Absolute value inequalities example 3

WEBVTT
Kind: captions
Language: en

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We're told to solve for y, and
we have this inequality that

00:00:03.790 --> 00:00:07.480
says that the absolute value of
y plus 22 is less than or

00:00:07.480 --> 00:00:11.210
equal to 13 and 1/2 or 13.5.

00:00:11.210 --> 00:00:14.260
So a good place to start is
maybe to just isolate the

00:00:14.260 --> 00:00:16.750
absolute value of y on the
left-hand side of this

00:00:16.750 --> 00:00:17.500
inequality.

00:00:17.500 --> 00:00:19.590
And the best way to do
that, we can subtract

00:00:19.590 --> 00:00:21.650
22 from both sides.

00:00:21.650 --> 00:00:25.050
So let's subtract 22
from both sides.

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The left-hand side, these guys
cancel out, that was the whole

00:00:27.580 --> 00:00:30.930
point, so you're just left with
the absolute value of y

00:00:30.930 --> 00:00:33.150
as being less than
or equal to.

00:00:33.150 --> 00:00:35.970
And then 13.5 minus 22,
let me do it over

00:00:35.970 --> 00:00:39.590
here, 13.5 minus 22.

00:00:39.590 --> 00:00:42.840
My brain imagines, or the way I
process it is, I say, well--

00:00:42.840 --> 00:00:45.760
I always like to put the larger
number first-- I say

00:00:45.760 --> 00:00:50.540
that's the negative
of 22 minus 13.5.

00:00:50.540 --> 00:00:56.140
And 22 minus 14 is 8, or the
difference between 22 and 14

00:00:56.140 --> 00:00:58.840
is 8, so the difference between
22 and 13 and 1/2 is

00:00:58.840 --> 00:01:00.430
going to be 1/2 more
than that.

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So this is going to be 8.5.

00:01:03.589 --> 00:01:05.260
So it's going to be
negative 8.5.

00:01:08.700 --> 00:01:11.780
So we get the absolute value of
y is less than or equal to

00:01:11.780 --> 00:01:14.220
negative 8.5.

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Now, this should cause you some
pause, because when you

00:01:18.990 --> 00:01:22.010
take the absolute value of
anything, what do you know

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you're going to get?

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If I tell you that the absolute
value of any number,

00:01:28.930 --> 00:01:30.410
oh, we'll just say the
absolute value of

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a is equal to x.

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What do you know about x?

00:01:34.330 --> 00:01:35.950
You know that x is
non-negative.

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It's either a positive
number or 0.

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Here we're saying that y is,
when we take the absolute

00:01:47.950 --> 00:01:50.460
value, has to be less than or
equal to a negative number.

00:01:50.460 --> 00:01:54.650
It's saying that it has to
be a negative number.

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This is implicitly saying that
the absolute value of y has to

00:02:02.070 --> 00:02:04.370
be negative.

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It not only has to be negative,
it has to be less

00:02:06.220 --> 00:02:08.530
than or equal to negative 8.5.

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We know that if you take the
absolute value of anything,

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you're going to get 0 or
a positive number.

00:02:12.690 --> 00:02:14.470
You're going to get a
non-negative number.

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There's no number you can put
here whose absolute value's

00:02:16.840 --> 00:02:19.240
going to give you a negative
number, especially one less

00:02:19.240 --> 00:02:21.530
than negative 8.5.

00:02:21.530 --> 00:02:24.460
So there is no solution
to this problem.

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You cannot find a y that
will satisfy this.

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So there is no solution.


Absolute inequalities 2

WEBVTT
Kind: captions
Language: en

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We have the absolute value of
2r minus 3 and 1/4 is less

00:00:05.500 --> 00:00:08.990
than 2 and 1/2, and we
want to solve for r.

00:00:08.990 --> 00:00:10.940
So right from the get go we
have to deal with this

00:00:10.940 --> 00:00:11.850
absolute value.

00:00:11.850 --> 00:00:14.290
And just as a bit of a review,
if I were to say that the

00:00:14.290 --> 00:00:18.220
absolute value of x is less
than, well let's just say,

00:00:18.220 --> 00:00:21.500
less than 2 and 1/2, that means
that the distance from x

00:00:21.500 --> 00:00:24.750
to 0 is less than 2 and 1/2.

00:00:24.750 --> 00:00:32.566
That means that x would have to
be less than 2 and 1/2, and

00:00:32.566 --> 00:00:36.850
x would have to be greater
than negative 2 and 1/2.

00:00:36.850 --> 00:00:38.620
And just think about
it for a second.

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If I were to draw it on a number
line right here, that

00:00:43.250 --> 00:00:50.030
is 0, that is 2 and 1/2, and
that is negative 2 and 1/2.

00:00:50.030 --> 00:00:53.600
These two numbers are exactly 2
and 1/2 away from 0, because

00:00:53.600 --> 00:00:56.470
both of their absolute
values is 2 and 1/2.

00:00:56.470 --> 00:00:58.940
Now, if we want all of the
numbers whose absolute value

00:00:58.940 --> 00:01:02.450
is less than 2 and 1/2, or that
are less than 2 and 1/2

00:01:02.450 --> 00:01:05.420
away from 0, it would be all
of the numbers in between.

00:01:08.490 --> 00:01:11.380
And that's exactly what these
two statements are saying. x

00:01:11.380 --> 00:01:15.410
has to be less than 2 and 1/2,
and it has to be greater than

00:01:15.410 --> 00:01:17.080
negative 2 and 1/2.

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If this absolute value were
the other way, that the

00:01:19.140 --> 00:01:21.313
absolute value of x has to be
greater than 2 and 1/2, then

00:01:21.313 --> 00:01:23.200
it would be the numbers
outside of this, and

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it would be an or.

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But we're dealing with the
less than situation right

00:01:26.590 --> 00:01:29.820
there, so let's just do what
we were able to figure out

00:01:29.820 --> 00:01:30.990
when it was just an x.

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The distance from this thing to
0 has to be less than 2 and

00:01:36.930 --> 00:01:45.990
1/2, so we can write that 2r
minus 3 and 1/4 has to be less

00:01:45.990 --> 00:01:55.610
than 2 and 1/2 and 2r minus 3
and 1/4 has to be greater than

00:01:55.610 --> 00:01:58.090
negative 2 and 1/2.

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Same exact reasoning here.

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Let me draw a line so we
don't get confused.

00:02:01.460 --> 00:02:03.050
Same exact reasoning here.

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This quantity right here
has to be between

00:02:06.940 --> 00:02:08.180
negative 2 and 1/2.

00:02:08.180 --> 00:02:09.820
It has to be greater
than negative 2

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and 1/2 right there.

00:02:12.470 --> 00:02:15.060
And it has to be less
than 2 and 1/2, so

00:02:15.060 --> 00:02:16.320
that's all I wrote there.

00:02:16.320 --> 00:02:19.450
So let's solve each of
these independently.

00:02:19.450 --> 00:02:22.340
Well, this first went over here,
you've learned before

00:02:22.340 --> 00:02:26.370
that I don't like improper
fractions, and I don't like

00:02:26.370 --> 00:02:27.300
fractions in general.

00:02:27.300 --> 00:02:29.710
So let's make all of
these fractions.

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Sorry, I don't like
mixed numbers.

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I want them to be improper
fractions.

00:02:32.490 --> 00:02:34.780
So let's turn all of these
into improper fractions.

00:02:34.780 --> 00:02:38.940
So if I were to rewrite it, we
get 2r minus 3 and 1/4 is the

00:02:38.940 --> 00:02:42.680
same thing as 3 times 4
is 12, plus 1 is 13.

00:02:42.680 --> 00:02:50.420
2r minus 13/4 is less than-- 2
times 2 is 4, plus 1 is five--

00:02:50.420 --> 00:02:51.600
is less than 5/2.

00:02:51.600 --> 00:02:53.280
So that's the first equation.

00:02:53.280 --> 00:02:56.320
And then the second question--
and do the same thing here--

00:02:56.320 --> 00:03:02.130
we have 2r minus 13 over
4 has to be a greater

00:03:02.130 --> 00:03:04.870
than negative 5/2.

00:03:04.870 --> 00:03:08.040
All right, now let's solve each
of these independently.

00:03:08.040 --> 00:03:10.640
To get rid of the fractions, the
easiest thing to do is to

00:03:10.640 --> 00:03:13.080
multiply both sides of
this equation by 4.

00:03:13.080 --> 00:03:15.730
That'll eliminate all of the
fractions, so let's do that.

00:03:15.730 --> 00:03:18.800
Let's multiply-- let me scroll
to the left a little bit--

00:03:18.800 --> 00:03:22.190
let's multiply both sides
of this equation by 4.

00:03:22.190 --> 00:03:22.760
And what do we get?

00:03:22.760 --> 00:03:28.840
4 times 2r is 8r, 4 times
negative 13 over 4 is negative

00:03:28.840 --> 00:03:32.010
13, is less than-- and I
multiplied by a positive

00:03:32.010 --> 00:03:34.270
number so I didn't have to
worry about swapping the

00:03:34.270 --> 00:03:40.620
inequality-- is less than 5/2
times 4 is 10, right?

00:03:40.620 --> 00:03:42.980
You get a 2 and a 1, it's 10.

00:03:42.980 --> 00:03:46.430
So you get 8r minus 13
is less than 10.

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Now we can add 13 to both sides
of this equation so that

00:03:50.090 --> 00:03:52.710
we get rid of it on the
left-hand side.

00:03:52.710 --> 00:03:56.460
Add 13 to both sides and we
get 8r-- these guys cancel

00:03:56.460 --> 00:04:02.010
out-- is less than 23,
and then we divide

00:04:02.010 --> 00:04:03.290
both sides by 8.

00:04:06.620 --> 00:04:08.790
And once again, we didn't have
to worry about the inequality

00:04:08.790 --> 00:04:11.380
because we're dividing
by a positive number.

00:04:11.380 --> 00:04:17.100
And we get r is less
than 23 over 8.

00:04:17.100 --> 00:04:20.160
Or, if you want to write that
as a mixed number, r is less

00:04:20.160 --> 00:04:26.070
than-- what is that--
2 and 7/8.

00:04:26.070 --> 00:04:28.390
So that's one condition, but we
still have to worry about

00:04:28.390 --> 00:04:29.230
this other condition.

00:04:29.230 --> 00:04:30.410
There was an and right here.

00:04:30.410 --> 00:04:31.820
Let's worry about it.

00:04:31.820 --> 00:04:36.120
So our other condition tells us
2r minus 13 over 4 has to

00:04:36.120 --> 00:04:38.540
be greater than negative 5/2.

00:04:38.540 --> 00:04:41.600
Let's multiply both sides
of this equation by 4.

00:04:41.600 --> 00:04:44.790
So 4 times 2r is 8r.

00:04:44.790 --> 00:04:49.080
4 times negative 13 over 4 is
negative 13, is greater than

00:04:49.080 --> 00:04:52.680
negative 5/2 times
4 is negative 10.

00:04:52.680 --> 00:04:55.013
Now we add 13 to both sides
of this equation.

00:04:59.640 --> 00:05:02.040
The left-hand side-- these guys
cancel out, you're just

00:05:02.040 --> 00:05:08.400
left with 8r-- is greater than
negative 10 plus 13 is 3.

00:05:08.400 --> 00:05:13.750
Or divide both sides of this by
8, and you're left with r

00:05:13.750 --> 00:05:16.450
has to be greater than 3/8.

00:05:16.450 --> 00:05:20.100
So our two conditions, r has to
be less than 2 and 7/8 and

00:05:20.100 --> 00:05:21.380
greater than 3/8.

00:05:21.380 --> 00:05:26.420
Or we can just write it like
this: r is greater than 3/8,

00:05:26.420 --> 00:05:31.980
so it's greater than-- maybe I
should say 3/8 is less than r,

00:05:31.980 --> 00:05:36.860
which is less than 2 and 7/8.

00:05:36.860 --> 00:05:39.045
So if we were to plot the
solution on the number line--

00:05:39.045 --> 00:05:44.490
which I'm about to do, so that's
my number line-- this

00:05:44.490 --> 00:05:51.990
is 0 right here, maybe
this is 1, 2, and 3.

00:05:51.990 --> 00:05:53.450
We have 2 and 7/8.

00:05:53.450 --> 00:05:55.980
We have to be less
than 2 and 7/8.

00:05:55.980 --> 00:05:59.000
Let's say that this is 2
and 7/8 right there.

00:05:59.000 --> 00:06:01.760
And we have to be greater
than 3/8.

00:06:01.760 --> 00:06:05.490
Say that is 3/8, so 3/8
will be some place

00:06:05.490 --> 00:06:06.790
right around there.

00:06:06.790 --> 00:06:09.415
And everything in between
is a valid solution.

00:06:12.520 --> 00:06:13.590
And we could try it out.

00:06:13.590 --> 00:06:15.970
Let's try out something that,
based on what I just drew,

00:06:15.970 --> 00:06:17.110
should be a valid solution.

00:06:17.110 --> 00:06:18.990
1 should be a valid solution.

00:06:18.990 --> 00:06:19.830
Let's try it out here.

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2 times 1 minus 3 and
1/4, what is that?

00:06:26.070 --> 00:06:29.570
That's 2 minus 3 and 1/4.

00:06:29.570 --> 00:06:30.890
And so what is that?

00:06:30.890 --> 00:06:36.690
2 minus 3 and 1/4 is-- well, 3
and 1/4 minus 2 is 1 and 1/4,

00:06:36.690 --> 00:06:39.790
so this will be negative
1 and 1/4.

00:06:39.790 --> 00:06:42.580
But we're taking the absolute
value of it, so we take the

00:06:42.580 --> 00:06:46.620
absolute value of it, which is
equal to 1 and 1/4, which is

00:06:46.620 --> 00:06:48.890
indeed less than 2 and 1/2.

00:06:48.890 --> 00:06:50.200
Now let's try another number.

00:06:50.200 --> 00:06:51.020
Let's try 0.

00:06:51.020 --> 00:06:53.590
Based on this, 0 should
not work.

00:06:53.590 --> 00:06:55.580
So what happens if
we put 0 here?

00:06:55.580 --> 00:06:58.930
You get 2 times 0, which
is 0, minus 3 and 1/4.

00:06:58.930 --> 00:07:02.420
If you take the absolute value
of negative 3 and 1/4, you'll

00:07:02.420 --> 00:07:05.580
get positive 3 and 1/4,
which won't work.

00:07:05.580 --> 00:07:08.250
3 and 1/4 is greater than
2 and 1/2, so that's

00:07:08.250 --> 00:07:09.850
true, that works out.

00:07:09.850 --> 00:07:11.480
And same thing for 3.

00:07:11.480 --> 00:07:18.870
2 times 3 is 6, minus 3
and 1/4 is 2 and 3/4.

00:07:18.870 --> 00:07:22.020
Take the absolute value, it's 2
and 3/4, still bigger than 2

00:07:22.020 --> 00:07:23.460
and 1/2, so it won't work.

00:07:23.460 --> 00:07:26.770
So at least the points that we
tried out seem to validate

00:07:26.770 --> 00:07:29.160
this solution that we got.


Absolute value inequalities example 1

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.810
We're told to graph all possible
values for h on the

00:00:04.810 --> 00:00:05.590
number line.

00:00:05.590 --> 00:00:08.510
And this is a especially
interesting inequality because

00:00:08.510 --> 00:00:11.060
we also have an absolute
value here.

00:00:11.060 --> 00:00:14.150
So the way we're going to do it,
we're going to solve this

00:00:14.150 --> 00:00:17.240
inequality in terms of the
absolute value of h, and from

00:00:17.240 --> 00:00:19.370
there we can solve it for h.

00:00:19.370 --> 00:00:21.430
So let's just get the absolute
value of h on

00:00:21.430 --> 00:00:22.890
one side of the equation.

00:00:22.890 --> 00:00:26.800
So the easiest way to do this
is to add 19 and 1/2 to both

00:00:26.800 --> 00:00:28.600
sides of this equation.

00:00:28.600 --> 00:00:32.159
I often like putting that as an
improper fraction, but 1/2

00:00:32.159 --> 00:00:35.670
is pretty easy to deal with,
so let's add 19 and 1/2 to

00:00:35.670 --> 00:00:37.680
both sides of this inequality.

00:00:37.680 --> 00:00:38.470
Did I just say equation?

00:00:38.470 --> 00:00:41.150
It's an inequality, not an
equation, it's an inequality

00:00:41.150 --> 00:00:42.600
sign, not an equal sign.

00:00:42.600 --> 00:00:46.380
So plus 19 and 1/2.

00:00:46.380 --> 00:00:49.210
On the left-hand side, these
guys obviously cancel out,

00:00:49.210 --> 00:00:52.290
that was the whole point, and we
are left with the absolute

00:00:52.290 --> 00:00:56.480
value of h on the left-hand
side is less than.

00:00:56.480 --> 00:01:01.660
And then if we have 19 and 1/2,
essentially minus 12, 19

00:01:01.660 --> 00:01:04.709
minus 12 is 7, so it's going
to be 7 and 1/2.

00:01:08.580 --> 00:01:13.260
So now we have that the absolute
value of h is less

00:01:13.260 --> 00:01:16.050
than 7 and 1/2.

00:01:16.050 --> 00:01:17.030
So what does this tell us?

00:01:17.030 --> 00:01:19.470
This means that the distance,
another way to interpret

00:01:19.470 --> 00:01:21.960
this-- remember, absolute value
is the same thing as

00:01:21.960 --> 00:01:25.340
distance from 0-- so another
way to interpret this

00:01:25.340 --> 00:01:37.210
statement is that the distance
from h to 0 has to be less

00:01:37.210 --> 00:01:43.130
than 7 and 1/2.

00:01:43.130 --> 00:01:46.200
So what values of h are going
to have less than a distance

00:01:46.200 --> 00:01:47.510
from 7 and 1/2?

00:01:47.510 --> 00:01:51.120
Well, it could be less than 7
and 1/2 and greater than 0, or

00:01:51.120 --> 00:01:51.765
equal to 0.

00:01:51.765 --> 00:01:53.580
So let me put it this way.

00:01:53.580 --> 00:01:59.020
So h could be less
than 7 and 1/2.

00:01:59.020 --> 00:02:01.950
But if it gets too far negative,
if it goes to

00:02:01.950 --> 00:02:04.240
negative 3, we're cool, negative
4, negative 5,

00:02:04.240 --> 00:02:07.420
negative 6, negative 7, we're
still cool, but then at

00:02:07.420 --> 00:02:10.090
negative 8, all of a sudden the
absolute value isn't going

00:02:10.090 --> 00:02:11.180
to be less than this.

00:02:11.180 --> 00:02:16.770
So it also has to be greater
than negative 7 and 1/2.

00:02:16.770 --> 00:02:19.540
If you give me any number in
this interval, its absolute

00:02:19.540 --> 00:02:22.320
value is going to be less than 7
and 1/2 because all of these

00:02:22.320 --> 00:02:25.650
numbers are less than 7
and 1/2 away from 0.

00:02:25.650 --> 00:02:27.550
Let me draw it on the number
line, which they

00:02:27.550 --> 00:02:29.820
want us to do anyway.

00:02:29.820 --> 00:02:37.540
So if this is the number line
right there, that is 0, and we

00:02:37.540 --> 00:02:42.200
draw some points, let's say
that this is 7, that is 8,

00:02:42.200 --> 00:02:47.070
that is negative 7, that
is negative 8.

00:02:47.070 --> 00:02:51.120
What numbers are less than
7 and 1/2 away from 0?

00:02:51.120 --> 00:02:54.170
Well, you have everything all
the way up to-- 7 and 1/2 is

00:02:54.170 --> 00:02:57.480
exactly 7 and 1/2 away, so you
can't count that, so 7 and

00:02:57.480 --> 00:02:59.220
1/2, you'll put a circle
around it.

00:02:59.220 --> 00:03:02.170
Same thing true for negative 7
and 1/2, the absolute value,

00:03:02.170 --> 00:03:03.810
it's exactly 7 and 1/2 away.

00:03:03.810 --> 00:03:06.940
We have to be less than 7 and
1/2 away, so neither of those

00:03:06.940 --> 00:03:09.330
points are going to be included,
positive 7 and 1/2

00:03:09.330 --> 00:03:11.060
or negative 7 and 1/2.

00:03:11.060 --> 00:03:14.630
Now, everything in between is
less than 7 and 1/2 away from

00:03:14.630 --> 00:03:19.310
0, so everything else counts.

00:03:19.310 --> 00:03:22.490
Everything outside of it is
clearly more than 7 and 1/2

00:03:22.490 --> 00:03:24.490
away from 0.

00:03:24.490 --> 00:03:26.000
And we're done.


Constructing and solving a one-step inequality

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.090
A contractor is purchasing
some stone

00:00:03.090 --> 00:00:04.780
tiles for a new patio.

00:00:04.780 --> 00:00:08.140
Each tile costs $3,
and he wants to

00:00:08.140 --> 00:00:10.470
spend less than $1,000.

00:00:10.470 --> 00:00:15.340
And it's less than $1,000, not
less than or equal to $1,000.

00:00:15.340 --> 00:00:18.750
The size of each tile
is one square foot.

00:00:18.750 --> 00:00:21.890
Write an inequality that
represents the number of tiles

00:00:21.890 --> 00:00:25.800
he can purchase with
a $1,000 limit.

00:00:25.800 --> 00:00:28.930
And then figure out how large
the stone patio can be.

00:00:28.930 --> 00:00:35.425
So let x be equal to the number
of tiles purchased.

00:00:39.460 --> 00:00:43.600
And so the cost of purchasing
x tiles, they're going to be

00:00:43.600 --> 00:00:46.520
$3 each, so it's
going to be 3x.

00:00:46.520 --> 00:00:50.200
So 3x is going to be the total
cost of purchasing the tiles.

00:00:50.200 --> 00:00:53.270
And he wants to spend
less than $1,000.

00:00:53.270 --> 00:00:55.990
3x is how much he spends
if he buys x tiles.

00:00:55.990 --> 00:01:01.880
It has to be less than $1,000,
we say it right there.

00:01:01.880 --> 00:01:04.060
If it was less than or equal to,
we'd have a little equal

00:01:04.060 --> 00:01:05.160
sign right there.

00:01:05.160 --> 00:01:08.190
So if we want to solve for x,
how many tiles can he buy?

00:01:08.190 --> 00:01:12.850
We can divide both sides of
this inequality by 3.

00:01:12.850 --> 00:01:15.710
And because we're dividing or
multiplying-- you could

00:01:15.710 --> 00:01:19.010
imagine we're multiplying by
1/3 or dividing by 3 --

00:01:19.010 --> 00:01:22.600
because this is a positive
number, we do not have to swap

00:01:22.600 --> 00:01:24.010
the inequality sign.

00:01:24.010 --> 00:01:30.110
So we are left with x is less
than 1,000 over three, which

00:01:30.110 --> 00:01:35.090
is 333 and 1/3.

00:01:35.090 --> 00:01:40.530
So he has to buy less than 333
and 1/3 tiles, that's how many

00:01:40.530 --> 00:01:43.690
tiles, and each tile
is one square foot.

00:01:43.690 --> 00:01:55.640
So if he can buy less than 333
and 1/3 tiles, then the patio

00:01:55.640 --> 00:02:03.320
also has to be less than 333
and 1/3 square feet.

00:02:03.320 --> 00:02:06.500
Feet squared, we could
say square feet.

00:02:06.500 --> 00:02:08.270
And we're done.


Solving a two-step inequality

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.490
We have the inequality 2/3
is greater than negative

00:00:04.490 --> 00:00:08.380
4y minus 8 and 1/3.

00:00:08.380 --> 00:00:10.410
Now, the first thing I want to
do here, just because mixed

00:00:10.410 --> 00:00:12.770
numbers bother me-- they're
actually hard to deal with

00:00:12.770 --> 00:00:13.470
mathematically.

00:00:13.470 --> 00:00:14.950
They're easy to think about--
oh, it's a little

00:00:14.950 --> 00:00:16.079
bit more than 8.

00:00:16.079 --> 00:00:18.910
Let's convert this to an
improper fraction.

00:00:18.910 --> 00:00:23.910
So 8 and 1/3 is equal to-- the
denominator's going to be 3.

00:00:23.910 --> 00:00:28.080
3 times 8 is 24, plus 1 is 25.

00:00:28.080 --> 00:00:30.920
So this thing over here is the
same thing as 25 over 3.

00:00:30.920 --> 00:00:32.240
Let me just rewrite
the whole thing.

00:00:32.240 --> 00:00:42.340
So it's 2/3 is greater than
negative 4y minus 25 over 3.

00:00:42.340 --> 00:00:44.730
Now, the next thing I want to
do, just because dealing with

00:00:44.730 --> 00:00:48.260
fractions are a bit of a pain,
is multiply both sides of this

00:00:48.260 --> 00:00:50.430
inequality by some
quantity that'll

00:00:50.430 --> 00:00:51.550
eliminate the fractions.

00:00:51.550 --> 00:00:53.610
And the easiest one I can think
of is multiply both

00:00:53.610 --> 00:00:54.480
sides by 3.

00:00:54.480 --> 00:00:56.870
That'll get rid of the 3's
in the denominator.

00:00:56.870 --> 00:00:59.430
So let's multiply both sides
of this equation by 3.

00:00:59.430 --> 00:01:00.850
That's the left-hand side.

00:01:00.850 --> 00:01:03.520
And then I'm going to multiply
the right-hand side.

00:01:03.520 --> 00:01:07.750
3, I'll put it in parentheses
like that.

00:01:07.750 --> 00:01:11.870
Well, one point that I want to
point out is that I did not

00:01:11.870 --> 00:01:15.160
have to swap the inequality
sign, because I multiplied

00:01:15.160 --> 00:01:17.300
both sides by a positive
number.

00:01:17.300 --> 00:01:19.600
If the 3 was a negative number,
if I multiplied both

00:01:19.600 --> 00:01:21.650
sides by negative 3, or negative
1, or negative

00:01:21.650 --> 00:01:24.920
whatever, I would have had to
swap the inequality sign.

00:01:24.920 --> 00:01:26.290
Anyway, let's simplify this.

00:01:26.290 --> 00:01:31.040
So the left-hand side, we have
3 times 2/3, which is just 2.

00:01:31.040 --> 00:01:32.660
2 is greater than.

00:01:32.660 --> 00:01:34.050
And then we can distribute
this 3.

00:01:34.050 --> 00:01:38.580
3 times negative 4y
is negative 12y.

00:01:38.580 --> 00:01:44.880
And then 3 times negative 25
over 3 is just negative 25.

00:01:44.880 --> 00:01:48.450
Now, we want to get all of our
constant terms on one side of

00:01:48.450 --> 00:01:50.830
the inequality and all of our
variable terms-- the only

00:01:50.830 --> 00:01:53.660
variable here is y on the other
side-- the y is already

00:01:53.660 --> 00:01:56.070
sitting here, so let's just get
this 25 on the other side

00:01:56.070 --> 00:01:56.820
of the inequality.

00:01:56.820 --> 00:01:59.490
And we can do that by
adding 25 to both

00:01:59.490 --> 00:02:01.310
sides of this equation.

00:02:01.310 --> 00:02:05.755
So let's add 25 to both sides
of this equation.

00:02:09.300 --> 00:02:13.642
And with the left-hand side, 2
plus 25 five is 27 and we're

00:02:13.642 --> 00:02:16.070
going to get 27 is
greater than.

00:02:16.070 --> 00:02:21.030
The right-hand side of the
inequality is negative 12y.

00:02:21.030 --> 00:02:23.930
And then negative 25 plus 25,
those cancel out, that was the

00:02:23.930 --> 00:02:27.300
whole point, so we're left
with 27 is greater than

00:02:27.300 --> 00:02:29.190
negative 12y.

00:02:29.190 --> 00:02:34.970
Now, to isolate the y, you can
either multiply both sides by

00:02:34.970 --> 00:02:38.710
negative 1/12 or you could say
let's just divide both sides

00:02:38.710 --> 00:02:40.180
by negative 12.

00:02:40.180 --> 00:02:43.630
Now, because I'm multiplying
or dividing by a negative

00:02:43.630 --> 00:02:46.950
number here, I'm going to need
to swap the inequality.

00:02:46.950 --> 00:02:48.030
So let me write this.

00:02:48.030 --> 00:02:51.690
If I divide both sides of this
equation by negative 12, then

00:02:51.690 --> 00:02:57.055
it becomes 27 over negative 12
is less than-- I'm swapping

00:02:57.055 --> 00:02:59.900
the inequality, let me do this
in a different color-- is less

00:02:59.900 --> 00:03:05.190
than negative 12y over
negative 12.

00:03:05.190 --> 00:03:08.280
Notice, when I divide both sides
of the inequality by a

00:03:08.280 --> 00:03:12.150
negative number, I swap the
inequality, the greater than

00:03:12.150 --> 00:03:13.950
becomes a less than.

00:03:13.950 --> 00:03:16.620
When it was positive, I didn't
have to swap it.

00:03:16.620 --> 00:03:20.280
So 27 divided by negative
12, well, they're both

00:03:20.280 --> 00:03:22.520
divisible by 3.

00:03:22.520 --> 00:03:24.850
So we're going to get, if we
divide the numerator and the

00:03:24.850 --> 00:03:30.960
denominator by 3, we get
negative 9 over 4 is less

00:03:30.960 --> 00:03:33.580
than-- these cancel out-- y.

00:03:33.580 --> 00:03:37.260
So y is greater than negative
9/4, or negative 9/4

00:03:37.260 --> 00:03:39.220
is less than y.

00:03:39.220 --> 00:03:40.740
And if you wanted to write
that-- just let me write

00:03:40.740 --> 00:03:45.110
this-- our answer is y is
greater than negative 9/4.

00:03:45.110 --> 00:03:47.660
I just swapped the order, you
could say negative 9/4

00:03:47.660 --> 00:03:48.870
is less than y.

00:03:48.870 --> 00:03:51.590
Or if you want to visualize that
a little bit better, 9/4

00:03:51.590 --> 00:03:55.140
is 2 and 1/4, so we could also
say y is greater than negative

00:03:55.140 --> 00:03:58.300
2 and 1/4 if we want to put
it as a mixed number.

00:03:58.300 --> 00:04:01.410
And if we wanted to graph it
on the number line-- let me

00:04:01.410 --> 00:04:04.620
draw a number line right here,
a real simple one.

00:04:04.620 --> 00:04:06.820
Maybe this is 0.

00:04:06.820 --> 00:04:10.870
Negative 2 is right over, let's
say negative 1, negative

00:04:10.870 --> 00:04:14.450
2, then say negative
3 is right there.

00:04:14.450 --> 00:04:17.610
Negative 2 and 1/4 is going
to be right here, and it's

00:04:17.610 --> 00:04:19.930
greater than, so we're not going
to include that in the

00:04:19.930 --> 00:04:21.170
solution set.

00:04:21.170 --> 00:04:23.700
So we're going to make an
open circle right there.

00:04:23.700 --> 00:04:27.560
And everything larger than that
is a valid y, is a y that

00:04:27.560 --> 00:04:30.450
will satisfy the inequality.


Constructing and solving a two-step inequality

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.850
We're told that for the past
few months, Old Maple Farms

00:00:04.850 --> 00:00:08.690
has grown about 1,000 more
apples than their chief rival

00:00:08.690 --> 00:00:11.000
in the region, River Orchards.

00:00:11.000 --> 00:00:14.180
Due to cold weather this year,
the harvests at both farms

00:00:14.180 --> 00:00:15.920
were down by about a third.

00:00:15.920 --> 00:00:19.050
However, both farms made up for
some of the shortfall by

00:00:19.050 --> 00:00:23.980
purchasing equal quantities
of apples from farms in

00:00:23.980 --> 00:00:26.540
neighboring states.

00:00:26.540 --> 00:00:28.950
What can you say about
the number of apples

00:00:28.950 --> 00:00:30.130
available at each farm?

00:00:30.130 --> 00:00:32.800
Does one farm have more than the
other, or do they have the

00:00:32.800 --> 00:00:33.670
same amount?

00:00:33.670 --> 00:00:35.200
How do I know?

00:00:35.200 --> 00:00:37.410
So let's define some
variables here.

00:00:37.410 --> 00:00:57.220
Let's let M be equal to number
of apples at Maple Farms. And

00:00:57.220 --> 00:00:58.310
then who's the other guy?

00:00:58.310 --> 00:00:59.260
River Orchards.

00:00:59.260 --> 00:01:03.750
So let's let R be equal
to the number of

00:01:03.750 --> 00:01:12.030
apples at River Orchards.

00:01:12.030 --> 00:01:14.690
So this first sentence, they
say-- let me do this in a

00:01:14.690 --> 00:01:17.540
different color-- they say for
the past few years, Old Maple

00:01:17.540 --> 00:01:22.560
Farms has grown about 1,000 more
apples than their chief

00:01:22.560 --> 00:01:25.650
rival in the region,
River Orchards.

00:01:25.650 --> 00:01:31.500
So we could say, hey, Maple is
approximately Old River, or M

00:01:31.500 --> 00:01:34.680
is approximately River
plus 1,000.

00:01:34.680 --> 00:01:37.190
Or since we don't know the exact
amount-- it says it's

00:01:37.190 --> 00:01:39.940
about 1,000 more, so we don't
know it's exactly 1,000 more--

00:01:39.940 --> 00:01:44.080
we can just say that in a normal
year, Old Maple Farms,

00:01:44.080 --> 00:01:49.880
which we denote by M, has a
larger amount of apples than

00:01:49.880 --> 00:01:51.170
River Orchard.

00:01:51.170 --> 00:01:53.750
So in a normal year, M is
greater than R, right?

00:01:53.750 --> 00:01:56.620
It has about 1,000 more apples
than Old Maple Farms.

00:01:56.620 --> 00:01:59.660
Now, they say due to cold
weather this year-- so let's

00:01:59.660 --> 00:02:02.570
talk about this year now-- the
harvests at both farms were

00:02:02.570 --> 00:02:03.995
down about a third.

00:02:07.930 --> 00:02:09.620
So this isn't a normal year.

00:02:09.620 --> 00:02:11.660
Let's talk about what's going
to happen this year.

00:02:11.660 --> 00:02:13.700
In this year, each of these
characters are going to be

00:02:13.700 --> 00:02:15.280
down by 1/3.

00:02:15.280 --> 00:02:18.710
Now if I go down by 1/3, that's
the same thing as being

00:02:18.710 --> 00:02:20.510
2/3 of what I was before.

00:02:20.510 --> 00:02:21.380
Let me do an example.

00:02:21.380 --> 00:02:26.700
If I'm at x, and I take away
1/3x, I'm left with 2/3x.

00:02:29.400 --> 00:02:33.800
So going down by 1/3 is the same
thing as multiplying the

00:02:33.800 --> 00:02:35.910
quantity by 2/3.

00:02:35.910 --> 00:02:38.760
So if we multiply each of these
quantities by 2/3, we

00:02:38.760 --> 00:02:40.880
can still hold this inequality,
because we're

00:02:40.880 --> 00:02:44.010
doing the same thing to both
sides of this inequality, and

00:02:44.010 --> 00:02:46.340
we're multiplying by
a positive number.

00:02:46.340 --> 00:02:48.680
If we were multiplying by a
negative number, we would have

00:02:48.680 --> 00:02:50.460
to swap the inequality.

00:02:50.460 --> 00:02:54.400
So we can multiply both
sides of this by 2/3.

00:02:54.400 --> 00:02:59.525
So 2/3 of M is still going to
be greater than 2/3 of R.

00:02:59.525 --> 00:03:02.620
And you could even draw that in
a number line if you like.

00:03:02.620 --> 00:03:04.280
Let's do this in
a number line.

00:03:04.280 --> 00:03:06.450
This all might be a little
intuitive for you, and if it

00:03:06.450 --> 00:03:08.980
is, I apologize, but if it's
not, it never hurts.

00:03:08.980 --> 00:03:10.440
So that's 0 on our
number line.

00:03:10.440 --> 00:03:15.490
So in a normal year, M is
has 1,000 more than R.

00:03:15.490 --> 00:03:18.330
So in a normal year, M might
be over here and

00:03:18.330 --> 00:03:20.310
maybe R is over here.

00:03:20.310 --> 00:03:23.950
I don't know, let's say
R is over there.

00:03:23.950 --> 00:03:29.050
Now, if we take 2/3 of M, that's
going to stick us some

00:03:29.050 --> 00:03:32.340
place around, oh, I
don't know, 2/3

00:03:32.340 --> 00:03:33.840
is right about there.

00:03:33.840 --> 00:03:40.250
So this is M-- let me write
this-- this is 2/3 M.

00:03:40.250 --> 00:03:41.910
And what's 2/3 of
R going to be?

00:03:41.910 --> 00:03:45.250
Well, if you take 2/3 of this,
you get to right about there,

00:03:45.250 --> 00:03:47.970
that is 2/3R.

00:03:47.970 --> 00:03:56.360
So you can see, 2/3R is still
less than 2/3M, or 2/3M is

00:03:56.360 --> 00:03:59.220
greater than 2/3R.

00:03:59.220 --> 00:04:04.890
Now, they say both farms made
up for some of the shortfall

00:04:04.890 --> 00:04:08.900
by purchasing equal quantities
of apples from farms in

00:04:08.900 --> 00:04:10.200
neighboring states.

00:04:10.200 --> 00:04:21.579
So let's let a be equal
to the quantity

00:04:21.579 --> 00:04:26.235
of apples both purchased.

00:04:29.240 --> 00:04:30.430
So they're telling
us that they both

00:04:30.430 --> 00:04:31.870
purchased the same amount.

00:04:31.870 --> 00:04:34.840
So we could add a to both sides
of this equation and it

00:04:34.840 --> 00:04:36.410
will not change the
inequality.

00:04:36.410 --> 00:04:39.670
As long as you add or subtract
the same value to both sides,

00:04:39.670 --> 00:04:41.640
it will not change
the inequality.

00:04:41.640 --> 00:04:46.000
So if you add a to both sides,
you have a plus 2/3M is a

00:04:46.000 --> 00:04:49.150
greater than 2/3R plus a.

00:04:49.150 --> 00:04:52.980
This is the amount that Old
Maple Farms has after

00:04:52.980 --> 00:04:57.080
purchasing the apples, and this
is the amount that River

00:04:57.080 --> 00:04:58.380
Orchards has.

00:04:58.380 --> 00:05:01.440
So after everything is said
and done, Old Maple Farms

00:05:01.440 --> 00:05:04.200
still has more apples, and
you can see that here.

00:05:04.200 --> 00:05:07.690
Maple Farms, a normal year, this
year they only had 2/3 of

00:05:07.690 --> 00:05:10.180
the production, but then they
purchased a apples.

00:05:10.180 --> 00:05:14.510
So let's say a is about, let's
say that a is that many

00:05:14.510 --> 00:05:18.040
apples, so they got back
to their normal amount.

00:05:18.040 --> 00:05:19.810
So let's say they got back
to their normal amount.

00:05:19.810 --> 00:05:21.610
So that's how many apples
they purchased, so

00:05:21.610 --> 00:05:23.590
he got back to M.

00:05:23.590 --> 00:05:29.360
Now, if R, if River Orchards
also purchased a apples, that

00:05:29.360 --> 00:05:33.110
same distance, a, if you go
along here gets you to right

00:05:33.110 --> 00:05:35.370
about over there.

00:05:35.370 --> 00:05:38.960
So once again, this is-- let
me do it a little bit

00:05:38.960 --> 00:05:42.010
different, because I don't like
it overlapping, so let me

00:05:42.010 --> 00:05:42.740
do it like this.

00:05:42.740 --> 00:05:46.260
So let's say this guy, M-- I
keep forgetting their names--

00:05:46.260 --> 00:05:50.900
Old Maple Farms purchases a
apples, gets them that far.

00:05:50.900 --> 00:05:52.360
So that's a apples.

00:05:52.360 --> 00:05:55.700
But River Orchards also
purchases a apples, so let's

00:05:55.700 --> 00:05:56.700
add that same amount.

00:05:56.700 --> 00:05:58.700
I'm just going to copy and paste
it so it's the exact

00:05:58.700 --> 00:05:59.950
same amount.

00:06:02.310 --> 00:06:07.450
So River Orchards also purchases
a, so it also

00:06:07.450 --> 00:06:09.470
purchases that same amount.

00:06:09.470 --> 00:06:12.980
So when all is said and done,
River Orchards is going to

00:06:12.980 --> 00:06:16.390
have this many apples in the
year that they had less

00:06:16.390 --> 00:06:17.800
production but they went
and purchased it.

00:06:17.800 --> 00:06:21.420
So this, right here, is--
this value right here

00:06:21.420 --> 00:06:26.120
is 2/3R plus a.

00:06:26.120 --> 00:06:27.380
That's what River
Orchards has.

00:06:27.380 --> 00:06:30.830
And then Old Maple Farms has
this value right here, which

00:06:30.830 --> 00:06:34.290
is 2/3M plus a.

00:06:34.290 --> 00:06:36.590
Everything said and done,
Old Maple Farms

00:06:36.590 --> 00:06:38.660
still has more apples.


Converting linear equations to slope-intercept form

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.430
We're asked to convert these
linear equations into

00:00:03.430 --> 00:00:08.870
slope-intercept form and then
graph them on a single

00:00:08.870 --> 00:00:09.740
coordinate plane.

00:00:09.740 --> 00:00:11.590
We have our coordinate
plane over here.

00:00:11.590 --> 00:00:14.720
And just as a bit of a review,
slope-intercept form is a form

00:00:14.720 --> 00:00:20.910
y is equal to mx plus b,
where m is the slope

00:00:20.910 --> 00:00:22.510
and b is the intercept.

00:00:22.510 --> 00:00:25.050
That's why it's called
slope-intercept form.

00:00:25.050 --> 00:00:27.020
So we just have to algebraically
manipulate these

00:00:27.020 --> 00:00:29.440
equations into this form.

00:00:29.440 --> 00:00:34.540
So let's start with line A,
so start with a line A.

00:00:34.540 --> 00:00:39.230
So line A, it's in standard form
right now, it's 4x plus

00:00:39.230 --> 00:00:42.570
2y is equal to negative 8.

00:00:42.570 --> 00:00:46.160
The first thing I'd like to do
is get rid of this 4x from the

00:00:46.160 --> 00:00:48.880
left-hand side, and the best way
to do that is to subtract

00:00:48.880 --> 00:00:51.460
4x from both sides
of this equation.

00:00:51.460 --> 00:00:55.420
So let me subtract 4x
from both sides.

00:00:55.420 --> 00:00:57.880
The left hand side of the
equation, these two 4x's

00:00:57.880 --> 00:01:02.440
cancel out, and I'm just left
with 2y is equal to.

00:01:02.440 --> 00:01:10.480
And on the right-hand side I
have negative 4x minus is 8,

00:01:10.480 --> 00:01:13.430
or negative 8 minus 4, however
you want to do it.

00:01:13.430 --> 00:01:15.890
Now we're almost at
slope-intercept form.

00:01:15.890 --> 00:01:18.330
We just have to get rid of this
2, and the best way to do

00:01:18.330 --> 00:01:20.770
that that I can think of is
divide both sides of this

00:01:20.770 --> 00:01:23.220
equation by 2.

00:01:23.220 --> 00:01:25.250
So let's divide both
sides by 2.

00:01:25.250 --> 00:01:28.190
So we divide the left-hand side
by 2 and then divide the

00:01:28.190 --> 00:01:29.700
right-hand side by 2.

00:01:29.700 --> 00:01:31.750
You have to divide
every term by 2.

00:01:31.750 --> 00:01:37.980
And then we are left with y is
equal to negative 4 divided by

00:01:37.980 --> 00:01:40.540
2 is negative 2x.

00:01:40.540 --> 00:01:45.070
Negative 8 divided by
2 is negative 4,

00:01:45.070 --> 00:01:47.480
negative 2x minus 4.

00:01:47.480 --> 00:01:52.900
So this is line A, let me
graph it right now.

00:01:52.900 --> 00:01:56.420
So line A, its y-intercept
is negative 4.

00:01:56.420 --> 00:01:59.100
So the point 0, negative
4 on this graph.

00:01:59.100 --> 00:02:01.790
If x is equal to 0, y is going
to be equal to negative 4, you

00:02:01.790 --> 00:02:04.150
can just substitute
that in the graph.

00:02:04.150 --> 00:02:07.330
So 0, 1, 2, 3, 4.

00:02:07.330 --> 00:02:10.979
That's the point
0, negative 4.

00:02:10.979 --> 00:02:14.300
That's the y-intercept
for line A.

00:02:14.300 --> 00:02:18.770
And then the slope
is negative 2x.

00:02:18.770 --> 00:02:23.830
So that means that if I change
x by positive 1 that y goes

00:02:23.830 --> 00:02:26.150
down by negative 2.

00:02:26.150 --> 00:02:27.460
So let's do that.

00:02:27.460 --> 00:02:30.250
So if I go over one in the
positive direction, I have to

00:02:30.250 --> 00:02:32.870
go down 2, that's what a
negative slope's going to do,

00:02:32.870 --> 00:02:33.920
negative 2 slope.

00:02:33.920 --> 00:02:37.750
If I go over 2, I'm going
to have to go down 4.

00:02:37.750 --> 00:02:42.460
If I go back negative 1, so
if I go in the x direction

00:02:42.460 --> 00:02:45.540
negative 1, that means in the y
direction I go positive two,

00:02:45.540 --> 00:02:48.620
because two divided by negative
one is still negative

00:02:48.620 --> 00:02:49.850
two, so I go over here.

00:02:49.850 --> 00:02:53.390
If I go back 2, I'm
going to go up 4.

00:02:53.390 --> 00:02:54.300
Let me just do that.

00:02:54.300 --> 00:02:56.490
Back 2 and then up 4.

00:02:56.490 --> 00:02:59.120
So this line is going
to look like this.

00:03:03.000 --> 00:03:06.160
Do my best to draw it,
that's a decent job.

00:03:06.160 --> 00:03:09.550
That is line A right there.

00:03:09.550 --> 00:03:13.710
All right, let's do line B.

00:03:13.710 --> 00:03:20.310
So line B, they say 4x is equal
to negative 8, and you

00:03:20.310 --> 00:03:22.850
might be saying hey, how do I
get that into slope-intercept

00:03:22.850 --> 00:03:24.040
form, I don't see a y.

00:03:24.040 --> 00:03:27.190
And the answer is you won't be
able to because you this can't

00:03:27.190 --> 00:03:29.900
be put into slope-intercept
form, but we can simplify it.

00:03:29.900 --> 00:03:33.240
So let's divide both sides
of this equation by 4.

00:03:33.240 --> 00:03:36.060
So you divide both sides
of this equation by 4.

00:03:36.060 --> 00:03:40.680
And you get x is equal
to negative 2.

00:03:40.680 --> 00:03:43.520
So this just means, I don't care
what your y is, x is just

00:03:43.520 --> 00:03:46.190
always going to be equal
to negative 2.

00:03:46.190 --> 00:03:49.110
So x is equal to negative 2 is
right there, negative 1,

00:03:49.110 --> 00:03:51.850
negative 2, and x is just always
going to be equal to

00:03:51.850 --> 00:03:59.620
negative 2 in both directions.

00:03:59.620 --> 00:04:02.180
And this is the x-axis,
that's the y-axis, I

00:04:02.180 --> 00:04:03.660
forgot to label them.

00:04:03.660 --> 00:04:05.880
Now let's do this last
character, 2y is equal to

00:04:05.880 --> 00:04:07.630
negative eight.

00:04:07.630 --> 00:04:12.310
So line C, we have 2y is
equal to negative 8.

00:04:12.310 --> 00:04:18.980
We can divide both sides of this
equation by 2, and we get

00:04:18.980 --> 00:04:22.570
y is equal to negative 4.

00:04:22.570 --> 00:04:25.020
So you might say hey, Sal, that
doesn't look like this

00:04:25.020 --> 00:04:28.130
form, slope-intercept
form, but it is.

00:04:28.130 --> 00:04:29.580
It's just that the slope is 0.

00:04:29.580 --> 00:04:34.960
We can rewrite this as y is
equal to 0x minus 4, where the

00:04:34.960 --> 00:04:38.260
y-intercept is negative
4 and the slope is 0.

00:04:47.070 --> 00:04:49.810
So if you move an arbitrary
amount in the x direction, the

00:04:49.810 --> 00:04:51.570
y is not going to change,
it's just going to

00:04:51.570 --> 00:04:52.760
stay at negative 4.

00:04:52.760 --> 00:04:54.760
Let me do a little bit neater.

00:04:54.760 --> 00:04:59.450
y is just going to stay
at negative 4.

00:04:59.450 --> 00:05:01.710
Or you can just interpret it as
y is equal to negative 4 no

00:05:01.710 --> 00:05:03.950
matter what x is.

00:05:03.950 --> 00:05:06.490
So then we are done.


Parallel lines from equation (example 3)

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.360
We're asked which of these
lines are parallel.

00:00:03.360 --> 00:00:06.000
So they give us three equations
of three different

00:00:06.000 --> 00:00:08.270
lines and if they're parallel,
then they have to have the

00:00:08.270 --> 00:00:08.975
same slope.

00:00:08.975 --> 00:00:11.570
So all we have to do over here
is figure out the slopes of

00:00:11.570 --> 00:00:12.830
each of these lines, and
if any of them are

00:00:12.830 --> 00:00:14.060
equal, they're parallel.

00:00:14.060 --> 00:00:15.970
So let's do line A.

00:00:15.970 --> 00:00:21.870
Line A, it's 2y is equal
to 12x plus 10.

00:00:21.870 --> 00:00:24.010
We're almost in slope-intercept
form, we can

00:00:24.010 --> 00:00:29.210
just divide both sides of
this equation by 2.

00:00:29.210 --> 00:00:33.420
We get y is equal to 6x--
right, 12 divided by

00:00:33.420 --> 00:00:36.330
2 -- 6x plus 5.

00:00:36.330 --> 00:00:38.860
So our slope in this case, we
have it in slope-intercept

00:00:38.860 --> 00:00:43.710
form, our slope in this
case is equal to 6.

00:00:43.710 --> 00:00:46.570
Let's try line B.

00:00:46.570 --> 00:00:51.810
Line B is y is equal to six.

00:00:51.810 --> 00:00:53.670
You might say this hey, this is
a bizarre character, how do

00:00:53.670 --> 00:00:54.920
I get this into slope-intercept

00:00:54.920 --> 00:00:56.790
form, where's the x?

00:00:56.790 --> 00:00:59.680
And my answer to you is
that it already is in

00:00:59.680 --> 00:01:00.640
slope-intercept form.

00:01:00.640 --> 00:01:05.670
I could just rewrite it as
y is equal to 0x plus 6.

00:01:05.670 --> 00:01:08.170
The x term is being multiplied
by 0 because the

00:01:08.170 --> 00:01:10.150
slope here is 0.

00:01:10.150 --> 00:01:12.130
y is going to be equal
to six no matter how

00:01:12.130 --> 00:01:13.280
much you change x.

00:01:13.280 --> 00:01:15.690
Change in y is always
going to be 0, it's

00:01:15.690 --> 00:01:17.880
always going to be 6.

00:01:17.880 --> 00:01:26.030
So here, our slope is 0, so
these two lines are definitely

00:01:26.030 --> 00:01:29.060
not parallel, they have
different slopes.

00:01:29.060 --> 00:01:31.690
So let's try line C.

00:01:31.690 --> 00:01:34.030
Line C-- I'll do it down here.

00:01:34.030 --> 00:01:40.780
Line C, so it's y minus 2 is
equal to 6 times x plus 2.

00:01:40.780 --> 00:01:43.120
And this is actually in
point-slope form, where the

00:01:43.120 --> 00:01:47.410
point x is equal to negative
2, y is equal to 2.

00:01:47.410 --> 00:01:50.910
So the point negative 2, 2, is
being represented here because

00:01:50.910 --> 00:01:52.870
you're subtracting the points.

00:01:52.870 --> 00:01:56.590
And the slope is 6, so we
already know that the slope is

00:01:56.590 --> 00:01:57.340
equal to 6.

00:01:57.340 --> 00:02:00.340
And sometimes people are
more comfortable with

00:02:00.340 --> 00:02:03.110
slope-intercept form, so let's
put it in slope-intercept form

00:02:03.110 --> 00:02:06.006
just to confirm that if we put
it in this form, the slope

00:02:06.006 --> 00:02:07.240
will still be equal to 6.

00:02:07.240 --> 00:02:11.320
So if we distribute the 6, we
get y minus 2 is equal to 6

00:02:11.320 --> 00:02:15.690
times x, 6x, plus
6 times 2 is 12.

00:02:15.690 --> 00:02:19.870
And if you add this 2 -- if you
add 2 to both sides of the

00:02:19.870 --> 00:02:24.410
equation, you get y-- because
these guys cancel out-- is

00:02:24.410 --> 00:02:27.330
equal to 6x plus 14.

00:02:27.330 --> 00:02:30.710
So you see, once again,
the slope is 6.

00:02:30.710 --> 00:02:35.700
So line A and line C have the
same the slope, so line A and

00:02:35.700 --> 00:02:37.100
line C are parallel.

00:02:37.100 --> 00:02:38.210
And they're different lines.

00:02:38.210 --> 00:02:39.460
If they had the same
y-intercept, then they would

00:02:39.460 --> 00:02:41.490
just be the same line.


Interpreting linear tables word problems example 1

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:04.750
Find the slope of the linear
function defined by the table.

00:00:04.750 --> 00:00:06.110
And they give us a table here.

00:00:06.110 --> 00:00:09.130
They define certain amount, I
guess these are shift lengths,

00:00:09.130 --> 00:00:12.720
and then they say how many hours
is a half a day, is a

00:00:12.720 --> 00:00:15.330
full day, is two days, is
a week, is a month.

00:00:15.330 --> 00:00:17.670
And then they tell us how much
money do we make in each of

00:00:17.670 --> 00:00:18.940
those time periods.

00:00:18.940 --> 00:00:22.800
If we work four hours, we make
$54, if we work eight hours,

00:00:22.800 --> 00:00:26.640
we make $108, so forth
and so on.

00:00:26.640 --> 00:00:27.880
And then they say what
does the slope

00:00:27.880 --> 00:00:29.900
represent in this situation?

00:00:29.900 --> 00:00:32.369
So we have to find the slope
and figure out what it

00:00:32.369 --> 00:00:33.780
represents.

00:00:33.780 --> 00:00:40.250
So just as a bit of review,
slope just equals the change

00:00:40.250 --> 00:00:43.080
in the dependent variable
divided by the change in the

00:00:43.080 --> 00:00:44.040
independent variable.

00:00:44.040 --> 00:00:46.330
So how much does a dependent
variable change for any amount

00:00:46.330 --> 00:00:48.310
of change of the independent
variable?

00:00:48.310 --> 00:00:50.240
In this situation, the dependent
variable is the

00:00:50.240 --> 00:00:52.660
amount of money you make because
it is dependent on how

00:00:52.660 --> 00:00:55.140
much time you work, this
is independent.

00:00:55.140 --> 00:00:58.110
So let's call the independent
variable x, the dependent

00:00:58.110 --> 00:00:59.860
variable y.

00:00:59.860 --> 00:01:07.600
So our slope in this situation
would be change in y divided

00:01:07.600 --> 00:01:09.170
by change in x.

00:01:13.510 --> 00:01:16.530
So how much does the amount of
money I make change when I

00:01:16.530 --> 00:01:19.560
work a certain number of hours,
when my hours worked

00:01:19.560 --> 00:01:21.450
change by a certain amount.

00:01:21.450 --> 00:01:23.640
So let's just take some
data points here.

00:01:23.640 --> 00:01:25.780
We could take really any of
these data points, I'll take

00:01:25.780 --> 00:01:27.740
some of the smaller numbers.

00:01:27.740 --> 00:01:31.750
So let's say if when I go from
four to eight hours, so my

00:01:31.750 --> 00:01:33.200
change in x is going
to be what?

00:01:33.200 --> 00:01:36.330
If I go from four to eight,
might change in x is going to

00:01:36.330 --> 00:01:39.660
be eight minus four,
four hours, right?

00:01:39.660 --> 00:01:41.280
So this is going to
be my change in x.

00:01:41.280 --> 00:01:43.070
I'm just picking these two
points, I could have picked

00:01:43.070 --> 00:01:45.490
four and forty if I wanted, but
the math would become more

00:01:45.490 --> 00:01:46.460
complicated.

00:01:46.460 --> 00:01:50.620
But how much does the amount of
money earn change if I go

00:01:50.620 --> 00:01:52.600
from four hours to
eight hours?

00:01:52.600 --> 00:01:57.110
Well, I go from $54 to $108,
so the difference in the

00:01:57.110 --> 00:02:03.730
amount of money I make
is $108 minus $54.

00:02:03.730 --> 00:02:07.380
So what is my change in
my dependent variable?

00:02:07.380 --> 00:02:11.630
Well, that's going to be $108
minus $54, that's just $54.

00:02:11.630 --> 00:02:13.130
And then what was the
change in the

00:02:13.130 --> 00:02:14.340
amount of hours I worked?

00:02:14.340 --> 00:02:17.300
Well, the change in the hours
I worked was four hours.

00:02:17.300 --> 00:02:20.940
So, if I work four more hours,
I make 54 more dollars.

00:02:20.940 --> 00:02:22.840
Let me put a little
equal sign there.

00:02:22.840 --> 00:02:27.860
So what is 54 divided by four?

00:02:27.860 --> 00:02:31.780
So four goes into 54-- looks
like there's going to be

00:02:31.780 --> 00:02:34.950
decimal here-- four goes
into five one time, one

00:02:34.950 --> 00:02:36.560
times four is four.

00:02:36.560 --> 00:02:40.950
Subtract, you get five minus
four is one, bring down this

00:02:40.950 --> 00:02:42.710
four you get 14.

00:02:42.710 --> 00:02:49.220
Four goes into 14 three times,
three times four is 12.

00:02:49.220 --> 00:02:54.880
Fourteen minus 12 is two, bring
down a 0 right here,

00:02:54.880 --> 00:02:58.460
four goes into 20 five times.

00:02:58.460 --> 00:02:59.900
And of course you have this
decimal right here.

00:02:59.900 --> 00:03:01.890
Five times four is 20.

00:03:01.890 --> 00:03:04.170
Subtract, no remainder.

00:03:04.170 --> 00:03:10.050
So this is equal to 13.5, but
since we're talking in terms

00:03:10.050 --> 00:03:15.530
of dollars, maybe say $13.50,
because that's

00:03:15.530 --> 00:03:16.630
our numerator, right?

00:03:16.630 --> 00:03:20.730
This is money earned, dollars
per hour, because that's our

00:03:20.730 --> 00:03:23.260
denominator, dollars per hour.

00:03:23.260 --> 00:03:25.040
So that essentially answers
our question.

00:03:25.040 --> 00:03:27.760
What does the slope represent
in this situation?

00:03:27.760 --> 00:03:35.050
It represents the hourly
wage for working at

00:03:35.050 --> 00:03:37.570
wherever this might be.

00:03:37.570 --> 00:03:39.160
Frankly, for this problem, you
didn't even have to take two

00:03:39.160 --> 00:03:39.690
data points.

00:03:39.690 --> 00:03:42.440
We could have said hey, if you
work four hours and make $54,

00:03:42.440 --> 00:03:45.530
54 divided by four is 13.50.

00:03:45.530 --> 00:03:49.080
Or we could have said hey, if
we work eight hours, we get

00:03:49.080 --> 00:03:52.870
$108, 108 divided by
eight is 13.50.

00:03:52.870 --> 00:03:54.670
So you didn't even have to take
two data points here, you

00:03:54.670 --> 00:03:56.950
could have just taken any of
these numbers divided by any

00:03:56.950 --> 00:03:57.860
of these numbers.

00:03:57.860 --> 00:03:59.790
But hopefully we also
learned a little bit

00:03:59.790 --> 00:04:02.260
about what slope is.


Equation of a line 2

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.030
We are asked to find
the equation that

00:00:03.030 --> 00:00:04.870
represents this line.

00:00:04.870 --> 00:00:09.020
And the easiest type of equation
to find out based on

00:00:09.020 --> 00:00:11.510
just being given a line
is an equation in the

00:00:11.510 --> 00:00:17.280
slope-intercept form, which is y
is equal to mx plus b, where

00:00:17.280 --> 00:00:23.940
m is the slope and b is the
y-intercept, the y-value of

00:00:23.940 --> 00:00:27.240
the coordinate where our graph
intersects the y-axis.

00:00:27.240 --> 00:00:29.340
So the y-intercept is
very easy to find.

00:00:29.340 --> 00:00:32.600
We can actually figure it out
just by inspecting our graph.

00:00:32.600 --> 00:00:34.300
We intersect the y-axis.

00:00:34.300 --> 00:00:35.840
That's the y-axis right there.

00:00:35.840 --> 00:00:37.040
It wasn't labeled.

00:00:37.040 --> 00:00:40.130
Right there, at the point
x is equal to 0.

00:00:40.130 --> 00:00:43.370
x is always going to be equal
to 0 at the y-axis.

00:00:43.370 --> 00:00:45.750
So x is equal to 0,
y is equal to 3.

00:00:45.750 --> 00:00:48.500
So our y-intercept is 3.

00:00:48.500 --> 00:00:49.640
And it makes complete sense.

00:00:49.640 --> 00:00:54.990
If x is equal to 0, y is equal
to 3, so b is equal to 3.

00:00:54.990 --> 00:00:57.680
Now the next thing we need to do
is figure out the slope of

00:00:57.680 --> 00:01:01.340
this line, figure out what m
is going to be equal to.

00:01:01.340 --> 00:01:06.870
And remember, slope, which is
the same thing as our m or

00:01:06.870 --> 00:01:11.320
what m represents, that is
our change in y for a

00:01:11.320 --> 00:01:13.390
given change in x.

00:01:13.390 --> 00:01:16.420
So let's go from this point to
some other point where we can

00:01:16.420 --> 00:01:18.290
definitely figure out its
coordinates, so that looks

00:01:18.290 --> 00:01:20.910
like a nice integer coordinate
right there.

00:01:20.910 --> 00:01:23.510
And let's figure out what our
change in y and our change in

00:01:23.510 --> 00:01:25.860
x was to get there.

00:01:25.860 --> 00:01:28.880
So our change in x, we're
starting here at x is equal to

00:01:28.880 --> 00:01:33.150
0, we're moving in the positive
x direction by five.

00:01:33.150 --> 00:01:37.160
So our change in x
is equal to 5.

00:01:37.160 --> 00:01:39.710
One, two, three, four, five.

00:01:39.710 --> 00:01:43.600
We went from x is equal to
0 to x is equal to 5.

00:01:43.600 --> 00:01:46.610
So our change in x
is equal to 5.

00:01:46.610 --> 00:01:47.990
That's our change in x.

00:01:47.990 --> 00:01:50.030
Now, what's our change in y?

00:01:50.030 --> 00:01:55.830
Well, we have to go down two, so
our change in y-- when our

00:01:55.830 --> 00:02:00.660
change in x is positive 5, our
change in y is negative 2.

00:02:03.200 --> 00:02:04.870
We're going down.

00:02:04.870 --> 00:02:09.690
We're decreasing in y, so our
change in y is negative 2.

00:02:09.690 --> 00:02:13.280
So our slope is negative 2/5.

00:02:13.280 --> 00:02:14.700
So we figured out the slope.

00:02:14.700 --> 00:02:16.010
We figured out the
y-intercept.

00:02:16.010 --> 00:02:19.710
We can now write the equation
that represents this line.

00:02:19.710 --> 00:02:26.130
It is y is equal to, this
character,, negative 2/5 times

00:02:26.130 --> 00:02:31.030
x, that's m right there,
plus b. b is 3.

00:02:33.600 --> 00:02:36.150
And we are done!


Writing equations of perpendicular lines

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.620
We're asked what is the
equation of line B?

00:00:03.620 --> 00:00:06.750
And they tell us that line A has
an equation y is equal to

00:00:06.750 --> 00:00:08.590
2x plus 11.

00:00:08.590 --> 00:00:10.770
And they say that the
line B contains the

00:00:10.770 --> 00:00:13.310
point 6, negative 7.

00:00:13.310 --> 00:00:18.520
And they tell us lines A and B
are perpendicular, so that

00:00:18.520 --> 00:00:25.680
means that slope of
B must be negative

00:00:25.680 --> 00:00:29.200
inverse of slope of A.

00:00:32.150 --> 00:00:34.490
So what we'll do is figure out
the slope of A, then take the

00:00:34.490 --> 00:00:35.690
negative inverse of it.

00:00:35.690 --> 00:00:38.000
Then we'll know the slope of B,
then we can use this point

00:00:38.000 --> 00:00:40.210
right here to fill in the
gaps and figure out B's

00:00:40.210 --> 00:00:41.560
y-intercept.

00:00:41.560 --> 00:00:42.590
So what's the slope of A?

00:00:42.590 --> 00:00:45.020
This is already in
slope-intercept form.

00:00:45.020 --> 00:00:49.570
The slope of A is right there,
it's the 2, mx plus b.

00:00:49.570 --> 00:00:52.540
So the slope here
is equal to 2.

00:00:52.540 --> 00:00:54.590
So the slope of A is 2.

00:00:54.590 --> 00:00:56.860
What is the slope of B?

00:00:56.860 --> 00:01:01.850
So what is B's slope going
to have to be?

00:01:01.850 --> 00:01:03.740
Well, it's perpendicular to
A, so it's going to be the

00:01:03.740 --> 00:01:05.420
negative inverse of this.

00:01:05.420 --> 00:01:08.040
The inverse of two is 1/2.

00:01:08.040 --> 00:01:10.700
The negative inverse of
that is negative 1/2.

00:01:10.700 --> 00:01:13.300
So B's slope is negative 1/2.

00:01:13.300 --> 00:01:17.520
So we know that B's equation
has to be y is equal to its

00:01:17.520 --> 00:01:23.290
slope, m times x plus
some y-intercept.

00:01:23.290 --> 00:01:27.720
We still don't know what the
y-intercept of B is, but we

00:01:27.720 --> 00:01:30.400
can use this information
to figure it out.

00:01:30.400 --> 00:01:37.210
We know that y is equal
to negative 7 when

00:01:37.210 --> 00:01:40.480
x is equal to 6.

00:01:40.480 --> 00:01:48.420
Negative 1/2 times
6 plus b, right?

00:01:48.420 --> 00:01:52.730
I just know that this is on the
point, so this point must

00:01:52.730 --> 00:01:55.700
satisfy the equation
of line B.

00:01:55.700 --> 00:02:00.420
So let's work out what b must
be-- or what b, the

00:02:00.420 --> 00:02:03.380
y-intercept, this is a lowercase
b, not the line B.

00:02:03.380 --> 00:02:07.320
So we have negative 7 is equal
to-- what's negative

00:02:07.320 --> 00:02:08.650
1/2 half times 6?

00:02:08.650 --> 00:02:10.840
That's not a b there,
that's a 6.

00:02:10.840 --> 00:02:12.500
What's negative 1/2 times 6?

00:02:12.500 --> 00:02:16.050
It's negative 3, is equal
to negative 3 plus our

00:02:16.050 --> 00:02:17.380
y-intercept.

00:02:17.380 --> 00:02:21.880
Let's add 3 to both sides of
this equation, so if we add 3

00:02:21.880 --> 00:02:25.150
to both sides-- I just want to
get rid of this 3 right here--

00:02:25.150 --> 00:02:25.800
what do we get?

00:02:25.800 --> 00:02:30.310
The left-hand side, negative
7 plus 3 is negative 4, and

00:02:30.310 --> 00:02:33.720
that's going to be equal to--
these guys cancel out-- that's

00:02:33.720 --> 00:02:36.170
equal to b, our y-intercept.

00:02:36.170 --> 00:02:39.900
So this right here
is a negative 4.

00:02:39.900 --> 00:02:45.980
So the equation of line B is y
is equal to-- its slope is

00:02:45.980 --> 00:02:49.160
a negative inverse of this
character-- so negative 1/2,

00:02:49.160 --> 00:02:50.770
negative 1/2 x.

00:02:50.770 --> 00:02:56.590
And its y-intercept we just
figured out is negative 4.

00:02:56.590 --> 00:03:00.160
And we are done.


Perpendicular lines from equation

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.460
We are asked which of these
lines are perpendicular.

00:00:03.460 --> 00:00:06.160
And it has to be perpendicular
to one of the other lines, you

00:00:06.160 --> 00:00:08.680
can't be just perpendicular
by yourself.

00:00:08.680 --> 00:00:11.970
And perpendicular line, just
so you have a visualization

00:00:11.970 --> 00:00:15.190
for what for perpendicular lines
look like, two lines are

00:00:15.190 --> 00:00:17.930
perpendicular if they intersect
at right angles.

00:00:17.930 --> 00:00:22.630
So if this is one line right
there, a perpendicular line

00:00:22.630 --> 00:00:23.850
will look like this.

00:00:23.850 --> 00:00:27.640
A perpendicular line will
intersect it, but it won't

00:00:27.640 --> 00:00:29.110
just be any intersection,
it will

00:00:29.110 --> 00:00:30.710
intersect at right angles.

00:00:33.900 --> 00:00:36.730
So these two lines are
perpendicular.

00:00:36.730 --> 00:00:39.570
Now, if two lines are
perpendicular, if the slope of

00:00:39.570 --> 00:00:44.330
this orange line is m-- so let's
say its equation is y is

00:00:44.330 --> 00:00:49.440
equal to mx plus, let's say
it's b 1, so it's some

00:00:49.440 --> 00:00:53.000
y-intercept-- then the equation
of this yellow line,

00:00:53.000 --> 00:00:56.360
its slope is going to be the
negative inverse of this guy.

00:00:56.360 --> 00:00:59.690
This guy right here is going to
be y is equal to negative 1

00:00:59.690 --> 00:01:03.950
over mx plus some other
y-intercept.

00:01:03.950 --> 00:01:06.230
Or another way to think about
it is if two lines are

00:01:06.230 --> 00:01:09.550
perpendicular, the product of
their slopes is going to be

00:01:09.550 --> 00:01:10.760
negative 1.

00:01:10.760 --> 00:01:14.440
And so you could write that
there. m times negative 1 over

00:01:14.440 --> 00:01:19.570
m, that's going to be-- these
two guys are going to cancel

00:01:19.570 --> 00:01:23.380
out-- that's going to be
equal to negative 1.

00:01:23.380 --> 00:01:26.200
So let's figure out the slopes
of each of these lines and

00:01:26.200 --> 00:01:29.090
figure out if any of them are
the negative inverse of any of

00:01:29.090 --> 00:01:31.060
the other ones.

00:01:31.060 --> 00:01:33.660
So line A, the slope is pretty
easy to figure out, it's

00:01:33.660 --> 00:01:38.850
already in slope-intercept
form, its slope is 3.

00:01:38.850 --> 00:01:42.070
So line A has a slope of 3.

00:01:42.070 --> 00:01:44.970
Line B, it's in standard form,
not too hard to put it in

00:01:44.970 --> 00:01:47.920
slope-intercept form, so
let's try to do it.

00:01:47.920 --> 00:01:49.950
So let's do line B over here.

00:01:49.950 --> 00:01:55.640
Line B, we have x plus 3y
is equal to negative 21.

00:01:55.640 --> 00:01:58.630
Let's subtract x from both sides
so that it ends up on

00:01:58.630 --> 00:02:00.500
the right-hand side.

00:02:00.500 --> 00:02:07.000
So we end up with 3y is equal
to negative x minus 21.

00:02:07.000 --> 00:02:12.070
And now let's divide both sides
of this equation by 3

00:02:12.070 --> 00:02:18.850
and we get y is equal to
negative 1/3 x minus 7.

00:02:18.850 --> 00:02:23.050
So this character's slope
is negative 1/3.

00:02:23.050 --> 00:02:25.660
So here m is equal
to negative 1/3.

00:02:25.660 --> 00:02:27.080
So we already see they
are the negative

00:02:27.080 --> 00:02:27.910
inverse of each other.

00:02:27.910 --> 00:02:30.970
You take the inverse of 3, it's
1/3, and then it's the

00:02:30.970 --> 00:02:31.820
negative of that.

00:02:31.820 --> 00:02:35.870
Or you take the inverse of
negative 1/3, it's negative 3,

00:02:35.870 --> 00:02:37.730
and then this is the
negative of that.

00:02:37.730 --> 00:02:41.820
So these two lines are
definitely perpendicular.

00:02:45.260 --> 00:02:47.320
Let's see the third
line over here.

00:02:47.320 --> 00:02:54.160
So line C is 3x plus
y is equal to 10.

00:02:54.160 --> 00:03:01.960
If we subtract 3x from both
sides, we get y is equal to

00:03:01.960 --> 00:03:04.850
negative 3x plus 10.

00:03:04.850 --> 00:03:07.480
So our slope in this
case is negative 3.

00:03:10.490 --> 00:03:14.020
Now this guy's the negative of
that guy, this guy's slope is

00:03:14.020 --> 00:03:16.560
a negative, but not the negative
inverse, so it's not

00:03:16.560 --> 00:03:17.600
perpendicular.

00:03:17.600 --> 00:03:20.370
And this guy is the inverse of
that guy but not the negative

00:03:20.370 --> 00:03:23.280
inverse, so this guy is not
perpendicular to either of the

00:03:23.280 --> 00:03:25.390
other two, but line
A and line B are

00:03:25.390 --> 00:03:28.170
perpendicular to each other.


Parallel lines from equation (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:03.480
We have three lines and we have
to figure out which of

00:00:03.480 --> 00:00:05.210
the three are parallel.

00:00:05.210 --> 00:00:08.250
So line A-- and it can't be
parallel on its own, it has to

00:00:08.250 --> 00:00:10.510
be parallel to another
of the three lines.

00:00:10.510 --> 00:00:13.740
So the equation for line
A is y is equal

00:00:13.740 --> 00:00:19.080
to 3/4 x minus four.

00:00:19.080 --> 00:00:29.940
Line B is 4y minus 20 is
equal to negative 3x.

00:00:29.940 --> 00:00:38.580
And then line C is negative
3x plus 4y is equal to 40.

00:00:38.580 --> 00:00:41.090
So to figure out if any of these
lines are parallel to

00:00:41.090 --> 00:00:42.450
any of the other lines,
we just have to

00:00:42.450 --> 00:00:45.210
compare their slopes.

00:00:45.210 --> 00:00:47.420
If any two of these lines have
the same slope and they're

00:00:47.420 --> 00:00:50.110
different lines, they have
different y-intercepts, then

00:00:50.110 --> 00:00:51.910
they're going to be parallel.

00:00:51.910 --> 00:00:53.970
Now line A, it's very easy
to figure out its slope.

00:00:53.970 --> 00:00:55.980
It's already in slope-intercept
form.

00:00:55.980 --> 00:01:02.910
This is mx plus b, the slope
is 3/4 and the y-intercept,

00:01:02.910 --> 00:01:04.880
which isn't as relevant when
you're figuring out parallel

00:01:04.880 --> 00:01:07.190
lines, is negative 4.

00:01:07.190 --> 00:01:10.180
So let's see what the other
character's slopes are.

00:01:10.180 --> 00:01:12.180
This isn't in any kind
of standard form.

00:01:12.180 --> 00:01:14.100
It's not a standard form,
slope-intercept, or

00:01:14.100 --> 00:01:16.380
point-slope form, but let's
see what the slope

00:01:16.380 --> 00:01:17.300
of this line is.

00:01:17.300 --> 00:01:19.900
So to get it into
slope-intercept form, which is

00:01:19.900 --> 00:01:23.050
really the easiest one to pick
out the slope from, let's add

00:01:23.050 --> 00:01:24.535
20 to both sides of
this equation.

00:01:28.800 --> 00:01:31.240
The left-hand side, those cancel
out, that was the whole

00:01:31.240 --> 00:01:37.650
point, you get 4y is equal
to negative 3x plus 20.

00:01:37.650 --> 00:01:39.670
And now we can divide
everything by 4.

00:01:44.770 --> 00:01:53.520
We are left with y is equal
to negative 3/4 x plus 5.

00:01:53.520 --> 00:01:56.130
So in this case, y-intercept
is 5, but most importantly,

00:01:56.130 --> 00:02:01.330
the slope is negative 3/4, so
it's different than this guy.

00:02:01.330 --> 00:02:04.090
This is negative 3/4, this is
positive 3/4, so these two

00:02:04.090 --> 00:02:06.360
guys definitely aren't
parallel.

00:02:06.360 --> 00:02:09.310
Let's move on of this guy
in standard form.

00:02:09.310 --> 00:02:11.250
So let's get the x term
on the other side.

00:02:11.250 --> 00:02:13.730
So let's add 3x to both sides
of this equation.

00:02:17.590 --> 00:02:19.130
Left-hand side, these
cancel out.

00:02:19.130 --> 00:02:24.800
We're just left with 4y is equal
to 3x plus 40, or 40

00:02:24.800 --> 00:02:26.620
plus 3x, either way.

00:02:26.620 --> 00:02:29.820
Now we can divide both sides
by 4 you have to divide

00:02:29.820 --> 00:02:31.980
every term by 4.

00:02:31.980 --> 00:02:34.450
The left-hand side, you're
left with y.

00:02:34.450 --> 00:02:39.720
The right-hand side, you
have 3/4 x plus 10.

00:02:39.720 --> 00:02:44.600
So here, our slope is 3/4 and
our y-intercept, if we care

00:02:44.600 --> 00:02:46.240
about it, is 10.

00:02:46.240 --> 00:02:50.590
So this line and this line have
the exact same slope,

00:02:50.590 --> 00:02:52.560
3/4, and they're different
lines because their

00:02:52.560 --> 00:02:54.280
y-intercept is different.

00:02:54.280 --> 00:02:58.480
So we know that A and C are
parallel lines and B is not

00:02:58.480 --> 00:03:02.220
parallel to either one
of the other two.


Parallel lines from equation

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:04.340
We are asked which of these
lines are parallel.

00:00:04.340 --> 00:00:07.490
So parallel lines are lines that
have the same slope, and

00:00:07.490 --> 00:00:11.000
they're different lines, so they
never, ever intersect.

00:00:11.000 --> 00:00:13.540
So we need to look for different
lines that have the

00:00:13.540 --> 00:00:15.030
exact same slope.

00:00:15.030 --> 00:00:20.230
And lucky for us, all of these
lines are in y equals mx plus

00:00:20.230 --> 00:00:23.990
b or slope-intercept form, so
you can really just look at

00:00:23.990 --> 00:00:26.330
these lines and figure
out their slope.

00:00:26.330 --> 00:00:31.870
The slope for line A,
m is equal to 2.

00:00:31.870 --> 00:00:34.100
We see it right over there.

00:00:34.100 --> 00:00:39.040
For line B, our slope is equal
to 3, so these two guys are

00:00:39.040 --> 00:00:39.890
not parallel.

00:00:39.890 --> 00:00:42.020
I'll graph it in a second
and you'll see that.

00:00:42.020 --> 00:00:48.360
And then finally, for line C--
I'll do it in purple-- the

00:00:48.360 --> 00:00:50.230
slope is 2.

00:00:50.230 --> 00:00:51.980
So m is equal to 2.

00:00:51.980 --> 00:00:54.690
I don't know if that purple
is too dark for you.

00:00:54.690 --> 00:00:58.040
So line C and line A have the
same slope, but they're

00:00:58.040 --> 00:01:00.530
different lines, they have
different y-intercepts, so

00:01:00.530 --> 00:01:01.960
they're going to be parallel.

00:01:01.960 --> 00:01:04.290
And to see that, let's actually
graph all of these

00:01:04.290 --> 00:01:05.300
characters.

00:01:05.300 --> 00:01:08.470
So line A, our y-intercept
is negative 6.

00:01:08.470 --> 00:01:14.080
So the point 0, 1,
2, 3, 4, 5, 6.

00:01:14.080 --> 00:01:15.680
And our slope is 2.

00:01:15.680 --> 00:01:18.730
So if we move 1 in the positive
x direction, we go up

00:01:18.730 --> 00:01:20.710
2 in the positive y direction.

00:01:20.710 --> 00:01:23.725
One in x, up 2 in y, if we
go to in x, we're going

00:01:23.725 --> 00:01:27.705
to go up 4 in y.

00:01:27.705 --> 00:01:32.030
And I can just do up 2, then
we're going to go 2, 4, and

00:01:32.030 --> 00:01:34.870
you're going to see it's all on
the same line, so line A is

00:01:34.870 --> 00:01:40.850
going to look something like--
do my best to draw it as

00:01:40.850 --> 00:01:42.180
straight as possible.

00:01:42.180 --> 00:01:44.910
Line A-- I can do a better
version than that-- line A is

00:01:44.910 --> 00:01:48.690
going to look like-- well,
that's about just as good as

00:01:48.690 --> 00:01:51.040
what I just drew--
that is line A.

00:01:51.040 --> 00:01:53.370
Now let's do line B.

00:01:53.370 --> 00:01:55.940
Line B, the y-intercept
is negative 6.

00:01:55.940 --> 00:01:56.650
0, negative 6.

00:01:56.650 --> 00:02:00.422
So it has the same y-intercept,
but its slope is

00:02:00.422 --> 00:02:05.320
3, so if x goes up by 1,
y will go up by 3.

00:02:05.320 --> 00:02:08.620
So x goes up by 1,
y goes up by 3.

00:02:08.620 --> 00:02:12.120
If x goes up by 2, y is
going to go up by 6.

00:02:12.120 --> 00:02:13.876
2, 4, 6.

00:02:18.990 --> 00:02:24.160
So this line is going to look
something like this.

00:02:24.160 --> 00:02:26.770
Trying my best to connect
the dots.

00:02:26.770 --> 00:02:30.550
It has a steeper slope, and you
see that when x increases,

00:02:30.550 --> 00:02:33.560
this blue line increases by
more in the y direction.

00:02:33.560 --> 00:02:35.940
So that is line B-- and notice,
they do intersect,

00:02:35.940 --> 00:02:38.780
there's definitely not
two parallel lines.

00:02:38.780 --> 00:02:41.560
And then finally, let's
look at line C.

00:02:41.560 --> 00:02:43.670
The y-intercept is 5.

00:02:43.670 --> 00:02:46.640
So 0, 1, 2, 3, 4, 5.

00:02:46.640 --> 00:02:48.690
The point 0, 5, its
y-intercept.

00:02:48.690 --> 00:02:50.370
And its slope is 2.

00:02:50.370 --> 00:02:53.240
So you increase by 1 in the x
direction, you're going to go

00:02:53.240 --> 00:02:55.640
up by 2 in the y direction.

00:02:55.640 --> 00:02:57.890
If you decrease by 1,
you're going to go

00:02:57.890 --> 00:03:00.560
down 2 in the y direction.

00:03:00.560 --> 00:03:03.390
If you increase by, well, you're
going to go to that

00:03:03.390 --> 00:03:05.100
point, you're going to have
a bunch of these points.

00:03:05.100 --> 00:03:07.840
And then if I were to graph the
line-- let me do it one

00:03:07.840 --> 00:03:09.810
more time-- if I were to
decrease by two, I'm going to

00:03:09.810 --> 00:03:11.930
have to go down by 4, right?

00:03:11.930 --> 00:03:14.010
Negative 4 over negative
2 still a slope of

00:03:14.010 --> 00:03:17.710
2, so 1, 2, 3, 4.

00:03:17.710 --> 00:03:20.780
And I can do that one more time,
get right over there.

00:03:20.780 --> 00:03:22.330
And then you'll see the line.

00:03:22.330 --> 00:03:28.120
The line will look like that,
it will look just like that.

00:03:28.120 --> 00:03:32.020
And notice that line C and
line A never intersect.

00:03:32.020 --> 00:03:36.560
They have the exact
same slope.

00:03:36.560 --> 00:03:40.000
Different y-intercepts, same
slope, so they're increasing

00:03:40.000 --> 00:03:42.780
at the exact same rate, but
they're never going to

00:03:42.780 --> 00:03:43.900
intersect each other.

00:03:43.900 --> 00:03:47.600
So line A and line
C are parallel.


Writing equations in all forms

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:06.250
A line passes through the points
negative 3, 6 and 6, 0.

00:00:06.250 --> 00:00:09.390
Find the equation of this line
in point slope form, slope

00:00:09.390 --> 00:00:11.860
intercept form, standard form.

00:00:11.860 --> 00:00:13.970
And the way to think about
these, these are just three

00:00:13.970 --> 00:00:16.030
different ways of writing
the same equation.

00:00:16.030 --> 00:00:18.220
So if you give me one of them,
we can manipulate it to get

00:00:18.220 --> 00:00:19.230
any of the other ones.

00:00:19.230 --> 00:00:23.890
But just so you know what these
are, point slope form,

00:00:23.890 --> 00:00:28.500
let's say the point x1, y1 are,
let's say that that is a

00:00:28.500 --> 00:00:30.160
point on the line.

00:00:30.160 --> 00:00:32.900
And when someone puts this
little subscript here, so if

00:00:32.900 --> 00:00:34.750
they just write an x, that means
we're talking about a

00:00:34.750 --> 00:00:36.690
variable that can take
on any value.

00:00:36.690 --> 00:00:39.770
If someone writes x with a
subscript 1 and a y with a

00:00:39.770 --> 00:00:45.150
subscript 1, that's like saying
a particular value x

00:00:45.150 --> 00:00:48.450
and a particular value of y,
or a particular coordinate.

00:00:48.450 --> 00:00:50.030
And you'll see that when
we do the example.

00:00:50.030 --> 00:00:52.480
But point slope form says
that, look, if I know a

00:00:52.480 --> 00:00:56.020
particular point, and if I know
the slope of the line,

00:00:56.020 --> 00:00:58.750
then putting that line in point
slope form would be y

00:00:58.750 --> 00:01:05.209
minus y1 is equal to
m times x minus x1.

00:01:05.209 --> 00:01:08.830
So, for example, and we'll do
that in this video, if the

00:01:08.830 --> 00:01:13.170
point negative 3 comma 6 is on
the line, then we'd say y

00:01:13.170 --> 00:01:19.510
minus 6 is equal to m times x
minus negative 3, so it'll end

00:01:19.510 --> 00:01:21.120
up becoming x plus 3.

00:01:21.120 --> 00:01:23.115
So this is a particular
x, and a particular y.

00:01:23.115 --> 00:01:25.130
It could be a negative
3 and 6.

00:01:25.130 --> 00:01:26.870
So that's point slope form.

00:01:26.870 --> 00:01:32.790
Slope intercept form is y is
equal to mx plus b, where once

00:01:32.790 --> 00:01:36.810
again m is the slope, b is the
y-intercept-- where does the

00:01:36.810 --> 00:01:40.180
line intersect the y-axis--
what value does y take

00:01:40.180 --> 00:01:41.680
on when x is 0?

00:01:41.680 --> 00:01:47.680
And then standard form is the
form ax plus by is equal to c,

00:01:47.680 --> 00:01:50.810
where these are just two
numbers, essentially.

00:01:50.810 --> 00:01:52.960
They really don't have
any interpretation

00:01:52.960 --> 00:01:54.430
directly on the graph.

00:01:54.430 --> 00:01:57.710
So let's do this, let's figure
out all of these forms. So the

00:01:57.710 --> 00:01:59.900
first thing we want to do
is figure out the slope.

00:01:59.900 --> 00:02:02.610
Once we figure out the slope,
then point slope form is

00:02:02.610 --> 00:02:05.820
actually very, very, very
straightforward to calculate.

00:02:05.820 --> 00:02:12.120
So, just to remind ourselves,
slope, which is equal to m,

00:02:12.120 --> 00:02:17.110
which is going to be equal to
the change in y over the

00:02:17.110 --> 00:02:18.480
change in x.

00:02:18.480 --> 00:02:19.940
Now what is the change in y?

00:02:19.940 --> 00:02:23.610
If we view this as our end
point, if we imagine that we

00:02:23.610 --> 00:02:26.190
are going from here to
that point, what is

00:02:26.190 --> 00:02:27.460
the change in y?

00:02:27.460 --> 00:02:30.860
Well, we have our end point,
which is 0, y ends up at the

00:02:30.860 --> 00:02:33.300
0, and y was at 6.

00:02:33.300 --> 00:02:38.650
So, our finishing y point is 0,
our starting y point is 6.

00:02:38.650 --> 00:02:41.870
What was our finishing x
point, or x-coordinate?

00:02:41.870 --> 00:02:44.810
Our finishing x-coordinate
was 6.

00:02:44.810 --> 00:02:46.690
Let me make this very clear, I
don't want to confuse you.

00:02:46.690 --> 00:02:53.720
So this 0, we have that 0, that
is that 0 right there.

00:02:53.720 --> 00:02:59.330
And then we have this 6, which
was our starting y point, that

00:02:59.330 --> 00:03:01.580
is that 6 right there.

00:03:01.580 --> 00:03:06.000
And then we want our finishing x
value-- that is that 6 right

00:03:06.000 --> 00:03:10.050
there, or that 6 right there--
and we want to subtract from

00:03:10.050 --> 00:03:14.080
that our starting x value.

00:03:14.080 --> 00:03:16.800
Well, our starting x value is
that right over there, that's

00:03:16.800 --> 00:03:19.910
that negative 3.

00:03:19.910 --> 00:03:21.880
And just to make sure we know
what we're doing, this

00:03:21.880 --> 00:03:25.440
negative 3 is that negative
3, right there.

00:03:25.440 --> 00:03:29.500
I'm just saying, if we go from
that point to that point, our

00:03:29.500 --> 00:03:31.640
y went down by 6, right?

00:03:31.640 --> 00:03:32.960
We went from 6 to 0.

00:03:32.960 --> 00:03:37.650
Our y went down by 6.

00:03:37.650 --> 00:03:40.420
So we get 0 minus
6 is negative 6.

00:03:40.420 --> 00:03:41.070
That makes sense.

00:03:41.070 --> 00:03:43.150
Y went down by 6.

00:03:43.150 --> 00:03:45.180
And, if we went from that point
to that point, what

00:03:45.180 --> 00:03:46.060
happened to x?

00:03:46.060 --> 00:03:49.560
We went from negative 3 to
6, it should go up by 9.

00:03:49.560 --> 00:03:54.870
And if you calculate this, take
your 6 minus negative 3,

00:03:54.870 --> 00:03:58.550
that's the same thing as
6 plus 3, that is 9.

00:03:58.550 --> 00:04:00.120
And what is negative 6/9?

00:04:00.120 --> 00:04:04.060
Well, if you simplify it,
it is negative 2/3.

00:04:04.060 --> 00:04:06.950
You divide the numerator and
the denominator by 3.

00:04:06.950 --> 00:04:09.860
So that is our slope,
negative 2/3.

00:04:09.860 --> 00:04:12.800
So we're pretty much ready
to use point slope form.

00:04:12.800 --> 00:04:15.760
We have a point, we could pick
one of these points, I'll just

00:04:15.760 --> 00:04:18.149
go with the negative 3, 6.

00:04:18.149 --> 00:04:19.269
And we have our slope.

00:04:19.269 --> 00:04:20.910
So let's put it in
point slope form.

00:04:29.050 --> 00:04:32.910
All we have to do is we say y
minus-- now we could have

00:04:32.910 --> 00:04:35.030
taken either of these points,
I'll take this one-- so y

00:04:35.030 --> 00:04:42.990
minus the y value over here, so
y minus 6 is equal to our

00:04:42.990 --> 00:04:51.480
slope, which is negative
2/3 times x minus our

00:04:51.480 --> 00:04:51.930
x-coordinate.

00:04:51.930 --> 00:04:55.490
Well, our x-coordinate, so x
minus our x-coordinate is

00:04:55.490 --> 00:05:00.370
negative 3, x minus negative
3, and we're done.

00:05:00.370 --> 00:05:01.570
We can simplify it
a little bit.

00:05:01.570 --> 00:05:08.650
This becomes y minus 6 is equal
to negative 2/3 times x.

00:05:08.650 --> 00:05:12.610
x minus negative 3 is the
same thing as x plus 3.

00:05:12.610 --> 00:05:15.760
This is our point slope form.

00:05:15.760 --> 00:05:19.270
Now, we can literally just
algebraically manipulate this

00:05:19.270 --> 00:05:23.500
guy right here to put it into
our slope intercept form.

00:05:23.500 --> 00:05:24.130
Let's do that.

00:05:24.130 --> 00:05:27.560
So let's do slope intercept
in orange.

00:05:27.560 --> 00:05:30.960
So we have slope intercept.

00:05:35.200 --> 00:05:37.040
So what can we do here
to simplify this?

00:05:37.040 --> 00:05:41.510
Well, we can multiply out the
negative 2/3, so you get y

00:05:41.510 --> 00:05:44.150
minus 6 is equal to-- I'm just
distributing the negative

00:05:44.150 --> 00:05:48.950
2/3-- so negative 2/3 times
x is negative 2/3 x.

00:05:48.950 --> 00:05:53.890
And then negative 2/3 times
3 is negative 2.

00:05:53.890 --> 00:05:56.150
And now to get it in slope
intercept form, we just have

00:05:56.150 --> 00:05:59.270
to add the 6 to both sides so
we get rid of it on the

00:05:59.270 --> 00:06:02.380
left-hand side, so let's
add 6 to both

00:06:02.380 --> 00:06:04.400
sides of this equation.

00:06:04.400 --> 00:06:06.010
Left-hand side of the equation,
we're just left with

00:06:06.010 --> 00:06:08.090
a y, these guys cancel out.

00:06:08.090 --> 00:06:12.830
You get a y is equal
to negative 2/3 x.

00:06:12.830 --> 00:06:15.810
Negative 2 plus 6 is plus 4.

00:06:15.810 --> 00:06:19.890
So there you have it, that is
our slope intercept form, mx

00:06:19.890 --> 00:06:22.590
plus b, that's our
y-intercept.

00:06:22.590 --> 00:06:24.480
Now the last thing we need
to do is get it into

00:06:24.480 --> 00:06:25.730
the standard form.

00:06:30.440 --> 00:06:32.840
So once again, we just have to
algebraically manipulate it so

00:06:32.840 --> 00:06:34.500
that the x's and the
y's are both on

00:06:34.500 --> 00:06:36.130
this side of the equation.

00:06:36.130 --> 00:06:40.140
So let's just add 2/3 x to both
sides of this equation.

00:06:40.140 --> 00:06:42.140
So I'll start it here.

00:06:42.140 --> 00:06:45.950
So we have y is equal to
negative 2/3 x plus 4, that's

00:06:45.950 --> 00:06:47.540
slope intercept form.

00:06:47.540 --> 00:06:53.790
Let's added 2/3 x, so
plus 2/3 x to both

00:06:53.790 --> 00:06:55.040
sides of this equation.

00:06:57.650 --> 00:07:00.060
I'm doing that so it I don't
have this 2/3 x on the

00:07:00.060 --> 00:07:02.350
right-hand side, this
negative 2/3 x.

00:07:02.350 --> 00:07:05.010
So the left-hand side of the
equation-- I scrunched it up a

00:07:05.010 --> 00:07:07.730
little bit, maybe more than I
should have-- the left-hand

00:07:07.730 --> 00:07:09.890
side of this equation is what?

00:07:09.890 --> 00:07:16.390
It is 2/3 x, because 2 over
3x, plus this y, that's my

00:07:16.390 --> 00:07:20.220
left-hand side, is equal to--
these guys cancel out-- is

00:07:20.220 --> 00:07:22.650
equal to 4.

00:07:22.650 --> 00:07:25.580
So this, by itself, we are in
standard form, this is the

00:07:25.580 --> 00:07:26.790
standard form of the equation.

00:07:26.790 --> 00:07:30.030
If we want it to look, make it
look extra clean and have no

00:07:30.030 --> 00:07:32.340
fractions here, we could
multiply both sides of this

00:07:32.340 --> 00:07:33.860
equation by 3.

00:07:33.860 --> 00:07:36.400
If we do that, what do we get?

00:07:36.400 --> 00:07:40.220
2/3 x times 3 is just 2x.

00:07:40.220 --> 00:07:43.440
y times 3 is 3y.

00:07:43.440 --> 00:07:45.700
And then 4 times 3 is 12.

00:07:45.700 --> 00:07:47.620
These are the same equations,
I just multiplied

00:07:47.620 --> 00:07:48.760
every term by 3.

00:07:48.760 --> 00:07:50.190
If you do it to the left-hand
side, you can do to the

00:07:50.190 --> 00:07:52.680
right-hand side-- or you have to
do to the right-hand side--

00:07:52.680 --> 00:07:56.160
and we are in standard form.


Graph from slope-intercept equation example

WEBVTT
Kind: captions
Language: en

00:00:00.390 --> 00:00:08.870
We are asked to graph y is
equal to 1/3x minus 2.

00:00:08.870 --> 00:00:11.740
Now, whenever you see an
equation in this form, this is

00:00:11.740 --> 00:00:14.680
called slope-intercept form.

00:00:14.680 --> 00:00:21.290
And the general way of writing
it is y is equal to mx plus b,

00:00:21.290 --> 00:00:23.140
where m is the slope.

00:00:23.140 --> 00:00:26.120
And here in this case, m is
equal to 1/3-- so let me write

00:00:26.120 --> 00:00:31.300
that down-- m is equal to 1/3,
and b is the y-intercept.

00:00:31.300 --> 00:00:35.310
So in this case, b is
equal to negative 2.

00:00:35.310 --> 00:00:38.090
And you know that b is the
y-intercept, because we know

00:00:38.090 --> 00:00:42.160
that the y-intercept occurs
when x is equal to 0.

00:00:42.160 --> 00:00:45.230
So if x is equal to 0 in either
of these situations,

00:00:45.230 --> 00:00:49.700
this term just becomes 0 and
y will be equal to b.

00:00:49.700 --> 00:00:53.380
So that's what we mean by
b is the y-intercept.

00:00:53.380 --> 00:00:55.690
So whenever you look at an
equation in this form, it's

00:00:55.690 --> 00:00:57.170
actually fairly straightforward

00:00:57.170 --> 00:00:58.690
to graph this line.

00:00:58.690 --> 00:01:00.220
b is the y-intercept.

00:01:00.220 --> 00:01:04.090
In this case it is negative 2,
so that means that this line

00:01:04.090 --> 00:01:09.810
must intersect the y-axis at y
is equal to negative 2, so

00:01:09.810 --> 00:01:11.760
it's this point right here.

00:01:11.760 --> 00:01:15.910
Negative 1, negative 2, this
is the point 0, negative 2.

00:01:15.910 --> 00:01:17.920
If you don't believe me, there's
nothing magical about

00:01:17.920 --> 00:01:22.410
this, try evaluating or
try solving for y when

00:01:22.410 --> 00:01:23.250
x is equal to 0.

00:01:23.250 --> 00:01:26.440
When x is equal to 0, this term
cancels out and you're

00:01:26.440 --> 00:01:28.900
just left with y is equal
to negative 2.

00:01:28.900 --> 00:01:31.510
So that's the y-intercept
right there.

00:01:31.510 --> 00:01:35.830
Now, this 1/3 tells us the
slope of the line.

00:01:35.830 --> 00:01:40.430
How much do we change in
y for any change in x?

00:01:40.430 --> 00:01:43.760
So this tells us that
1/3, so that right

00:01:43.760 --> 00:01:46.230
there, is the slope.

00:01:46.230 --> 00:01:54.380
So it tells us that 1/3 is equal
to the change in y over

00:01:54.380 --> 00:01:57.840
the change in x.

00:01:57.840 --> 00:02:01.760
Or another way to think about
it, if x changes by 3, then y

00:02:01.760 --> 00:02:03.890
would change by 1.

00:02:03.890 --> 00:02:05.660
So let me graph that.

00:02:05.660 --> 00:02:08.930
So we know that this point is
on the graph, that's the

00:02:08.930 --> 00:02:09.960
y-intercept.

00:02:09.960 --> 00:02:13.530
The slope tells us that if x
changes by 3-- so let me go 3

00:02:13.530 --> 00:02:21.180
three to the right, 1, 2, 3--
that y will change by 1.

00:02:21.180 --> 00:02:23.820
So this must also be a
point on the graph.

00:02:23.820 --> 00:02:24.720
And we could keep doing that.

00:02:24.720 --> 00:02:27.960
If x changes by 3,
y changes by 1.

00:02:27.960 --> 00:02:34.030
If x goes down by 3, y
will go down by 1.

00:02:34.030 --> 00:02:37.710
If x goes down by 6, y
will go down by 2.

00:02:37.710 --> 00:02:43.600
It's that same ratio, so
1, 2, 3, 4, 5, 6, 1, 2.

00:02:43.600 --> 00:02:45.810
And you can see all of these
points are on the line, and

00:02:45.810 --> 00:02:49.080
the line is the graph of
this equation up here.

00:02:49.080 --> 00:02:51.690
So let me graph it.

00:02:51.690 --> 00:02:59.220
So it'll look something
like that.

00:02:59.220 --> 00:03:00.810
And you're done.


x- and y-intercepts 2

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.680
We're told to find the x- and
y-intercepts for the graph of

00:00:04.680 --> 00:00:09.030
this equation: 2 y plus
1/3x is equal to 12.

00:00:09.030 --> 00:00:14.400
And just as a bit of a
refresher, the x-intercept is

00:00:14.400 --> 00:00:17.240
the point on the graph that
intersects the x-axis.

00:00:17.240 --> 00:00:21.110
So we're not above or below
the x-axis, so our y value

00:00:21.110 --> 00:00:22.910
must be equal to 0.

00:00:22.910 --> 00:00:29.330
And by the exact same argument,
the y-intercept

00:00:29.330 --> 00:00:32.290
occurs when we're not to the
right or the left of the

00:00:32.290 --> 00:00:36.070
y-axis, so that's when
x is equal to 0.

00:00:36.070 --> 00:00:38.580
So let's set each of these
values to 0 and then solve for

00:00:38.580 --> 00:00:41.340
what the other one has
to be at that point.

00:00:41.340 --> 00:00:46.080
So for the x-intercept,
when y is equal to

00:00:46.080 --> 00:00:47.400
0, let's solve this.

00:00:47.400 --> 00:00:52.720
So we get 2 times 0, plus
1/3x is equal to 12.

00:00:52.720 --> 00:00:55.530
I just set y is equal to
0 right there, right?

00:00:55.530 --> 00:00:56.800
I put 0 for y.

00:00:56.800 --> 00:00:59.870
Well, anything times 0 is just
0, so you're just left with

00:00:59.870 --> 00:01:03.760
1/3x is equal to 12.

00:01:03.760 --> 00:01:08.120
To solve for x, you can think of
it as either dividing both

00:01:08.120 --> 00:01:11.410
sides by 1/3, or we can multiply
both sides by the

00:01:11.410 --> 00:01:13.020
reciprocal of 1/3.

00:01:13.020 --> 00:01:15.720
And the reciprocal of 1/3 is 3,
or you can even think of it

00:01:15.720 --> 00:01:17.185
as 3 over 1.

00:01:17.185 --> 00:01:19.890
So times 3 over 1.

00:01:19.890 --> 00:01:24.250
And so we're left with 3 times
1/3, that just cancels out, so

00:01:24.250 --> 00:01:28.690
you're left with x is equal
to 12 times 3, or

00:01:28.690 --> 00:01:30.550
x is equal to 36.

00:01:30.550 --> 00:01:33.340
So when y is equal
to 0, x is 36.

00:01:33.340 --> 00:01:40.520
So the point 36 comma 0 is on
the graph of this equation.

00:01:40.520 --> 00:01:43.360
And this is also the
x-intercept.

00:01:43.360 --> 00:01:46.640
Now, let's do the same thing
for the y-intercept.

00:01:46.640 --> 00:01:55.060
So let's set x equal 0, so you
get 2y plus 1/3, times 0 is

00:01:55.060 --> 00:01:56.320
equal to 12.

00:01:56.320 --> 00:01:59.600
Once again, anything
times 0 is 0.

00:01:59.600 --> 00:02:04.470
So that's 0, and you're just
left with 2y is equal to 12.

00:02:04.470 --> 00:02:09.009
Divide both sides by 2 to solve
for y, and you're left

00:02:09.009 --> 00:02:13.640
with y is equal to
12 over 2, is 6.

00:02:13.640 --> 00:02:17.636
So the y-intercept is when
x is equal to 0 and

00:02:17.636 --> 00:02:20.650
y is equal to 6.

00:02:20.650 --> 00:02:22.770
So let's plot these
two points.

00:02:22.770 --> 00:02:24.780
I'll just do a little hand-drawn
graph, and make it

00:02:24.780 --> 00:02:28.180
clear what the x- and the
y-intercepts are.

00:02:28.180 --> 00:02:32.150
So let me draw-- that's my
vertical axis, and that is my

00:02:32.150 --> 00:02:36.630
horizontal axis-- and we have
the point 36 comma 0.

00:02:36.630 --> 00:02:39.660
So this is the origin right
here, that's the x-axis,

00:02:39.660 --> 00:02:40.670
that's the y-axis.

00:02:40.670 --> 00:02:43.750
The point 36 comma 0 might
be all the way over here.

00:02:43.750 --> 00:02:47.270
So that's the point
36 comma 0.

00:02:47.270 --> 00:02:54.460
And if that's 36, then the
point 0, 6 might be right

00:02:54.460 --> 00:02:55.210
about there.

00:02:55.210 --> 00:02:57.940
So that's the point 0, 6.

00:02:57.940 --> 00:03:01.390
And the line will look
something like this.

00:03:04.810 --> 00:03:06.810
I'm trying my best to draw
a straight line.

00:03:06.810 --> 00:03:11.360
And notice where the line
intercepted or intersected the

00:03:11.360 --> 00:03:15.330
y-axis, that's the y-intercept,
x is 0, because

00:03:15.330 --> 00:03:17.190
we're not to the right
or the left of it.

00:03:17.190 --> 00:03:21.750
Where the line intersected the
x-axis, y is 0, because we're

00:03:21.750 --> 00:03:24.300
not above or below it.


Finding intercepts from an equation

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:08.960
We have the equation negative 5x
plus 4y is equal to 20, and

00:00:08.960 --> 00:00:11.820
we're told to find the
intercepts of this equation.

00:00:11.820 --> 00:00:18.500
So we have to find the
intercepts and then use the

00:00:18.500 --> 00:00:22.310
intercepts to graph this line
on the coordinate plane.

00:00:22.310 --> 00:00:25.850
So then graph the line.

00:00:25.850 --> 00:00:28.120
So whenever someone talks about
intercepts, they're

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talking about where you're
intersecting

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the x and the y-axes.

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So let me label my axes here, so
this is the x-axis and that

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is the y-axis there.

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And when I intersect the x-axis,
what's going on?

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What is my y value when
I'm at the x-axis?

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Well, my y value is 0, I'm not
above or below the x-axis.

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Let me write this down.

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The x-intercept is when y
is equal to 0, right?

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And then by that same
argument, what's the

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y-intercept?

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Well, if I'm somewhere along the
y-axis, what's my x value?

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Well, I'm not to the right or
the left, so my x value has to

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be 0, so the y-intercept occurs
when x is equal to 0.

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So to figure out the intercepts,
let's set y equal

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to 0 in this equation and solve
for x, and then let's

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set x is equal to 0 and
then solve for y.

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So when y is equal to 0, what
does this equation become?

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I'll do it in orange.

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You get negative 5x plus 4y.

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Well we're saying y is 0, so
4 times 0 is equal to 20.

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4 times 0 is just 0, so we
can just not write that.

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So let me just rewrite it.

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So we have negative
5x is equal to 20.

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We can divide both sides of this
equation by negative 5.

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The negative 5 cancel out,
that was the whole point

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behind dividing by negative 5,
and we get x is equal to 20

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divided by negative
5 is negative 4.

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So when y is equal to 0, we
saw that right there, x is

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equal to negative 4.

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Or if we wanted to plot that
point, we always put the x

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coordinate first, so that
would be the point

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negative 4 comma 0.

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So let me graph that.

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So if we go 1, 2, 3, 4.

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That's a negative 4.

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And then the y value is
just 0, so that point

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is right over there.

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That is the x-intercept, y
is 0, x is negative 4.

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Notice we're intersecting
the x-axis.

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Now let's do the exact same
thing for the y-intercept.

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Let's set x equal to 0, so if
we set x is equal to 0, we

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have negative 5 times 0 plus
4y is equal to 20.

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Well, anything times 0 is
0, so we can just put

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that out of the way.

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And remember, this was setting
x is equal to 0, we're doing

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the y-intercept now.

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So this just simplifies
to 4y is equal to 20.

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We can divide both sides of this
equation by 4 to get rid

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of this 4 right there, and you
get y is equal to 20 over 4,

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which is 5.

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So when x is equal to
0, y is equal to 5.

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So the point 0, 5 is on the
graph for this line.

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So 0, 5.

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x is 0 and y is 1, 2, 3,
4, 5, right over there.

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And notice, when x is 0, we're
right on the y-axis, this is

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our y-intercept right
over there.

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And if we graph the line, all
you need is two points to

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graph any line, so we just have
to connect the dots and

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that is our line.

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So let me connect the dots,
trying my best to draw as

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straight of a line is I can--
well, I can do a better job

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than that-- to draw as straight
of a line as I can.

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And that's the graph of this
equation using the x-intercept

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and the y-intercept.