.

00:02:53.890 --> 00:02:57.670
Divide the numerator and the
denominator by 7, so this

00:02:57.670 --> 00:03:01.080
becomes a 1, this becomes a 2.

00:03:01.080 --> 00:03:03.920
Then we can divide the numerator
and denominator by

00:03:03.920 --> 00:03:07.220
9, this becomes a 1,
this becomes a 3.

00:03:07.220 --> 00:03:15.380
So this becomes 1 times 1
over 3 times 2 or 1/6.

00:03:15.380 --> 00:03:20.740
Or you could have said just 7
times 9, which is 63 over 27

00:03:20.740 --> 00:03:23.040
times 14, which is a huge number
that I don't know the

00:03:23.040 --> 00:03:25.200
exact number to and it would
have taken you forever to

00:03:25.200 --> 00:03:30.560
actually put it in a
most simple form.

00:03:30.560 --> 00:03:33.030
But this way when you divide it
ahead of time it simplified

00:03:33.030 --> 00:03:34.930
things a good bit.

00:03:34.930 --> 00:03:37.540
So let's do a word problem.

00:03:37.540 --> 00:03:39.860
And it involves monkeys.

00:03:39.860 --> 00:03:43.280
Three monkeys spend
a day gathering

00:03:43.280 --> 00:03:44.820
this coconuts together.

00:03:44.820 --> 00:03:46.820
When they have finished
they're very

00:03:46.820 --> 00:03:48.750
tired and fall asleep.

00:03:48.750 --> 00:03:51.460
The following morning the first
monkey wakes up, not

00:03:51.460 --> 00:03:55.590
wishing to disturb his friends,
he decides to divide

00:03:55.590 --> 00:03:58.980
the coconuts into three
equal piles.

00:03:58.980 --> 00:04:03.630
There is one left over, so he
throws this odd one away,

00:04:03.630 --> 00:04:05.850
helps himself to his share
and goes home.

00:04:05.850 --> 00:04:10.000
So this is the third monkey.

00:04:10.000 --> 00:04:13.330
So he divides what he sees
into three piles.

00:04:13.330 --> 00:04:14.780
We don't know how many
are in each pile.

00:04:14.780 --> 00:04:20.200
Pile 1, pile 2, pile 3.

00:04:20.200 --> 00:04:25.660
And then he had one coconut
left over that he chucks.

00:04:25.660 --> 00:04:28.240
That's a coconut right there.

00:04:28.240 --> 00:04:31.290
That's with that first, he
divides the coconuts into

00:04:31.290 --> 00:04:33.190
three equal piles.

00:04:33.190 --> 00:04:35.840
There was one left over so he
throws that away, helps

00:04:35.840 --> 00:04:37.230
himself to his share
and goes home.

00:04:37.230 --> 00:04:42.200
So his share is, let's say,
he takes this share and

00:04:42.200 --> 00:04:44.980
he throws this away.

00:04:44.980 --> 00:04:47.060
That's what this tells us.

00:04:47.060 --> 00:04:49.340
He decides to divide the
coconuts into three equal

00:04:49.340 --> 00:04:52.860
piles, there is one left over,
so he throws it away and he

00:04:52.860 --> 00:04:55.100
helps himself to his share.

00:04:55.100 --> 00:04:56.180
All right.

00:04:56.180 --> 00:04:58.880
Now a few minutes later the
second monkey awakes, not

00:04:58.880 --> 00:05:02.830
realizing that the first has
already gone, he, too, divides

00:05:02.830 --> 00:05:05.820
the coconuts into three
equal heaps.

00:05:05.820 --> 00:05:10.020
He finds one left over, throws
the odd away, and helps

00:05:10.020 --> 00:05:11.740
himself to his fair share.

00:05:11.740 --> 00:05:15.280
So this is he finds one left
over, throws the odd away,

00:05:15.280 --> 00:05:16.550
helps himself to
his fair share.

00:05:16.550 --> 00:05:21.430
So this is the second monkey.

00:05:21.430 --> 00:05:22.510
So this is all he sees.

00:05:22.510 --> 00:05:25.350
He just sees these two
piles together.

00:05:25.350 --> 00:05:28.040
So he divides these into
three equal heaps.

00:05:28.040 --> 00:05:31.980
So maybe that's one heap right
there, that is-- let me make

00:05:31.980 --> 00:05:33.930
it a little bit more equal.

00:05:33.930 --> 00:05:39.650
So this is one heap, that is
two heaps, and that is--

00:05:39.650 --> 00:05:42.130
actually, he actually sees
one left over, too.

00:05:42.130 --> 00:05:43.180
So let me make that clear.

00:05:43.180 --> 00:05:46.600
He finds one left over, so he
finds one coconut left over,

00:05:46.600 --> 00:05:49.040
and then the rest of this--
remember, this is gone and

00:05:49.040 --> 00:05:49.610
this is gone.

00:05:49.610 --> 00:05:52.550
So this is all we're dealing
with at this point.

00:05:52.550 --> 00:05:55.040
One left over and three
equal heaps.

00:05:55.040 --> 00:06:01.660
So one heap, two heaps,
and three heaps.

00:06:01.660 --> 00:06:04.810
And then he takes
one for himself.

00:06:04.810 --> 00:06:08.220
He takes one for himself, and he
takes the extra coconut and

00:06:08.220 --> 00:06:11.310
he throws it away.

00:06:11.310 --> 00:06:13.130
So this is all we
have left now.

00:06:16.630 --> 00:06:18.690
And then you could guess
what happens.

00:06:18.690 --> 00:06:21.460
In the morning, the third monkey
awakes to find that he

00:06:21.460 --> 00:06:22.530
is all alone.

00:06:22.530 --> 00:06:26.690
He spots the two discarded
coconuts and puts

00:06:26.690 --> 00:06:28.700
them with the pile.

00:06:28.700 --> 00:06:29.420
Interesting.

00:06:29.420 --> 00:06:31.440
OK, so he puts them with
the pile, giving him

00:06:31.440 --> 00:06:33.530
a total of 12 coconuts.

00:06:33.530 --> 00:06:36.720
So he spots that one
and that one, puts

00:06:36.720 --> 00:06:39.000
them back in the pile.

00:06:39.000 --> 00:06:43.150
So that is the first-- or maybe
I should say-- yeah,

00:06:43.150 --> 00:06:47.490
this is the first monkey or
three monkeys-- actually, I

00:06:47.490 --> 00:06:48.870
should call this the
first monkey.

00:06:48.870 --> 00:06:53.150
The first monkey did this, the
second monkey, and now we're

00:06:53.150 --> 00:06:55.980
dealing with the third monkey.

00:06:55.980 --> 00:06:59.650
He spots the two discarded
coconuts and puts them with

00:06:59.650 --> 00:07:01.210
the pile giving a total of 12.

00:07:01.210 --> 00:07:03.770
So this is the third monkey,
the third one to wake up.

00:07:03.770 --> 00:07:05.190
Sorry if I confused you.

00:07:05.190 --> 00:07:08.560
This was the first one to wake
up, has one left over with

00:07:08.560 --> 00:07:09.620
three equal piles.

00:07:09.620 --> 00:07:11.860
One left over, keeps
one for himself.

00:07:11.860 --> 00:07:17.010
Second monkey, he sees these
coconuts, he makes three equal

00:07:17.010 --> 00:07:18.810
piles, has one left over.

00:07:18.810 --> 00:07:23.300
Then the third monkey wakes up,
sees these two coconuts,

00:07:23.300 --> 00:07:26.490
and then sees what's in
the pile and he has

00:07:26.490 --> 00:07:31.960
a total of 12 coconuts.

00:07:31.960 --> 00:07:35.350
And so they're saying how many
coconuts did the first and a

00:07:35.350 --> 00:07:37.660
second monkey take?

00:07:37.660 --> 00:07:38.630
All right.

00:07:38.630 --> 00:07:41.300
Actually, let me
be very clear.

00:07:41.300 --> 00:07:43.170
The total is 12 coconuts.

00:07:43.170 --> 00:07:46.690
This right here is 12 coconuts
after taking

00:07:46.690 --> 00:07:48.120
the discarded ones.

00:07:48.120 --> 00:07:54.900
So this is 12, this right here
must be 10 coconuts.

00:07:54.900 --> 00:07:58.750
And so this is 10 coconuts, and
if you look at these two

00:07:58.750 --> 00:08:03.010
piles that are left over, this
must be 5 coconuts, 5

00:08:03.010 --> 00:08:06.070
coconuts, and he took
5 coconuts.

00:08:06.070 --> 00:08:09.760
So if you go to this point over
here, how many coconuts

00:08:09.760 --> 00:08:10.710
were there?

00:08:10.710 --> 00:08:15.560
Well, there must have been 15
plus this 1 coconut over here.

00:08:15.560 --> 00:08:18.520
So the first monkey must
have had 16 coconuts

00:08:18.520 --> 00:08:21.000
in these two piles.

00:08:21.000 --> 00:08:23.600
So this must have
been 8 and 8.

00:08:23.600 --> 00:08:25.630
And then he took an
equal share of 8.

00:08:25.630 --> 00:08:31.130
So to start off with we had 8
times 3, 24 coconuts plus 1.

00:08:31.130 --> 00:08:34.320
So we started with
25 coconuts.

00:08:34.320 --> 00:08:39.340
This guy took 8, then
we had 16 coconuts.

00:08:39.340 --> 00:08:42.940
He took 8, got rid of 1,
so he only had 16 left.

00:08:42.940 --> 00:08:45.960
The second guy comes along and
he said, OK, I'm going to

00:08:45.960 --> 00:08:47.400
divide it into 3 groups of 5.

00:08:47.400 --> 00:08:48.400
I have one left over.

00:08:48.400 --> 00:08:49.690
He took 5.

00:08:49.690 --> 00:08:54.940
Then you had 10 left over plus
the 2 discarded ones was 12.


Subraction of Rational Numbers

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.170
Let's do some problems that
involve subtracting fractions,

00:00:04.170 --> 00:00:05.760
or instead of saying fractions,
we could say

00:00:05.760 --> 00:00:08.710
rational numbers, either way.

00:00:08.710 --> 00:00:11.390
I'll do every other
of these up here.

00:00:11.390 --> 00:00:14.110
Let's start with b.

00:00:14.110 --> 00:00:18.540
We have 2/3 minus 1/4.

00:00:18.540 --> 00:00:21.220
Just as in the case with adding
fractions, you need to

00:00:21.220 --> 00:00:22.890
have a common denominator
here.

00:00:22.890 --> 00:00:24.960
You have to know how many
pieces of the pie we're

00:00:24.960 --> 00:00:26.200
dealing with.

00:00:26.200 --> 00:00:29.320
So the common denominator of 3
and 4, or the least common

00:00:29.320 --> 00:00:32.020
multiple of 3 and 4, the
smallest number that's

00:00:32.020 --> 00:00:35.180
divisible by both of
them, that's 12.

00:00:35.180 --> 00:00:37.240
So let me write it this way.

00:00:37.240 --> 00:00:37.900
Let me do it this way.

00:00:37.900 --> 00:00:39.310
This will be more fun.

00:00:39.310 --> 00:00:46.390
2/3, if we write 12 in the
denominator, 3 times 4 is 12,

00:00:46.390 --> 00:00:49.680
so 2 times 4 is 8.

00:00:49.680 --> 00:00:51.230
These are equivalent.

00:00:51.230 --> 00:00:54.330
We multiplied the numerator
and the denominator by 4.

00:00:54.330 --> 00:00:58.520
And then 1/4, if we put
12 in the denominator,

00:00:58.520 --> 00:01:01.310
4 times 3 is 12.

00:01:01.310 --> 00:01:04.080
1 times 3 is 3.

00:01:04.080 --> 00:01:06.400
3/12 and 1/4 are
the same thing.

00:01:06.400 --> 00:01:08.700
And we have a minus
sign right there.

00:01:08.700 --> 00:01:12.710
So this is going to be equal to
8 minus 3 over 12, now that

00:01:12.710 --> 00:01:14.660
we have our common denominator,

00:01:14.660 --> 00:01:18.450
which is equal to 5/12.

00:01:18.450 --> 00:01:23.960
Now let's do d.

00:01:23.960 --> 00:01:25.210
Let me do it in a
different color.

00:01:27.680 --> 00:01:34.980
15 over 11, 15/11 minus 9/7.

00:01:34.980 --> 00:01:38.140
Let's get a common
denominator.

00:01:38.140 --> 00:01:40.810
Let's see, between 11 and 7, I
think 77 is going to be the

00:01:40.810 --> 00:01:42.380
first number you're going
to find that is

00:01:42.380 --> 00:01:43.180
divisible by both.

00:01:43.180 --> 00:01:44.860
You just multiply the two.

00:01:44.860 --> 00:01:47.170
So you have 77.

00:01:47.170 --> 00:01:51.390
To go from 11 to 77, you have
to multiply it by 7.

00:01:51.390 --> 00:01:57.230
So 15 times 7 is 70 plus 35.

00:01:57.230 --> 00:02:01.000
That is 105.

00:02:01.000 --> 00:02:04.930
And then we have
minus over 77.

00:02:04.930 --> 00:02:07.400
7 times 11 is 77.

00:02:07.400 --> 00:02:11.730
9 times 11 is 99.

00:02:11.730 --> 00:02:18.170
So 105 minus-- this is going
to be 105 minus 99 over 77.

00:02:18.170 --> 00:02:19.840
What's 105 minus 99?

00:02:19.840 --> 00:02:23.620
It is 6, and we are done.

00:02:23.620 --> 00:02:24.800
You can't reduce
this any more.

00:02:24.800 --> 00:02:29.390
6 and 77 aren't divisible by--
don't have any common factors.

00:02:29.390 --> 00:02:33.220
Let's do f.

00:02:33.220 --> 00:02:36.470
Let me do it in a new color.

00:02:36.470 --> 00:02:46.640
So f has us doing
7/27 minus 9/39.

00:02:46.640 --> 00:02:51.200
All right, so what is the
common denominator here?

00:02:51.200 --> 00:02:52.480
And actually, there's
something we

00:02:52.480 --> 00:02:54.440
might be able to simplify.

00:02:54.440 --> 00:02:57.460
We might be able to simplify
this a little bit.

00:02:57.460 --> 00:03:00.170
9/39, they're both
divisible by 3.

00:03:00.170 --> 00:03:06.220
So I can rewrite 9/39 as divide
9 by 3, it becomes 3.

00:03:06.220 --> 00:03:10.290
You divide 39 by 3,
it becomes 13.

00:03:10.290 --> 00:03:15.360
So this becomes 7/27
minus 3/13.

00:03:15.360 --> 00:03:18.850
So what's the common denominator
between 27 and 13?

00:03:18.850 --> 00:03:20.650
Well, they don't have any common
factors, so we're just

00:03:20.650 --> 00:03:22.330
going to have to multiply
the two.

00:03:22.330 --> 00:03:25.290
So let's see, 27 times 13--
let me do it over on

00:03:25.290 --> 00:03:26.290
the side over here.

00:03:26.290 --> 00:03:30.690
So 27 times 13.

00:03:30.690 --> 00:03:37.370
3 times 7 is 21, carry the 2,
3 times 2 is 6 plus 2 is 8.

00:03:37.370 --> 00:03:38.660
Let's put the 0 down here.

00:03:38.660 --> 00:03:40.160
1 times 7 is 7.

00:03:40.160 --> 00:03:42.600
1 times 2 is 2.

00:03:42.600 --> 00:03:44.000
And we add.

00:03:44.000 --> 00:03:45.210
1 plus 0 is 1.

00:03:45.210 --> 00:03:49.650
8 plus 7 is 15.

00:03:49.650 --> 00:03:52.170
1 plus 2 is 3.

00:03:52.170 --> 00:03:57.160
So our common denominator
is going to be 351 minus

00:03:57.160 --> 00:04:00.560
something over 351.

00:04:00.560 --> 00:04:05.480
To go from 27 to 351, we
have to multiply by 13.

00:04:05.480 --> 00:04:08.422
So to go from 7 to whatever
numerator here, we're going to

00:04:08.422 --> 00:04:09.750
have to multiply by 13.

00:04:09.750 --> 00:04:14.930
7 times 13 is 70,
plus 21 is 91.

00:04:14.930 --> 00:04:17.310
This is 91/351.

00:04:17.310 --> 00:04:23.650
And to go from this 13 to 351,
we have to multiply by 27.

00:04:23.650 --> 00:04:26.810
So on the numerator, we have
to multiply by 27.

00:04:26.810 --> 00:04:33.140
3 times 27, that is 60
plus 21, that is 81.

00:04:33.140 --> 00:04:39.830
And so our answer is going
to be over 351.

00:04:39.830 --> 00:04:43.550
91 minus 81, well,
that's just 10.

00:04:43.550 --> 00:04:45.230
And then we are done.

00:04:45.230 --> 00:04:46.960
And then let's do one more.

00:04:46.960 --> 00:04:49.420
These are getting a little
hairy, but we can power

00:04:49.420 --> 00:04:50.470
through them.

00:04:50.470 --> 00:05:02.380
h says 13/64 minus 7/40.

00:05:02.380 --> 00:05:06.980
So let's get a common
denominator between 64 and 40.

00:05:06.980 --> 00:05:09.380
So 80 won't work.

00:05:09.380 --> 00:05:12.360
We could actually try to--
actually, let's do this the

00:05:12.360 --> 00:05:13.150
old-fashioned way.

00:05:13.150 --> 00:05:14.990
Let's look at the factors.

00:05:14.990 --> 00:05:18.655
Let's look at the prime
factorization of the two and

00:05:18.655 --> 00:05:20.580
then we could find the least
common multiple.

00:05:20.580 --> 00:05:21.720
So this'll be interesting.

00:05:21.720 --> 00:05:29.780
So 64-- let me do it in a
different color here-- 64, we

00:05:29.780 --> 00:05:34.340
can factor that as 2 times 32,
which is the same thing as 2

00:05:34.340 --> 00:05:40.260
times 16, which is the same
thing as 2 times 8, which is

00:05:40.260 --> 00:05:43.030
the same thing as 2 times
4, which is the same

00:05:43.030 --> 00:05:48.410
thing as 2 times 2.

00:05:48.410 --> 00:05:51.540
So this is 2 times itself,
one, two, three,

00:05:51.540 --> 00:05:54.500
four, five, six times.

00:05:54.500 --> 00:05:59.250
So we could say 64 is the same
thing as 2 times 2 times 2

00:05:59.250 --> 00:06:01.560
times 2 times 2 times 2.

00:06:01.560 --> 00:06:02.740
If you know exponents,
this is the same

00:06:02.740 --> 00:06:04.210
thing as 2 to the sixth.

00:06:04.210 --> 00:06:06.340
That's 64.

00:06:06.340 --> 00:06:09.730
Now, 40 you do the same thing.

00:06:09.730 --> 00:06:13.690
40-- let me do that in
a different color.

00:06:13.690 --> 00:06:22.560
40 is 2 times 20, which is 2
times 10, which is 2 times 5.

00:06:22.560 --> 00:06:28.480
So 40 is 2 times 2 times 2
times 2 times 2 times 5.

00:06:28.480 --> 00:06:31.220
This is the prime factorization
of 40.

00:06:31.220 --> 00:06:36.280
Now, if we compare the two, this
2 times 2 times 2, this

00:06:36.280 --> 00:06:39.180
is inside of 64.

00:06:39.180 --> 00:06:41.040
So in order to find the least
common multiple, we don't have

00:06:41.040 --> 00:06:43.260
a 5 in the prime factorization
of 64.

00:06:43.260 --> 00:06:45.230
We have the 2 times 2 times 2.

00:06:45.230 --> 00:06:48.990
So to get there, the least
common multiple of these will

00:06:48.990 --> 00:06:52.100
be if we just multiply
this times 5.

00:06:52.100 --> 00:06:55.740
So we take 64 and we multiply it
times 5 because this number

00:06:55.740 --> 00:07:01.670
now is going to have 64 in it,
that's 64, and it's also going

00:07:01.670 --> 00:07:04.470
to have 40 in it right there.

00:07:04.470 --> 00:07:06.980
So 64 times 5.

00:07:06.980 --> 00:07:10.440
So what is 64 times 5?

00:07:10.440 --> 00:07:14.900
So if we take 64-- I'm running
out of real estate; let me

00:07:14.900 --> 00:07:22.990
scroll over to the right--
so 64 times 5 is 320.

00:07:22.990 --> 00:07:28.760
So it's 320, so that
is times 5.

00:07:28.760 --> 00:07:30.660
So if we multiply 64
times 5, we have to

00:07:30.660 --> 00:07:32.280
multiply 13 times 5.

00:07:32.280 --> 00:07:36.380
So that's 50 plus 15.

00:07:36.380 --> 00:07:42.070
13 times 5, so it's
65, so that is 65.

00:07:42.070 --> 00:07:50.050
And then to go from 40 to 320,
we have to multiply by 8.

00:07:50.050 --> 00:07:50.940
And you could see that there.

00:07:50.940 --> 00:07:53.870
You have this other three 2's,
2 to the third power.

00:07:53.870 --> 00:07:56.730
So we have to multiply
it by 8 over here.

00:07:56.730 --> 00:08:02.350
So 7 times 8 is 56.

00:08:02.350 --> 00:08:14.170
So we have a minus 56 is equal
to 65 minus 56 is 9, is equal

00:08:14.170 --> 00:08:18.600
to 9 over 320.

00:08:18.600 --> 00:08:20.100
That was a little bit of a hairy
problem, but we were

00:08:20.100 --> 00:08:21.210
able to power through it.

00:08:21.210 --> 00:08:26.400
Let's do this last problem
down here.

00:08:26.400 --> 00:08:29.720
Consider the equation
2/3x plus 1/2.

00:08:29.720 --> 00:08:34.770
All right, y is equal
to 2/3x plus 1/2.

00:08:34.770 --> 00:08:37.880
Determine the change in y
between x is equal to 1 and x

00:08:37.880 --> 00:08:39.330
is equal to 2.

00:08:39.330 --> 00:08:40.340
All right.

00:08:40.340 --> 00:08:44.630
When x-- let's make little
table: x and y.

00:08:44.630 --> 00:08:46.950
When x is equal to
1, what is y?

00:08:46.950 --> 00:08:54.400
It's 2/3 times 1 plus 1/2, which
is equal to 2/3 plus

00:08:54.400 --> 00:08:58.040
1/2, which is equal to a common
denominator of 6.

00:08:58.040 --> 00:09:05.680
2/3 is 4/6-- multiply both of
them by 2-- plus 1/2 is 3/6.

00:09:05.680 --> 00:09:07.055
So this is equal to 7/6.

00:09:09.770 --> 00:09:10.270
All right.

00:09:10.270 --> 00:09:12.880
When x is equal to 2.

00:09:12.880 --> 00:09:19.880
When x is equal to 2, you have
2/3 times 2 plus 1/2, so it's

00:09:19.880 --> 00:09:26.470
going to be equal to 4/3 plus
1/2, which is equal to a

00:09:26.470 --> 00:09:28.290
common denominator of 6.

00:09:28.290 --> 00:09:35.620
This is equal to 8/6-- 3 times
2 is 6; 4 times 2 is 8-- plus

00:09:35.620 --> 00:09:39.570
3/6, which is equal to 11/6.

00:09:39.570 --> 00:09:43.030
And they say determine the
change in y between x equal 1

00:09:43.030 --> 00:09:44.000
and x equal to 2.

00:09:44.000 --> 00:09:47.330
So how much larger is this--
this is our new y when x is

00:09:47.330 --> 00:09:49.850
equal to 2-- relative to this?

00:09:49.850 --> 00:10:00.690
So we could take our 11/6 minus
7/6, and our change in y

00:10:00.690 --> 00:10:06.440
between x is equal to 1 and x is
equal to 2, 11 minus 7 is 4

00:10:06.440 --> 00:10:09.570
over 6, which is equal to 2/3.

00:10:09.570 --> 00:10:12.800
And that makes complete sense
because every time you

00:10:12.800 --> 00:10:16.260
increase x by 1, you're
increasing this expression by

00:10:16.260 --> 00:10:17.490
another 2/3.

00:10:17.490 --> 00:10:20.435
When you go from 1 to 2, you
should increase by 2/3.