Introduction to graphing systems of linear inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.430
Graph the solution
set for this system.

00:00:03.430 --> 00:00:04.930
It's a system of inequalities.

00:00:04.930 --> 00:00:09.480
We have y is greater than x
minus 8, and y is less than 5

00:00:09.480 --> 00:00:10.180
minus x.

00:00:10.180 --> 00:00:13.370
Let's graph the solution set
for each of these inequalities,

00:00:13.370 --> 00:00:15.230
and then essentially
where they overlap

00:00:15.230 --> 00:00:19.380
is the solution set for the
system, the set of coordinates

00:00:19.380 --> 00:00:21.020
that satisfy both.

00:00:21.020 --> 00:00:24.360
So let me draw a
coordinate axes here.

00:00:24.360 --> 00:00:31.730
So that is my x-axis, and
then I have my y-axis.

00:00:31.730 --> 00:00:34.540
And that is my y-axis.

00:00:34.540 --> 00:00:36.490
And now let me draw
the boundary line,

00:00:36.490 --> 00:00:39.059
the boundary for this
first inequality.

00:00:39.059 --> 00:00:40.600
So the boundary line
is going to look

00:00:40.600 --> 00:00:43.596
like y is equal to x minus 8.

00:00:43.596 --> 00:00:44.970
But it's not going
to include it,

00:00:44.970 --> 00:00:46.966
because it's only
greater than x minus 8.

00:00:46.966 --> 00:00:49.170
But let's just graph x minus 8.

00:00:49.170 --> 00:00:52.310
So the y-intercept
here is negative 8.

00:00:52.310 --> 00:00:55.030
When x is 0, y is
going to be negative 8.

00:00:55.030 --> 00:01:01.030
So just go negative 1,
negative 2, 3, 4, 5, 6, 7, 8.

00:01:01.030 --> 00:01:02.780
So that is negative 8.

00:01:02.780 --> 00:01:05.850
So the point 0, negative
8 is on the line.

00:01:05.850 --> 00:01:08.890
And then it has a slope of 1.

00:01:08.890 --> 00:01:11.330
You don't see it right there,
but I could write it as 1x.

00:01:11.330 --> 00:01:13.752
So the slope here
is going to be 1.

00:01:13.752 --> 00:01:15.710
I could just draw a line
that goes straight up,

00:01:15.710 --> 00:01:20.240
or you could even say that it'll
intersect if y is equal to 0,

00:01:20.240 --> 00:01:22.890
if y were equal to 0,
x would be equal to 8.

00:01:22.890 --> 00:01:27.810
So 1, 2, 3, 4, 5, 6, 7, 8.

00:01:27.810 --> 00:01:29.707
And so this is x is equal to 8.

00:01:29.707 --> 00:01:31.290
If it has a slope
of 1, for every time

00:01:31.290 --> 00:01:33.414
you move to the right 1,
you're going to move up 1.

00:01:33.414 --> 00:01:35.770
So the line is going to
look something like this.

00:01:35.770 --> 00:01:37.840
And actually, let me not
draw it as a solid line.

00:01:37.840 --> 00:01:39.270
If I did it as a
solid line, that

00:01:39.270 --> 00:01:41.600
would actually be this
equation right here.

00:01:41.600 --> 00:01:43.750
But we're not going
to include that line.

00:01:43.750 --> 00:01:47.352
We care about the y values that
are greater than that line.

00:01:47.352 --> 00:01:49.060
So what we want to do
is do a dotted line

00:01:49.060 --> 00:01:51.268
to show that that's just
the boundary, that we're not

00:01:51.268 --> 00:01:54.020
including that in
our solution set.

00:01:54.020 --> 00:01:55.270
Let me do this in a new color.

00:01:55.270 --> 00:01:59.310
So this will be the
color for that line,

00:01:59.310 --> 00:02:02.030
or for that inequality,
I should say.

00:02:02.030 --> 00:02:04.260
So that is the boundary line.

00:02:04.260 --> 00:02:06.960
And this says y is
greater than x minus 8.

00:02:06.960 --> 00:02:09.563
So you pick an x,
and then x minus 8

00:02:09.563 --> 00:02:10.979
would get us on
the boundary line.

00:02:10.979 --> 00:02:13.100
And then y is greater than that.

00:02:13.100 --> 00:02:17.010
So it's all the y values above
the line for any given x.

00:02:17.010 --> 00:02:22.646
So it'll be this region above
the line right over here.

00:02:22.646 --> 00:02:24.020
And if that confuses
you, I mean,

00:02:24.020 --> 00:02:25.890
in general I like to just
think, oh, greater than,

00:02:25.890 --> 00:02:26.970
it's going to be above the line.

00:02:26.970 --> 00:02:28.980
If it's less than, it's
going to be below a line.

00:02:28.980 --> 00:02:30.355
But if you want
to make sure, you

00:02:30.355 --> 00:02:32.130
can just test on either
side of this line.

00:02:32.130 --> 00:02:34.370
So you could try the
point 0, 0, which

00:02:34.370 --> 00:02:36.050
should be in our solution set.

00:02:36.050 --> 00:02:39.460
And if you say, 0 is
greater than 0 minus 8,

00:02:39.460 --> 00:02:41.431
or 0 is greater than
negative 8, that works.

00:02:41.431 --> 00:02:43.680
So this definitely should
be part of the solution set.

00:02:43.680 --> 00:02:47.110
And you could try something
out here like 10 comma 0

00:02:47.110 --> 00:02:48.320
and see that it doesn't work.

00:02:48.320 --> 00:02:51.340
Because you would have 10 minus
8, which would be 2, and then

00:02:51.340 --> 00:02:52.080
you'd have 0.

00:02:52.080 --> 00:02:53.800
And 0 is not greater than 2.

00:02:53.800 --> 00:02:55.310
So when you test
something out here,

00:02:55.310 --> 00:02:56.962
you also see that it won't work.

00:02:56.962 --> 00:02:58.420
But in general, I
like to just say,

00:02:58.420 --> 00:03:00.040
hey look, this is
the boundary line,

00:03:00.040 --> 00:03:03.690
and we're greater than the
boundary line for any given x.

00:03:03.690 --> 00:03:05.990
Now let's do this one over here.

00:03:05.990 --> 00:03:07.080
Let's do this one.

00:03:07.080 --> 00:03:09.260
The boundary line for
it is going to be y

00:03:09.260 --> 00:03:11.480
is equal to 5 minus x.

00:03:11.480 --> 00:03:15.510
So the boundary line is
y is equal to 5 minus x.

00:03:15.510 --> 00:03:17.790
So once again, if x
is equal to 0, y is 5.

00:03:17.790 --> 00:03:21.460
So 1, 2, 3, 4, 5.

00:03:21.460 --> 00:03:23.310
And then it has a
slope of negative 1.

00:03:23.310 --> 00:03:27.850
We could write this as y is
equal to negative 1x plus 5.

00:03:27.850 --> 00:03:29.620
That's a little bit
more traditional.

00:03:29.620 --> 00:03:31.430
So once again, y-intercept at 5.

00:03:31.430 --> 00:03:33.170
And it has a slope
of negative 1.

00:03:33.170 --> 00:03:38.720
Or another way to think
about it, when y is 0,

00:03:38.720 --> 00:03:40.330
x will be equal to 5.

00:03:40.330 --> 00:03:43.176
So 1, 2, 3, 4, 5.

00:03:43.176 --> 00:03:44.800
So every time we move
to the right one,

00:03:44.800 --> 00:03:47.730
we go down one because we
have a negative 1 slope.

00:03:47.730 --> 00:03:50.190
So it will look like this.

00:03:50.190 --> 00:03:53.780
And once again, I want
to do a dotted line

00:03:53.780 --> 00:03:57.727
because we are-- so
that is our dotted line.

00:03:57.727 --> 00:03:59.310
And I'm doing a
dotted line because it

00:03:59.310 --> 00:04:01.640
says y is less than 5 minus x.

00:04:01.640 --> 00:04:04.870
If it was y is
equal to 5 minus x,

00:04:04.870 --> 00:04:07.000
I would have included the line.

00:04:07.000 --> 00:04:10.360
If it was y is less than
or equal to 5 minus x,

00:04:10.360 --> 00:04:12.930
I also would have
made this line solid.

00:04:12.930 --> 00:04:17.110
But it's only less than,
so for any x value,

00:04:17.110 --> 00:04:19.250
this is what 5
minus x-- 5 minus x

00:04:19.250 --> 00:04:20.790
will sit on that boundary line.

00:04:20.790 --> 00:04:23.160
But we care about the y values
that are less than that,

00:04:23.160 --> 00:04:28.250
so we want everything
that is below the line.

00:04:28.250 --> 00:04:30.650
And once again, you can test
on either side of the line.

00:04:30.650 --> 00:04:34.240
0, 0 should work for this
second inequality right here.

00:04:34.240 --> 00:04:37.370
0 is indeed less than 5 minus 0.

00:04:37.370 --> 00:04:38.710
0 is less than 5.

00:04:38.710 --> 00:04:41.140
And then you could try
something like 0, 10

00:04:41.140 --> 00:04:44.270
and see that it doesn't
work, because if you had 10

00:04:44.270 --> 00:04:46.610
is less than 5 minus
0, that doesn't work.

00:04:46.610 --> 00:04:49.070
So it is everything
below the line like that.

00:04:49.070 --> 00:04:51.230
And like we said,
the solution set

00:04:51.230 --> 00:04:56.810
for this system are all
of the x's and y's, all

00:04:56.810 --> 00:05:00.000
of the coordinates that
satisfy both of them.

00:05:00.000 --> 00:05:02.030
So all of this shaded
in purple satisfies

00:05:02.030 --> 00:05:03.330
the second inequality.

00:05:03.330 --> 00:05:06.680
All of this shaded in green
satisfies the first inequality.

00:05:06.680 --> 00:05:10.470
So the stuff that satisfies
both of them is their overlap.

00:05:10.470 --> 00:05:14.460
So it's all of this
region in blue.

00:05:14.460 --> 00:05:17.320
Hopefully this isn't
making it too messy.

00:05:17.320 --> 00:05:21.340
All of this region in blue
where the two overlap,

00:05:21.340 --> 00:05:24.470
below the magenta dotted
line on the left-hand side,

00:05:24.470 --> 00:05:26.340
and above the
green magenta line.

00:05:26.340 --> 00:05:27.840
That's only where they overlap.

00:05:27.840 --> 00:05:32.350
So it's only this
region over here,

00:05:32.350 --> 00:05:35.398
and you're not including
the boundary lines.


Checking solutions of systems of inequalities example

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.208
Is two comma five a solution of this system?

00:00:04.267 --> 00:00:06.533
And we have a system of inequalities right over here.

00:00:06.533 --> 00:00:12.600
We have Y is greater than or equal to 2x plus 1 and X is greater than 1.

00:00:12.600 --> 00:00:19.148
In order for two comma five to be a solution of this system, it just has to satisfy both inequalities.

00:00:19.164 --> 00:00:20.738
So, lets just try it out.

00:00:20.933 --> 00:00:25.267
So when X is equal to two and Y is equal to five, it has to satify both of these.

00:00:25.267 --> 00:00:27.067
So lets try it with the first one.

00:00:27.067 --> 00:00:29.467
So if we assume X is two and Y is five,

00:00:29.467 --> 00:00:37.400
we would get an inequality that says that five is greater than or equal to two times two plus one.

00:00:37.400 --> 00:00:39.467
X is two; Y is five.

00:00:39.467 --> 00:00:44.933
This gives us five is greater than or equal to two times two is four plus one is five.

00:00:44.933 --> 00:00:50.133
Y is greater than or equal to five. That's true! Five is equal to five.

00:00:50.133 --> 00:00:52.800
So that equal part of the greater than or equal saves us.

00:00:52.800 --> 00:00:57.400
That definitely satisfies the first inequality. Lets see the second one.

00:00:57.400 --> 00:00:59.600
X needs to be greater than one.

00:00:59.600 --> 00:01:03.067
So in two comma five, X is two.

00:01:03.067 --> 00:01:06.200
So two is greater than one.

00:01:06.200 --> 00:01:09.533
So it actually satisfies both of these inequalities.

00:01:09.533 --> 00:01:14.333
So two comma five is a solution for this system.


Linear function example: spending money

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:02.700
Jill just received $40.

00:00:02.700 --> 00:00:06.610
The number of dollars she
has left, y, after x days,

00:00:06.610 --> 00:00:12.670
is approximated by the formula
y is equal to 40 minus 2.5x.

00:00:12.670 --> 00:00:14.660
Graph the equation
and use the graph

00:00:14.660 --> 00:00:18.950
to estimate how much money
Jill will have 8 days later.

00:00:18.950 --> 00:00:21.080
So let's just make a
table of x and y values.

00:00:21.080 --> 00:00:24.730
Then we can use that table
to actually plot the graph.

00:00:24.730 --> 00:00:27.150
And then ask us to do
everything else they want us.

00:00:27.150 --> 00:00:28.840
How much money she'll
have after 8 days.

00:00:28.840 --> 00:00:30.430
And we should actually just
put that right in the equation,

00:00:30.430 --> 00:00:32.119
we might as well do that.

00:00:32.119 --> 00:00:34.142
So we're doing numbers of days.

00:00:34.142 --> 00:00:35.600
We're not going to
go back in time.

00:00:35.600 --> 00:00:38.090
She starts with $40, so
we can start with 0 days.

00:00:38.090 --> 00:00:39.870
So 0 days.

00:00:39.870 --> 00:00:41.786
So she just received
the $40, you don't even

00:00:41.786 --> 00:00:43.202
have to look at
the equation here.

00:00:43.202 --> 00:00:45.586
What's 0 days after that,
how much money will she have?

00:00:45.586 --> 00:00:47.460
Well she hasn't had a
chance to spend it yet.

00:00:47.460 --> 00:00:48.834
So you could just
think about it.

00:00:48.834 --> 00:00:49.546
She'll have $40.

00:00:49.546 --> 00:00:50.920
Or you could look
at the equation

00:00:50.920 --> 00:00:52.970
and see that the
equation verifies this.

00:00:52.970 --> 00:00:58.460
When x is 0, so the y value
is going to be 40 minus 2.5

00:00:58.460 --> 00:01:02.100
times 0, which is just 40.

00:01:02.100 --> 00:01:04.099
Because that part
right over there is 0.

00:01:04.099 --> 00:01:07.950
So at time, at 0 number of
days, she will have $40.

00:01:07.950 --> 00:01:10.060
Now we could do one day
later, but then we're

00:01:10.060 --> 00:01:11.930
going to have these
decimal points in here.

00:01:11.930 --> 00:01:14.044
So that this part
of the equation

00:01:14.044 --> 00:01:15.460
always ends up
with clean numbers,

00:01:15.460 --> 00:01:17.840
let's multiply it
by multiples of 2.

00:01:17.840 --> 00:01:22.920
So then at 2 days, how
much money will she have?

00:01:22.920 --> 00:01:26.930
Well, it's going to be
40 minus 2 times-- I'll

00:01:26.930 --> 00:01:30.600
do the same order--
minus 2.5 times 2.

00:01:30.600 --> 00:01:33.230
2.5 times 2 is 5.

00:01:33.230 --> 00:01:37.020
So 40 minus 5 is $35.

00:01:37.020 --> 00:01:39.391
After 4 days, it's
going to be 40--

00:01:39.391 --> 00:01:40.890
let me do this in
a different color,

00:01:40.890 --> 00:01:42.264
so when I plot
the points, you'll

00:01:42.264 --> 00:01:44.330
see where I got my
information from.

00:01:44.330 --> 00:01:49.560
After 4 days she's going to
have 40 minus 2.5 times 4.

00:01:49.560 --> 00:01:51.690
2.5 times 4 is 10.

00:01:51.690 --> 00:01:54.070
So 40 minus 10 is 30.

00:01:54.070 --> 00:01:59.450
You see, every 2 days that
goes by, she is spending $5.

00:01:59.450 --> 00:02:01.540
$5 every 2 days,
or $2.50 every day.

00:02:01.540 --> 00:02:03.290
And you actually see
that right over here.

00:02:03.290 --> 00:02:04.090
She is spending.

00:02:04.090 --> 00:02:05.260
That's a negative sign.

00:02:05.260 --> 00:02:07.960
$2.50 dollars every day.

00:02:07.960 --> 00:02:11.290
Every time you increment
x by 1, $2.50 goes away.

00:02:11.290 --> 00:02:12.570
So let's keep going.

00:02:12.570 --> 00:02:17.910
So then after-- I'll look for
another color here-- 6 days,

00:02:17.910 --> 00:02:22.350
it's going to be 40
minus 2.5 times 6.

00:02:22.350 --> 00:02:25.595
2.5 times 6 is 15.

00:02:25.595 --> 00:02:29.260
40 minus 15 is equal to 25.

00:02:29.260 --> 00:02:31.960
Then finally we could
do after 8 days.

00:02:31.960 --> 00:02:37.760
After 8 days, she'll have
40 minus 2.5 times 8.

00:02:37.760 --> 00:02:39.830
2.5 times 8 is 20.

00:02:39.830 --> 00:02:42.679
So 40 minus 20 is $20.

00:02:42.679 --> 00:02:44.220
So we actually
answered our question.

00:02:44.220 --> 00:02:46.710
Our estimate for how much
money Jill will have 8 days

00:02:46.710 --> 00:02:48.862
later is actually $20.

00:02:48.862 --> 00:02:50.070
But let's do this first part.

00:02:50.070 --> 00:02:52.710
Let's actually graph the
equation, see it visually.

00:02:52.710 --> 00:02:54.207
So let me draw some axes here.

00:02:54.207 --> 00:02:55.790
This will be a
hand-drawn graph, but I

00:02:55.790 --> 00:02:57.770
think it'll get the job done.

00:02:57.770 --> 00:03:00.340
So let's make that our
y-axis, or in this case

00:03:00.340 --> 00:03:02.140
it's the number of
dollars she has.

00:03:02.140 --> 00:03:05.480
And let's make this my x-axis.

00:03:05.480 --> 00:03:06.930
This is our x-axis.

00:03:06.930 --> 00:03:09.480
And we only need to focus
on the first quadrant.

00:03:09.480 --> 00:03:11.282
Because at least
in this context,

00:03:11.282 --> 00:03:13.740
we're assuming she won't have
a negative number of dollars.

00:03:13.740 --> 00:03:15.660
So the y values
will be positive.

00:03:15.660 --> 00:03:18.190
And we assume that the days
are only going to be positive.

00:03:18.190 --> 00:03:19.760
We're not to deal
with negative time.

00:03:19.760 --> 00:03:21.410
So the x values are always
going to be positive.

00:03:21.410 --> 00:03:22.410
So we're only going
to be operating

00:03:22.410 --> 00:03:23.380
in the first quadrant.

00:03:23.380 --> 00:03:25.050
So that's all I have to draw.

00:03:25.050 --> 00:03:26.880
And so she starts off at $40.

00:03:26.880 --> 00:03:31.600
Let me mark off the y-axis
in increments of 10 first.

00:03:31.600 --> 00:03:38.290
So this would be $10, this
would be $20, this would be $30,

00:03:38.290 --> 00:03:40.750
this would be $40.

00:03:40.750 --> 00:03:46.320
And then I could do the $35, the
$25, the $15, and then the $5.

00:03:46.320 --> 00:03:49.290
And then let me
mark off the days.

00:03:49.290 --> 00:03:52.240
So this is-- let me do it, I'll
do that same yellow color--

00:03:52.240 --> 00:03:58.420
so this is after 2 days, this
is 4 days, this is 6 days,

00:03:58.420 --> 00:03:59.950
this is 8 days.

00:03:59.950 --> 00:04:02.100
We could keep going if we like.

00:04:02.100 --> 00:04:07.760
So after 0 days, so this
right over here, after 0 days,

00:04:07.760 --> 00:04:09.030
she has $40.

00:04:09.030 --> 00:04:10.950
So that's this point
right over here.

00:04:10.950 --> 00:04:12.690
That's that right over there.

00:04:12.690 --> 00:04:14.400
Then after 2 days, she has $35.

00:04:18.860 --> 00:04:20.640
2 in the x direction,
then we go up 35.

00:04:20.640 --> 00:04:22.550
So that's that point
right over there.

00:04:22.550 --> 00:04:26.460
Then after 4 days, she has $30.

00:04:26.460 --> 00:04:30.830
So you go x 4, remember the days
are in x, or x are the days,

00:04:30.830 --> 00:04:33.680
actually you should
mark it, these are days.

00:04:33.680 --> 00:04:36.720
And the y-axis is
the dollar axis.

00:04:36.720 --> 00:04:38.180
So after 4 days, she has $30.

00:04:41.460 --> 00:04:44.530
Then after 6 days-- I'm
going to do the same color--

00:04:44.530 --> 00:04:47.060
after 6 days she has $25.

00:04:47.060 --> 00:04:51.790
So x-coordinate is 6,
y-coordinate is 25.

00:04:51.790 --> 00:04:54.426
And then finally, after
8 days, she has $20.

00:04:59.574 --> 00:05:00.990
And so we've plotted
those points,

00:05:00.990 --> 00:05:01.725
and we could connect them.

00:05:01.725 --> 00:05:03.190
We could actually just,
if we had a nice ruler,

00:05:03.190 --> 00:05:04.370
we could just
connect two of those,

00:05:04.370 --> 00:05:05.550
and we would have the line.

00:05:05.550 --> 00:05:06.966
But our line looks
something-- let

00:05:06.966 --> 00:05:09.410
me do this in a new color--
our line would look something

00:05:09.410 --> 00:05:13.791
like that.

00:05:13.791 --> 00:05:16.676
That shows how much she
has after every day.

00:05:16.676 --> 00:05:17.300
And we're done.

00:05:17.300 --> 00:05:18.716
We've graphed the
equation, and we

00:05:18.716 --> 00:05:21.950
know she'll have $20
left after 8 days.


Graphing solutions to two-variable linear equations example 1

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:04.560
Create a graph of the
linear equation 5x plus 2y

00:00:04.560 --> 00:00:06.154
is equal to 20.

00:00:06.154 --> 00:00:07.570
So the line is
essentially the set

00:00:07.570 --> 00:00:10.650
of all coordinate,
all x's and y's, that

00:00:10.650 --> 00:00:13.024
satisfy this relationship
right over here.

00:00:13.024 --> 00:00:14.940
To make things simpler,
what we're going to do

00:00:14.940 --> 00:00:16.050
is set up a table
where we're going

00:00:16.050 --> 00:00:18.091
to put a bunch of x values
in and then figure out

00:00:18.091 --> 00:00:20.421
the corresponding y value
based on this relationship.

00:00:20.421 --> 00:00:21.920
But to make it a
little bit simpler,

00:00:21.920 --> 00:00:23.760
I'm going to solve for y here.

00:00:23.760 --> 00:00:26.740
So it becomes easier to
solve for y for any given x.

00:00:26.740 --> 00:00:32.130
So we have 5x plus
2y is equal to 20.

00:00:32.130 --> 00:00:33.990
If we want to solve
for y, let's just

00:00:33.990 --> 00:00:35.800
get rid of the 5x on
the left-hand side.

00:00:35.800 --> 00:00:40.880
So let's subtract 5x from
both sides of this equation.

00:00:40.880 --> 00:00:42.800
The left-hand side,
these guys cancel out,

00:00:42.800 --> 00:00:45.680
so we get 2y is equal
to the right hand side,

00:00:45.680 --> 00:00:48.530
you have 20 minus 5x.

00:00:48.530 --> 00:00:52.430
And then you can divide both
sides of this equation by 2.

00:00:52.430 --> 00:00:54.950
So you divide both sides by 2.

00:00:54.950 --> 00:00:57.330
The left-hand side, we
just have a y, and then

00:00:57.330 --> 00:00:59.590
the right-hand side, we
could leave it that way.

00:00:59.590 --> 00:01:01.714
That actually would be a
pretty straightforward way

00:01:01.714 --> 00:01:04.260
to leave it, or we could
call this 20 divided by 2

00:01:04.260 --> 00:01:10.376
is 10 minus 5x over 2
or minus 5/2 times x.

00:01:10.376 --> 00:01:11.750
And so now using
this, let's just

00:01:11.750 --> 00:01:13.473
come up with a bunch
of x values and see

00:01:13.473 --> 00:01:17.150
what the corresponding y values
are, and then just plot them.

00:01:17.150 --> 00:01:19.210
So let me do this
in a new color.

00:01:19.210 --> 00:01:21.470
So let me-- a slightly
different shade of yellow.

00:01:21.470 --> 00:01:23.630
So we have x values,
and then let's

00:01:23.630 --> 00:01:25.910
think about what
the corresponding y

00:01:25.910 --> 00:01:27.347
value is going to be.

00:01:27.347 --> 00:01:29.180
So I'll start, well, I
could start anywhere.

00:01:29.180 --> 00:01:31.180
I'll start at x is equal
to 0, just because that

00:01:31.180 --> 00:01:32.960
tends to keep things
pretty simple.

00:01:32.960 --> 00:01:39.440
If x is 0, then y is equal
to 10 minus 5/2 times

00:01:39.440 --> 00:01:43.600
0, which is equal to
5/2 times 0 is just a 0.

00:01:43.600 --> 00:01:46.290
So it's just 10 minus 0 or 10.

00:01:46.290 --> 00:01:50.800
So that gives us the coordinate,
the point, 0 comma 10.

00:01:50.800 --> 00:01:54.280
When x is 0, y is 10.

00:01:54.280 --> 00:01:55.310
So x is 0.

00:01:55.310 --> 00:01:57.870
So it's going to be right here
at the middle of the x-axis.

00:01:57.870 --> 00:02:00.050
And you go up 10 for
the y-coordinate.

00:02:00.050 --> 00:02:06.390
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

00:02:06.390 --> 00:02:08.360
So it's right over here.

00:02:08.360 --> 00:02:12.390
So that's the point 0 comma 10.

00:02:12.390 --> 00:02:13.940
Let's do another point.

00:02:13.940 --> 00:02:16.550
Let's say that x is 2.

00:02:16.550 --> 00:02:18.140
I'm going to pick
multiples of 2 here

00:02:18.140 --> 00:02:20.760
just so that I get a
nice clean answer here.

00:02:20.760 --> 00:02:28.640
So when x is 2, then y is
equal to 10 minus 5/2 times 2,

00:02:28.640 --> 00:02:30.680
and the 2 in the
denominator cancels out

00:02:30.680 --> 00:02:33.510
with this 2 in the numerator.

00:02:33.510 --> 00:02:36.990
So it simplifies to
10 minus 5, or just 5.

00:02:36.990 --> 00:02:41.720
So that tells us the point
x equals 2, y is equal to 5,

00:02:41.720 --> 00:02:43.890
is on the line.

00:02:43.890 --> 00:02:47.600
So 2x is equal to 1,
2 right over here.

00:02:47.600 --> 00:02:48.920
And then y is equal to 5.

00:02:48.920 --> 00:02:49.900
You go up 5.

00:02:49.900 --> 00:02:54.100
1, 2, 3, 4, 5, just like that.

00:02:54.100 --> 00:02:55.749
So that's the point 2, 5.

00:02:55.749 --> 00:02:57.540
And when you're drawing
a line you actually

00:02:57.540 --> 00:02:58.415
just need two points.

00:02:58.415 --> 00:03:00.456
If you have a ruler or
any kind of straight edge,

00:03:00.456 --> 00:03:02.140
we could just connect
these two points.

00:03:02.140 --> 00:03:05.200
And if we do it neatly,
every point on that line

00:03:05.200 --> 00:03:07.830
should satisfy this
relationship right here.

00:03:07.830 --> 00:03:11.010
Just so we get practice,
I'll do more points.

00:03:11.010 --> 00:03:14.280
So let me do, let's say
when x is equal to 4,

00:03:14.280 --> 00:03:19.750
then y is equal to
10 minus 5/2 times 4.

00:03:19.750 --> 00:03:22.100
This is equal to 5/2 times 4.

00:03:22.100 --> 00:03:23.760
This is equal to 10, right?

00:03:23.760 --> 00:03:27.110
Because the 2, divide the
denominator by 2 you get 1,

00:03:27.110 --> 00:03:31.062
divide the numerator by
2 you get 2, or 4 over 2

00:03:31.062 --> 00:03:32.020
is the same thing as 2.

00:03:32.020 --> 00:03:36.050
So it becomes 2 times 5
is 10, 10 minus 10 is 0.

00:03:36.050 --> 00:03:40.850
So the point 4 comma
0 is on our line.

00:03:40.850 --> 00:03:45.130
So x is 1, 2, 3,
4, and then y is 0.

00:03:45.130 --> 00:03:50.990
So we don't move up at
all, so we have 4 comma 0.

00:03:50.990 --> 00:03:52.060
And I could keep going.

00:03:52.060 --> 00:03:53.147
I could try other points.

00:03:53.147 --> 00:03:55.230
You could do them if you
like, but this is plenty.

00:03:55.230 --> 00:03:57.646
Just two of these would have
been enough to draw the line.

00:03:57.646 --> 00:04:00.250
So let me just draw it.

00:04:00.250 --> 00:04:01.740
So I'll do it in white.

00:04:01.740 --> 00:04:03.840
So the line will look
something like this.

00:04:12.660 --> 00:04:15.246
And I could keep going
in both directions.

00:04:15.246 --> 00:04:16.120
So there you have it.

00:04:16.120 --> 00:04:19.410
That is the graph of
our linear equation.

00:04:19.410 --> 00:04:21.480
Let me make my line
a little bit bolder,

00:04:21.480 --> 00:04:24.197
just in case you found that
first line hard to read.

00:04:24.197 --> 00:04:25.780
So let me make it a
little bit bolder.

00:04:25.780 --> 00:04:28.520
And I think you get
the general idea.


Coordinate plane: plot ordered pairs

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:04.660
Plot the ordered pair
6, comma negative 8

00:00:04.660 --> 00:00:06.360
into the coordinate plane.

00:00:06.360 --> 00:00:08.630
So this is a coordinate
plane right over here.

00:00:08.630 --> 00:00:12.000
The horizontal axis
here, this is the x-axis.

00:00:12.000 --> 00:00:16.160
The vertical axis
here is the y-axis.

00:00:16.160 --> 00:00:21.160
And the convention, when we
get an ordered pair like this,

00:00:21.160 --> 00:00:24.540
is that the first coordinate
is the x-coordinate,

00:00:24.540 --> 00:00:27.890
and the second coordinate
is the y-coordinate.

00:00:27.890 --> 00:00:32.610
So they're telling us that
we have an x-coordinate of 6.

00:00:32.610 --> 00:00:36.350
That means we count
up 6 on the x-axis.

00:00:36.350 --> 00:00:37.950
So let's count up 1.

00:00:37.950 --> 00:00:41.139
I could even write it down--
1, and then we have 2.

00:00:41.139 --> 00:00:41.930
That's right there.

00:00:41.930 --> 00:00:43.200
Then we have 3.

00:00:43.200 --> 00:00:44.800
Then we have 4.

00:00:44.800 --> 00:00:45.990
Then we have 5.

00:00:45.990 --> 00:00:49.950
And then we've moved
up 6 along the x-axis.

00:00:49.950 --> 00:00:53.770
We started at the
origin, at 0, 0.

00:00:53.770 --> 00:00:55.040
That's the origin.

00:00:55.040 --> 00:00:56.860
And we moved 6 to the right.

00:00:56.860 --> 00:00:58.920
That's this part
right over here.

00:00:58.920 --> 00:01:01.680
Now, our y-coordinate
is negative 8.

00:01:01.680 --> 00:01:05.440
That means we move down 8
in the vertical direction.

00:01:05.440 --> 00:01:06.930
So we can see it
right over here.

00:01:06.930 --> 00:01:08.930
So this right here, I'll
just do that every two.

00:01:08.930 --> 00:01:10.360
So this is down 1.

00:01:10.360 --> 00:01:13.670
This right here is
negative 2, negative 3.

00:01:13.670 --> 00:01:18.600
This right here is negative
4, negative 5, negative 6,

00:01:18.600 --> 00:01:21.480
negative 7, negative 8.

00:01:21.480 --> 00:01:25.590
So we move to the right 6,
and then we move down 8.

00:01:25.590 --> 00:01:27.420
We move to the
right 6, and then we

00:01:27.420 --> 00:01:34.070
move down 1, 2, 3,
4, 5, 6, 7, and 8.

00:01:34.070 --> 00:01:36.100
And we are right over there.

00:01:36.100 --> 00:01:41.592
So that is the point
, 6 negative 8.

00:01:41.592 --> 00:01:43.050
You could also look
at it this way.

00:01:43.050 --> 00:01:44.030
The x-coordinate is 6.

00:01:44.030 --> 00:01:45.620
So you move 6 there.

00:01:45.620 --> 00:01:48.630
And so we're going to be along
this line right over here.

00:01:48.630 --> 00:01:51.320
And then you can go, and then
the y-coordinate is negative 8.

00:01:51.320 --> 00:01:52.250
So you go down to negative 8.

00:01:52.250 --> 00:01:54.625
And you say, we're going to
be along this line over here.

00:01:54.625 --> 00:01:57.004
And the place that they
intersect is this coordinate.

00:01:57.004 --> 00:01:58.670
The easiest way I'd
like to think of it,

00:01:58.670 --> 00:02:01.280
move 6 to the right,
move along the x-axis,

00:02:01.280 --> 00:02:03.430
and then move along in
the vertical direction

00:02:03.430 --> 00:02:04.540
along the y-axis.

00:02:04.540 --> 00:02:07.080
So you go down 8.


Coordinate plane: quadrants

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:05.520
In which quadrant is the point
negative 7 comma 7 located?

00:00:05.520 --> 00:00:07.890
So let's just review
what a quadrant is.

00:00:07.890 --> 00:00:10.680
A quadrant are each
of the four sections

00:00:10.680 --> 00:00:11.750
of the coordinate plane.

00:00:11.750 --> 00:00:13.250
And when we talk
about the sections,

00:00:13.250 --> 00:00:15.870
we're talking about the sections
as divided by the coordinate

00:00:15.870 --> 00:00:16.960
axes.

00:00:16.960 --> 00:00:20.600
So this right here is the
x-axis and this up-down axis

00:00:20.600 --> 00:00:21.726
is the y-axis.

00:00:21.726 --> 00:00:23.600
And you can see it
divides a coordinate plane

00:00:23.600 --> 00:00:25.140
into four sections.

00:00:25.140 --> 00:00:27.950
We call each of these
sections quadrants.

00:00:27.950 --> 00:00:31.920
This one over here, where both
the x-values and the y-values

00:00:31.920 --> 00:00:34.910
are positive, we call
the first quadrant.

00:00:34.910 --> 00:00:37.980
And we use the Roman numeral I.

00:00:37.980 --> 00:00:40.170
Then if we kind of
move counterclockwise

00:00:40.170 --> 00:00:43.120
around the coordinate
plane, this quadrant where

00:00:43.120 --> 00:00:47.070
the x-values are negative and
the y-values are positive,

00:00:47.070 --> 00:00:48.825
we call this the
second quadrant.

00:00:51.370 --> 00:00:52.840
I could write it.

00:00:52.840 --> 00:00:56.050
We call this the
second quadrant.

00:00:56.050 --> 00:00:59.120
Then we go down here where
both the x-values are negative

00:00:59.120 --> 00:01:00.880
and the y-values are negative.

00:01:00.880 --> 00:01:03.110
We call this the third
quadrant, once again

00:01:03.110 --> 00:01:04.660
using Roman numerals.

00:01:04.660 --> 00:01:07.430
Then finally the quadrant
where the x-values are positive

00:01:07.430 --> 00:01:09.180
but the y-values
are negative, we

00:01:09.180 --> 00:01:11.560
call this the fourth quadrant.

00:01:11.560 --> 00:01:14.310
So let's see which quadrant
the point negative 7 comma

00:01:14.310 --> 00:01:14.927
7 is located.

00:01:14.927 --> 00:01:16.510
So there's two ways
to think about it.

00:01:16.510 --> 00:01:20.180
You could just say, look,
we have a negative x-value.

00:01:20.180 --> 00:01:22.030
Our x-value is
negative, so we're

00:01:22.030 --> 00:01:23.240
going to move to the left.

00:01:23.240 --> 00:01:24.730
So we're going to
be on this side.

00:01:24.730 --> 00:01:26.438
We're going to be on
this side right here

00:01:26.438 --> 00:01:28.220
of the coordinate plane.

00:01:28.220 --> 00:01:30.280
Just by the fact that
the x-value is negative,

00:01:30.280 --> 00:01:33.040
we're going to be either in the
second or the third quadrant.

00:01:33.040 --> 00:01:36.150
Now, we know that the
y-value is positive.

00:01:36.150 --> 00:01:39.230
We know that the
y-value is positive.

00:01:39.230 --> 00:01:42.430
So if the x-value is negative
and the y-value is positive,

00:01:42.430 --> 00:01:44.410
we're going to land
someplace right over here

00:01:44.410 --> 00:01:45.537
in the second quadrant.

00:01:45.537 --> 00:01:47.120
The other way to
think about it is you

00:01:47.120 --> 00:01:48.860
could literally
just plot this point

00:01:48.860 --> 00:01:51.030
and see that it falls
in the second quadrant.

00:01:51.030 --> 00:01:52.080
So let's do that.

00:01:52.080 --> 00:01:53.400
If x is negative 7.

00:01:53.400 --> 00:01:57.040
So that's negative 1, negative
2, negative 3, negative 4,

00:01:57.040 --> 00:01:59.607
negative 5, negative
6, negative 7.

00:01:59.607 --> 00:02:00.440
Did I do that right?

00:02:00.440 --> 00:02:03.690
1, 2, 3 4, 5, 6, 7.

00:02:03.690 --> 00:02:05.360
So this is x is negative 7.

00:02:05.360 --> 00:02:07.190
And then we have to
go up 7 because y

00:02:07.190 --> 00:02:08.410
is equal to positive 7.

00:02:08.410 --> 00:02:14.510
So 1, 2, 3, 4, 5, 6 7.

00:02:14.510 --> 00:02:20.100
So the point negative 7
comma 7 is right over here,

00:02:20.100 --> 00:02:23.323
clearly lies in the
second quadrant.


Graphing using x- and y-intercepts

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.620
Identify the x and
y-intercepts of the line

00:00:03.620 --> 00:00:06.880
y is equal to 3x minus 9.

00:00:06.880 --> 00:00:09.150
Then graph the line.

00:00:09.150 --> 00:00:11.450
So the x-intercept,
I'll just abbreviate it

00:00:11.450 --> 00:00:15.820
as x-int, that is where the
line intersects the x-axis.

00:00:15.820 --> 00:00:17.920
So where it
intersects the x-axis.

00:00:17.920 --> 00:00:20.830
Remember, this horizontal
axis is the x-axis.

00:00:20.830 --> 00:00:22.919
So when something
intersects the x-axis,

00:00:22.919 --> 00:00:24.460
what do we know
about its coordinate?

00:00:24.460 --> 00:00:28.325
Its x-value could be anything,
but we know it's y-value is 0.

00:00:28.325 --> 00:00:29.950
If we're intersecting,
if we're sitting

00:00:29.950 --> 00:00:32.280
on the x-axis
someplace, that means

00:00:32.280 --> 00:00:34.380
that we haven't moved
in the y-direction.

00:00:34.380 --> 00:00:36.100
That means that y is 0.

00:00:36.100 --> 00:00:39.610
So this means,
literally, that y is 0.

00:00:39.610 --> 00:00:42.190
So we need to find
the x-value defined

00:00:42.190 --> 00:00:45.400
by this relationship
when y is equal to 0.

00:00:45.400 --> 00:00:47.912
Similarly, when we talk
about the y-intercept,

00:00:47.912 --> 00:00:49.370
I'll do it down
here-- when we talk

00:00:49.370 --> 00:00:52.126
about the y-intercept,
what does that mean?

00:00:52.126 --> 00:00:53.500
Well, y-intercept
means-- so this

00:00:53.500 --> 00:00:56.160
is the y-axis right over
here running up and down.

00:00:56.160 --> 00:00:58.020
The y-intercept is
the point at which

00:00:58.020 --> 00:01:01.239
the line intercepts the y-axis.

00:01:01.239 --> 00:01:02.030
So what's going on?

00:01:02.030 --> 00:01:04.650
If we're at the
y-axis, our y-value

00:01:04.650 --> 00:01:07.600
could be anything depending on
where we intersect the y-axis.

00:01:07.600 --> 00:01:10.470
But we know that we haven't
moved to the right or the left.

00:01:10.470 --> 00:01:14.470
We know that our x-value
is 0 at the y-intercept.

00:01:14.470 --> 00:01:17.400
So over here, our
x-value is going to be 0.

00:01:17.400 --> 00:01:19.140
And to find the
actual point, we just

00:01:19.140 --> 00:01:21.760
have to find the
corresponding y-value defined

00:01:21.760 --> 00:01:24.440
by this relationship
or this equation.

00:01:24.440 --> 00:01:25.850
So let's do the first one first.

00:01:25.850 --> 00:01:28.010
The x-intercept is
when y is equal to 0.

00:01:28.010 --> 00:01:31.130
So we set y is equal to 0,
and then we'll solve for x.

00:01:31.130 --> 00:01:34.650
So we get 0 is
equal to 3x minus 9.

00:01:34.650 --> 00:01:37.630
We can add 9 to both
sides of this equation

00:01:37.630 --> 00:01:39.120
to isolate the x-term.

00:01:39.120 --> 00:01:42.000
So we get 9 is equal to 3x.

00:01:42.000 --> 00:01:43.460
These cancel out.

00:01:43.460 --> 00:01:46.410
We could divide both sides by 3.

00:01:46.410 --> 00:01:48.260
Divide both sides by 3.

00:01:48.260 --> 00:01:52.930
We get 3 is equal to
x or x is equal to 3.

00:01:52.930 --> 00:01:56.925
So the point y is equal to 0, x
is equal to 3 is on this line.

00:01:56.925 --> 00:01:58.050
And let me put it in order.

00:01:58.050 --> 00:01:59.535
x-coordinate always goes first.

00:01:59.535 --> 00:02:02.260
So it's 3 comma 0.

00:02:02.260 --> 00:02:03.470
So this is the origin.

00:02:03.470 --> 00:02:06.440
1, 2, 3 is right over here.

00:02:06.440 --> 00:02:08.900
That is 3 comma 0.

00:02:08.900 --> 00:02:12.030
This right here is
the x-intercept.

00:02:12.030 --> 00:02:15.720
And remember, notice that
point lies on the x-axis,

00:02:15.720 --> 00:02:17.190
but the y-value is 0.

00:02:17.190 --> 00:02:18.414
We haven't moved up or down.

00:02:18.414 --> 00:02:20.080
When you think
x-intercept, you say, OK,

00:02:20.080 --> 00:02:22.380
that means my y-value is 0.

00:02:22.380 --> 00:02:24.170
So I have to solve
for the x-value.

00:02:24.170 --> 00:02:26.660
Now we do the opposite
for the y-intercept.

00:02:26.660 --> 00:02:29.090
And the y-intercept, we're
sitting on this line,

00:02:29.090 --> 00:02:30.380
x-value must be 0.

00:02:30.380 --> 00:02:34.100
So let's figure out what y is
equal to when x is equal to 0.

00:02:34.100 --> 00:02:37.110
So y is equal to-- I want
to do it in that pink color.

00:02:37.110 --> 00:02:42.060
y is equal to-- y is equal
to 3 times-- x is 0 now.

00:02:42.060 --> 00:02:44.620
3 times 0 minus 9.

00:02:44.620 --> 00:02:46.230
Well, 3 times 0 is just 0.

00:02:46.230 --> 00:02:47.560
So 0 minus 9.

00:02:47.560 --> 00:02:49.520
Well that's, just
equal to negative 9.

00:02:49.520 --> 00:02:52.810
So we have the point
0 comma negative 9.

00:02:52.810 --> 00:02:57.870
So when x is 0, we
go down 9 for y.

00:02:57.870 --> 00:03:04.720
1, 2, 3, 4, 5, 6, 7, 8, 9.

00:03:04.720 --> 00:03:08.620
So right there is the
point 0 comma negative 9.

00:03:08.620 --> 00:03:10.740
Notice, it sits on the y-axis.

00:03:10.740 --> 00:03:12.340
That's why it's the y-intercept.

00:03:12.340 --> 00:03:13.800
And the x-value is 0.

00:03:13.800 --> 00:03:15.480
We haven't moved to
the left or right.

00:03:15.480 --> 00:03:17.110
All you need is two
points for a line,

00:03:17.110 --> 00:03:18.290
so we're now ready to graph.

00:03:18.290 --> 00:03:20.350
We essentially just have
to connect the dots.

00:03:20.350 --> 00:03:22.430
So it's going to look
something like this.

00:03:22.430 --> 00:03:25.909
It's going to look
something-- our line.

00:03:25.909 --> 00:03:27.450
I don't have a good
line tool, so I'm

00:03:27.450 --> 00:03:30.240
going to try my best
to draw it nicely--

00:03:30.240 --> 00:03:33.860
is going to look
something like that.

00:03:33.860 --> 00:03:34.950
And you just keep going.

00:03:34.950 --> 00:03:36.040
You just keep going.

00:03:36.040 --> 00:03:37.870
You want to do a straight line.

00:03:37.870 --> 00:03:41.940
So it just keeps going on
and on and on like that.

00:03:41.940 --> 00:03:45.752
It just keeps going.

00:03:45.752 --> 00:03:47.960
And I could keep going all
the way in that direction,

00:03:47.960 --> 00:03:50.293
and then-- but then my line
doesn't look as straight all

00:03:50.293 --> 00:03:50.835
of a sudden.

00:03:50.835 --> 00:03:52.210
I think you get
the general idea.

00:03:52.210 --> 00:03:54.200
I can keep going like that,
and then keep going like that.

00:03:54.200 --> 00:03:56.010
I don't have a nice
ruler to do it with.

00:03:56.010 --> 00:03:57.850
And we're done.


Slope from two ordered pairs example 1

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.850
Find the slope of
the line that goes

00:00:02.850 --> 00:00:07.450
through the ordered pairs
4 comma 2 and negative 3

00:00:07.450 --> 00:00:08.640
comma 16.

00:00:08.640 --> 00:00:16.270
So just as a reminder, slope
is defined as rise over run.

00:00:16.270 --> 00:00:20.700
Or, you could view that rise
is just change in y and run

00:00:20.700 --> 00:00:22.520
is just change in x.

00:00:22.520 --> 00:00:24.920
The triangles here,
that's the delta symbol.

00:00:24.920 --> 00:00:26.810
It literally means "change in."

00:00:26.810 --> 00:00:29.097
Or another way, and you
might see this formula,

00:00:29.097 --> 00:00:30.680
and it tends to be
really complicated.

00:00:30.680 --> 00:00:33.310
But just remember it's just
these two things over here.

00:00:33.310 --> 00:00:37.740
Sometimes, slope will be
specified with the variable m.

00:00:37.740 --> 00:00:40.099
And they'll say that
m is the same thing--

00:00:40.099 --> 00:00:42.140
and this is really the
same thing as change in y.

00:00:42.140 --> 00:00:48.520
They'll write y2 minus
y1 over x2 minus x1.

00:00:48.520 --> 00:00:51.180
And this notation tends
to be kind of complicated,

00:00:51.180 --> 00:00:53.290
but all this means
is, is you take

00:00:53.290 --> 00:00:56.180
the y-value of your endpoint
and subtract from it

00:00:56.180 --> 00:00:58.010
the y-value of your
starting point.

00:00:58.010 --> 00:01:00.430
That will essentially
give you your change in y.

00:01:00.430 --> 00:01:03.840
And it says take the
x-value of your endpoint

00:01:03.840 --> 00:01:07.140
and subtract from that the
x-value of your starting point.

00:01:07.140 --> 00:01:08.620
And that'll give
you change in x.

00:01:08.620 --> 00:01:11.210
So whatever of
these work for you,

00:01:11.210 --> 00:01:13.970
let's actually figure out
the slope of the line that

00:01:13.970 --> 00:01:16.360
goes through these two points.

00:01:16.360 --> 00:01:18.484
So we're starting
at-- and actually, we

00:01:18.484 --> 00:01:19.400
could do it both ways.

00:01:19.400 --> 00:01:21.260
We could start at this
point and go to that point

00:01:21.260 --> 00:01:23.551
and calculate the slope or
we could start at this point

00:01:23.551 --> 00:01:25.950
and go to that point
and calculate the slope.

00:01:25.950 --> 00:01:27.350
So let's do it both ways.

00:01:27.350 --> 00:01:35.230
So let's say that our starting
point is the point 4 comma 2.

00:01:35.230 --> 00:01:42.980
And let's say that our endpoint
is negative 3 comma 16.

00:01:42.980 --> 00:01:45.160
So what is the change
in x over here?

00:01:45.160 --> 00:01:47.750
What is the change in
x in this scenario?

00:01:47.750 --> 00:01:50.620
So we're going from
4 to negative 3.

00:01:50.620 --> 00:01:52.550
If something goes
from 4 to negative 3,

00:01:52.550 --> 00:01:54.850
what was it's change?

00:01:54.850 --> 00:01:57.740
You have to go
down 4 to get to 0,

00:01:57.740 --> 00:02:00.650
and then you have to go down
another 3 to get to negative 3.

00:02:00.650 --> 00:02:05.460
So our change in x
here is negative 7.

00:02:05.460 --> 00:02:09.850
Actually, let me
write it this way.

00:02:09.850 --> 00:02:15.030
Our change in x is equal
to negative 3 minus 4,

00:02:15.030 --> 00:02:18.280
which is equal to negative 7.

00:02:18.280 --> 00:02:22.570
If I'm going from 4 to
negative 3, I went down by 7.

00:02:22.570 --> 00:02:25.370
Our change in x is negative 7.

00:02:25.370 --> 00:02:27.680
Let's do the same thing
for the change in y.

00:02:27.680 --> 00:02:30.110
And notice, I implicitly
use this formula over here.

00:02:30.110 --> 00:02:36.160
Our change in x was this value,
our endpoint, our end x-value

00:02:36.160 --> 00:02:38.790
minus our starting x-value.

00:02:38.790 --> 00:02:41.490
Let's do the same thing
for our change in y.

00:02:41.490 --> 00:02:43.510
Our change in y.

00:02:43.510 --> 00:02:46.390
If we're starting at
2 and we go to 16,

00:02:46.390 --> 00:02:48.681
that means we moved up 14.

00:02:48.681 --> 00:02:50.180
Or another way you
could say it, you

00:02:50.180 --> 00:02:53.110
could take your ending
y-value and subtract from that

00:02:53.110 --> 00:02:56.530
your starting y-value
and you get 14.

00:02:56.530 --> 00:02:58.490
So what is the slope over here?

00:02:58.490 --> 00:03:02.200
Well, the slope is just
change in y over change in x.

00:03:02.200 --> 00:03:05.310
So the slope over
here is change in y

00:03:05.310 --> 00:03:10.590
over change in x, which
is-- our change in y is 14.

00:03:10.590 --> 00:03:13.990
And our change in
x is negative 7.

00:03:13.990 --> 00:03:17.540
And then if we want to simplify
this, 14 divided by negative 7

00:03:17.540 --> 00:03:21.302
is negative 2.

00:03:21.302 --> 00:03:22.760
Now, what I want
to show you is, is

00:03:22.760 --> 00:03:24.760
that we could have done
it the other way around.

00:03:24.760 --> 00:03:26.740
We could have made
this the starting point

00:03:26.740 --> 00:03:28.000
and this the endpoint.

00:03:28.000 --> 00:03:30.390
And what we would have
gotten is the negative values

00:03:30.390 --> 00:03:32.598
of each of these, but then
they would've canceled out

00:03:32.598 --> 00:03:34.290
and we would still
get negative 2.

00:03:34.290 --> 00:03:35.960
Let's try it out.

00:03:35.960 --> 00:03:40.990
So let's say that our start
point was negative 3 comma 16.

00:03:40.990 --> 00:03:45.170
And let's say that our
endpoint is the 4 comma 2.

00:03:45.170 --> 00:03:46.700
4 comma 2.

00:03:46.700 --> 00:03:49.700
So in this situation,
what is our change in x?

00:03:49.700 --> 00:03:51.670
Our change in x.

00:03:51.670 --> 00:03:54.370
If I start at negative
3 and I go to 4,

00:03:54.370 --> 00:03:55.990
that means I went up 7.

00:03:55.990 --> 00:03:57.720
Or if you want to
just calculate that,

00:03:57.720 --> 00:04:00.450
you would do 4 minus negative 3.

00:04:00.450 --> 00:04:02.730
4 minus negative 3.

00:04:02.730 --> 00:04:05.760
But needless to say,
we just went up 7.

00:04:05.760 --> 00:04:08.010
And what is our change in y?

00:04:08.010 --> 00:04:12.280
Our change in y over here,
or we could say our rise.

00:04:12.280 --> 00:04:17.720
If we start at 16 and we end at
2, that means we went down 14.

00:04:17.720 --> 00:04:22.140
Or you could just say 2
minus 16 is negative 14.

00:04:22.140 --> 00:04:23.570
We went down by 14.

00:04:23.570 --> 00:04:24.900
This was our run.

00:04:24.900 --> 00:04:28.180
So if you say rise
over run, which

00:04:28.180 --> 00:04:30.350
is the same thing as change
in y over change in x,

00:04:30.350 --> 00:04:34.385
our rise is negative 14
and our run here is 7.

00:04:34.385 --> 00:04:36.010
So notice, these are
just the negatives

00:04:36.010 --> 00:04:38.870
of these values from
when we swapped them.

00:04:38.870 --> 00:04:42.560
So once again, this is
equal to negative 2.

00:04:42.560 --> 00:04:43.820
And let's just visualize this.

00:04:43.820 --> 00:04:46.810
Let me do a quick
graph here just

00:04:46.810 --> 00:04:50.560
to show you what a downward
slope would look like.

00:04:50.560 --> 00:04:53.410
So let me draw our two points.

00:04:53.410 --> 00:04:55.620
So this is my x-axis.

00:04:55.620 --> 00:04:57.620
That is my y-axis.

00:04:57.620 --> 00:05:00.310
So this point over
here, 4 comma 2.

00:05:00.310 --> 00:05:01.910
So let me graph it.

00:05:01.910 --> 00:05:03.690
So we're going to go
all the way up to 16.

00:05:03.690 --> 00:05:04.981
So let me save some space here.

00:05:04.981 --> 00:05:08.030
So we have 1, 2, 3, 4.

00:05:08.030 --> 00:05:10.970
It's 4 comma-- 1, 2.

00:05:10.970 --> 00:05:12.950
So 4 comma 2 is right over here.

00:05:12.950 --> 00:05:14.750
4 comma 2.

00:05:14.750 --> 00:05:17.421
Then we have the point
negative 3 comma 16.

00:05:17.421 --> 00:05:18.670
So let me draw that over here.

00:05:18.670 --> 00:05:21.170
So we have negative 1, 2, 3.

00:05:21.170 --> 00:05:22.790
And we have to go up 16.

00:05:22.790 --> 00:05:30.630
So this is 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16.

00:05:30.630 --> 00:05:32.690
So it goes right over here.

00:05:32.690 --> 00:05:35.660
So this is negative 3 comma 16.

00:05:35.660 --> 00:05:37.890
Negative 3 comma 16.

00:05:37.890 --> 00:05:39.610
So the line that
goes between them

00:05:39.610 --> 00:05:42.020
is going to look
something like this.

00:05:42.020 --> 00:05:47.120
Try my best to draw a
relatively straight line.

00:05:47.120 --> 00:05:49.300
That line will keep going.

00:05:49.300 --> 00:05:51.260
So the line will keep going.

00:05:51.260 --> 00:05:52.820
So that's my best attempt.

00:05:52.820 --> 00:05:55.000
And now notice, it's
downward sloping.

00:05:55.000 --> 00:05:57.980
As you increase an x-value,
the line goes down.

00:05:57.980 --> 00:06:00.800
It's going from the top
left to the bottom right.

00:06:00.800 --> 00:06:03.430
As x gets bigger,
y gets smaller.

00:06:03.430 --> 00:06:06.100
That's what a downward-sloping
line looks like.

00:06:06.100 --> 00:06:08.650
And just to visualize our
change in x's and our change

00:06:08.650 --> 00:06:13.630
in y's that we dealt with
here, when we started at 4

00:06:13.630 --> 00:06:16.200
and we ended at-- or when
we started at 4 comma 2

00:06:16.200 --> 00:06:18.660
and ended at
negative 3 comma 16,

00:06:18.660 --> 00:06:21.810
that was analogous to starting
here and ending over there.

00:06:21.810 --> 00:06:25.420
And we said our change
in x was negative 7.

00:06:25.420 --> 00:06:26.790
We had to move back.

00:06:26.790 --> 00:06:30.680
Our run we had to move in
the left direction by 7.

00:06:30.680 --> 00:06:32.720
That's why it was a negative 7.

00:06:32.720 --> 00:06:34.750
And then we had to move
in the y-direction.

00:06:34.750 --> 00:06:37.550
We had to move in the
y-direction positive 14.

00:06:37.550 --> 00:06:39.860
So that's why our
rise was positive.

00:06:39.860 --> 00:06:42.470
So it's 14 over negative
7, or negative 2.

00:06:42.470 --> 00:06:45.880
When we did it the other way,
we started at this point.

00:06:45.880 --> 00:06:48.330
We started at this point,
and then ended at this point.

00:06:48.330 --> 00:06:51.410
Started at negative 3, 16
and ended at that point.

00:06:51.410 --> 00:06:56.450
So in that situation,
our run was positive 7.

00:06:56.450 --> 00:06:58.980
And now we have to go
down in the y-direction

00:06:58.980 --> 00:07:00.980
since we switched the
starting and the endpoint.

00:07:00.980 --> 00:07:03.880
And now we had to
go down negative 14.

00:07:03.880 --> 00:07:07.900
Our run is now positive 7 and
our rise is now negative 14.

00:07:07.900 --> 00:07:11.204
Either way, we got
the same slope.


Ordered pair solutions to equations

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.390
Is 3 comma negative 4 a
solution to the equation

00:00:03.390 --> 00:00:06.060
5x plus 2y is equal to 7?

00:00:06.060 --> 00:00:09.270
So they're saying, does x
equal 3, y equal negative 4,

00:00:09.270 --> 00:00:12.600
satisfy this equation, or
this relationship right here?

00:00:12.600 --> 00:00:15.080
So one way to do it is just
to substitute x is equal to 3

00:00:15.080 --> 00:00:16.670
and y equals
negative 4 into this

00:00:16.670 --> 00:00:21.230
and see if 5 times x plus 2
times y does indeed equal 7.

00:00:21.230 --> 00:00:26.660
So we have 5 times 3
plus 2 times negative 4.

00:00:26.660 --> 00:00:29.456
This is equal to 15.

00:00:29.456 --> 00:00:34.240
15 plus negative 8, which
does indeed equal 7.

00:00:34.240 --> 00:00:36.280
So it does satisfy the equation.

00:00:36.280 --> 00:00:37.970
So it is on the line.

00:00:37.970 --> 00:00:39.120
It is a solution.

00:00:39.120 --> 00:00:43.109
x equals 3, y equals negative 4
is a solution to this equation.

00:00:43.109 --> 00:00:44.900
So we've essentially
answered our question.

00:00:44.900 --> 00:00:45.960
It is.

00:00:45.960 --> 00:00:47.614
Now, another way to
do it, and I'm not

00:00:47.614 --> 00:00:49.030
going to go into
the details here,

00:00:49.030 --> 00:00:50.820
is you could actually
graph the line.

00:00:50.820 --> 00:00:52.870
So maybe the line might
look something like this,

00:00:52.870 --> 00:00:54.244
I'm not going to
do it in detail.

00:00:54.244 --> 00:00:57.400
And you see, if you have
a very good drawing of it,

00:00:57.400 --> 00:00:59.570
you see whether the
point lies on the line.

00:00:59.570 --> 00:01:02.310
If the point, when you graph
the point, does lie on the line,

00:01:02.310 --> 00:01:03.490
it would be a solution.

00:01:03.490 --> 00:01:06.080
If the point somehow ends
up not being on the line

00:01:06.080 --> 00:01:07.650
then you'd know it
isn't a solution.

00:01:07.650 --> 00:01:10.066
But to do this, you would have
to have a very good drawing

00:01:10.066 --> 00:01:12.250
and so you could very
precisely determine

00:01:12.250 --> 00:01:13.540
whether it's on the line.

00:01:13.540 --> 00:01:15.030
If you do the
substitution method,

00:01:15.030 --> 00:01:17.360
if you just substitute the
values into the equation

00:01:17.360 --> 00:01:19.010
and see if it comes
out mathematically,

00:01:19.010 --> 00:01:20.690
this will always be exact.

00:01:20.690 --> 00:01:24.890
So this is all we really
had to do in this example.

00:01:24.890 --> 00:01:28.500
So it definitely is a
solution to the equation.


Slope from two ordered pairs example 2

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.140
Find the slope of the line
that goes through the ordered

00:00:03.140 --> 00:00:07.600
pairs 7, negative 1 and
negative 3, negative 1.

00:00:07.600 --> 00:00:09.900
Let me just do a
quick graph of these

00:00:09.900 --> 00:00:12.830
just so we can visualize
what they look like.

00:00:12.830 --> 00:00:19.180
So let me draw a
quick graph over here.

00:00:19.180 --> 00:00:22.200
So our first point
is 7, negative 1.

00:00:22.200 --> 00:00:25.330
So 1, 2, 3, 4, 5, 6, 7.

00:00:25.330 --> 00:00:27.030
This is the x-axis.

00:00:27.030 --> 00:00:28.640
7, negative 1.

00:00:28.640 --> 00:00:31.300
So it's 7, negative 1
is right over there.

00:00:31.300 --> 00:00:33.800
7, negative 1.

00:00:33.800 --> 00:00:36.070
This, of course, is the y-axis.

00:00:36.070 --> 00:00:39.680
And then the next point
is negative 3, negative 1.

00:00:39.680 --> 00:00:42.710
So we go back 3 in the
horizontal direction.

00:00:42.710 --> 00:00:49.140
Negative 3 for the y-coordinate
is still negative 1.

00:00:49.140 --> 00:00:53.600
So the line that connects these
two points will look like this.

00:00:53.600 --> 00:00:56.830
It will look like that.

00:00:56.830 --> 00:00:59.370
Now, they're asking us to find
the slope of the line that

00:00:59.370 --> 00:01:00.661
goes through the ordered pairs.

00:01:00.661 --> 00:01:01.890
Find the slope of this line.

00:01:01.890 --> 00:01:03.890
And just to give a little
bit of intuition here,

00:01:03.890 --> 00:01:06.980
slope is a measure of
a line's inclination.

00:01:06.980 --> 00:01:09.170
And the way that
it's defined-- slope

00:01:09.170 --> 00:01:18.610
is defined as rise over run, or
change in y over change in x,

00:01:18.610 --> 00:01:24.630
or sometimes you'll see it
defined as the variable m.

00:01:24.630 --> 00:01:26.800
And then they'll
define change in y

00:01:26.800 --> 00:01:29.640
as just being the
second y-coordinate

00:01:29.640 --> 00:01:31.750
minus the first
y-coordinate and then

00:01:31.750 --> 00:01:34.430
the change in x as the
second x-coordinate

00:01:34.430 --> 00:01:35.980
minus the first x-coordinate.

00:01:35.980 --> 00:01:37.814
These are all different
variations in slope,

00:01:37.814 --> 00:01:39.563
but hopefully you'll
appreciate that these

00:01:39.563 --> 00:01:40.800
are measuring inclination.

00:01:40.800 --> 00:01:43.530
If I rise a ton when
I run a little bit,

00:01:43.530 --> 00:01:46.800
if I move a little bit in the x
direction, and I rise a bunch,

00:01:46.800 --> 00:01:48.360
then I have a very steep line.

00:01:48.360 --> 00:01:50.510
I have a very steep
upward-sloping line.

00:01:50.510 --> 00:01:54.190
If I don't change at
all when I run a bit,

00:01:54.190 --> 00:01:55.630
then I have a very low slope.

00:01:55.630 --> 00:01:57.380
And that's actually
what's happening here.

00:01:57.380 --> 00:01:59.910
I'm going from-- you
could either view

00:01:59.910 --> 00:02:01.470
this as the starting
point or view

00:02:01.470 --> 00:02:02.600
this as the starting point.

00:02:02.600 --> 00:02:04.350
But let's view this
as the starting point.

00:02:04.350 --> 00:02:06.450
So this negative 3, 1.

00:02:06.450 --> 00:02:13.340
If I go from negative 3,
negative 1 to 7, negative 1,

00:02:13.340 --> 00:02:14.820
I'm running a good bit.

00:02:14.820 --> 00:02:16.190
I'm going from negative 3.

00:02:16.190 --> 00:02:19.630
My x value is negative 3 here,
and it goes all the way to 7.

00:02:19.630 --> 00:02:24.250
So my change in x here is 10.

00:02:24.250 --> 00:02:27.890
To go from negative 3 to 7,
I changed my x value by 10.

00:02:27.890 --> 00:02:29.710
But what's my change in y?

00:02:29.710 --> 00:02:32.490
Well, my y value
here is negative 1,

00:02:32.490 --> 00:02:35.030
and my y value over here
is still negative 1.

00:02:35.030 --> 00:02:37.950
So my change in y is a 0.

00:02:37.950 --> 00:02:40.570
My change in y is going to be 0.

00:02:40.570 --> 00:02:42.910
My y value does not
change no matter

00:02:42.910 --> 00:02:45.380
how much I change my x value.

00:02:45.380 --> 00:02:49.910
So the slope here
is going to be--

00:02:49.910 --> 00:02:54.040
when we run 10,
what was our rise?

00:02:54.040 --> 00:02:55.840
How much did we change in y?

00:02:55.840 --> 00:02:57.140
Well, we didn't rise at all.

00:02:57.140 --> 00:02:58.780
We didn't go up or down.

00:02:58.780 --> 00:03:01.220
So the slope here is 0.

00:03:01.220 --> 00:03:04.160
Or another way to think about
is this line has no inclination.

00:03:04.160 --> 00:03:08.620
It's a completely flat-- it's
a completely horizontal line.

00:03:08.620 --> 00:03:10.240
So this should make sense.

00:03:10.240 --> 00:03:12.230
This is a 0.

00:03:12.230 --> 00:03:14.570
The slope here is 0.

00:03:14.570 --> 00:03:16.090
And just to make
sure that this gels

00:03:16.090 --> 00:03:18.380
with all of these other
formulas that you might know--

00:03:18.380 --> 00:03:19.755
but I want to make
it very clear.

00:03:19.755 --> 00:03:21.860
These are all just
telling you rise over run

00:03:21.860 --> 00:03:25.750
or change in y over change in
x, a way to measure inclination.

00:03:25.750 --> 00:03:27.370
But let's just
apply them just so,

00:03:27.370 --> 00:03:28.970
hopefully, it all
makes sense to you.

00:03:28.970 --> 00:03:33.010
So we could also say slope is
change in y over change in x.

00:03:33.010 --> 00:03:37.170
If we take this to be
our start and if we

00:03:37.170 --> 00:03:39.520
take this to be our
end point, then we

00:03:39.520 --> 00:03:43.176
would call this over here x1.

00:03:43.176 --> 00:03:44.300
And then this is over here.

00:03:44.300 --> 00:03:45.550
This is y1.

00:03:45.550 --> 00:03:47.990
And then we would
call this x2 and we

00:03:47.990 --> 00:03:51.310
would call this y2, if
this is our start point

00:03:51.310 --> 00:03:52.860
and that is our end point.

00:03:52.860 --> 00:03:56.960
And so the slope here, the
change in y, y2 minus y1.

00:03:56.960 --> 00:04:02.880
So it's negative 1
minus negative 1,

00:04:02.880 --> 00:04:10.600
all of that over x2, negative
3, minus x1, minus 7.

00:04:10.600 --> 00:04:13.520
So the numerator, negative
1 minus negative 1,

00:04:13.520 --> 00:04:16.589
that's the same thing
as negative 1 plus 1.

00:04:16.589 --> 00:04:19.700
And our denominator
is negative 3 minus 7,

00:04:19.700 --> 00:04:22.200
which is negative 10.

00:04:22.200 --> 00:04:26.980
So once again, negative 1
plus 1 is 0 over negative 10.

00:04:26.980 --> 00:04:29.329
And this is still going to be 0.

00:04:29.329 --> 00:04:31.370
And the only reason why
we got a negative 10 here

00:04:31.370 --> 00:04:33.850
and a positive 10
there is because we

00:04:33.850 --> 00:04:36.110
swapped the starting
and the ending point.

00:04:36.110 --> 00:04:39.820
In this example right over here,
we took this as the start point

00:04:39.820 --> 00:04:42.870
and made this coordinate
over here as the end point.

00:04:42.870 --> 00:04:44.830
Over here, we
swapped them around.

00:04:44.830 --> 00:04:47.460
7, negative 1 was
our start point,

00:04:47.460 --> 00:04:49.880
and negative 3, negative
1 is our end point.

00:04:49.880 --> 00:04:54.170
So if we start over
here, our change in x

00:04:54.170 --> 00:04:56.070
is going to be negative 10.

00:04:56.070 --> 00:04:58.170
But our change in y is
still going to be 0.

00:04:58.170 --> 00:05:01.370
So regardless of how you do it,
the slope of this line is 0.

00:05:01.370 --> 00:05:03.866
It's a horizontal line.


Slope of a line: negative slope

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:04.340
Find the slope of the line
pictured on the graph.

00:00:04.340 --> 00:00:11.160
So the slope of a line is
defined to be rise over run.

00:00:11.160 --> 00:00:22.020
Or you could also view it as
change in y over change in x.

00:00:22.020 --> 00:00:25.450
And let me show you
what that means.

00:00:25.450 --> 00:00:28.120
So let's start at some
arbitrary point on this line,

00:00:28.120 --> 00:00:30.320
and they highlight
some of these points.

00:00:30.320 --> 00:00:32.570
So let's start at one of
these points right over here.

00:00:32.570 --> 00:00:35.020
So if we wanted to start one
of these points-- and let's

00:00:35.020 --> 00:00:37.840
say we want to change our x
in the positive direction.

00:00:37.840 --> 00:00:40.220
So we want to go to the right.

00:00:40.220 --> 00:00:42.160
So let's say we want
to go from this point

00:00:42.160 --> 00:00:43.360
to this point over here.

00:00:43.360 --> 00:00:45.830
How much do we
have to move in x?

00:00:45.830 --> 00:00:48.277
So if we want to move in x,
we have to go from this point

00:00:48.277 --> 00:00:48.860
to this point.

00:00:48.860 --> 00:00:51.970
We're going from
negative 3 to 0.

00:00:55.580 --> 00:00:59.370
So our change in x-- and
this triangle, that's delta.

00:00:59.370 --> 00:01:00.580
That means "change in."

00:01:00.580 --> 00:01:03.600
Our change in x is equal to 3.

00:01:07.960 --> 00:01:12.830
So what was our change in y when
our change in x is equal to 3?

00:01:12.830 --> 00:01:16.840
Well, when we moved from
this point to this point,

00:01:16.840 --> 00:01:20.370
our x-value changed by 3, but
what happened to our y-value?

00:01:20.370 --> 00:01:22.140
Well, our y-value went down.

00:01:22.140 --> 00:01:24.840
It went from positive
3 to positive 2.

00:01:24.840 --> 00:01:27.880
Our y-value went down by 1.

00:01:27.880 --> 00:01:32.350
So our change in y is
equal to negative 1.

00:01:32.350 --> 00:01:34.290
So we rose negative 1.

00:01:34.290 --> 00:01:35.570
We actually went down.

00:01:35.570 --> 00:01:38.430
So our rise is
negative 1 when our

00:01:38.430 --> 00:01:40.890
run-- when our
change in x-- is 3.

00:01:40.890 --> 00:01:44.850
So change in y over change
in x is negative 1 over 3,

00:01:44.850 --> 00:01:50.710
or we could say that our
slope is negative 1/3.

00:01:50.710 --> 00:01:52.800
Let me scroll over a little bit.

00:01:52.800 --> 00:01:55.270
It is negative 1/3.

00:01:55.270 --> 00:01:57.980
And I want to show you that we
can do this with any two points

00:01:57.980 --> 00:01:59.010
on the line.

00:01:59.010 --> 00:02:01.990
We could even go further
than 3 in the x-direction.

00:02:01.990 --> 00:02:03.190
So let's go the other way.

00:02:03.190 --> 00:02:04.690
Let's start at this
point right over

00:02:04.690 --> 00:02:07.550
here and then move backwards
to this point over here,

00:02:07.550 --> 00:02:10.180
just to show you that we'll
still get the same result.

00:02:10.180 --> 00:02:14.780
So to go from this point to that
point, what is our change in x?

00:02:14.780 --> 00:02:17.870
So our change in x is
this right over here.

00:02:17.870 --> 00:02:21.260
Our change in x is that
distance right over there.

00:02:21.260 --> 00:02:24.120
We started at 3, and
we went to negative 3.

00:02:24.120 --> 00:02:26.000
We went back 6.

00:02:26.000 --> 00:02:29.840
Over here, our change in
x is equal to negative 6.

00:02:29.840 --> 00:02:31.800
We're starting at
this point now.

00:02:31.800 --> 00:02:35.780
So over here our change
in x is negative 6.

00:02:35.780 --> 00:02:37.970
And then when our change
in x is negative 6, when

00:02:37.970 --> 00:02:39.660
we start at this
point and we move 6

00:02:39.660 --> 00:02:43.330
back, what is our change
of y to get to that point?

00:02:43.330 --> 00:02:45.570
Well, our y-value went from 1.

00:02:45.570 --> 00:02:47.290
That was our y-value
at this point.

00:02:47.290 --> 00:02:49.955
And then when we go back to
this point, our y-value is 3.

00:02:52.901 --> 00:02:53.650
So what did we do?

00:02:53.650 --> 00:02:55.160
We moved up by 2.

00:02:55.160 --> 00:02:57.775
Our change in y is equal to 2.

00:03:02.050 --> 00:03:06.890
Slope is change in y over
change in x, or rise over run.

00:03:06.890 --> 00:03:08.690
Change in y is just rise.

00:03:08.690 --> 00:03:10.580
Change in x is
just run, how much

00:03:10.580 --> 00:03:13.480
you're moving in the
horizontal direction.

00:03:13.480 --> 00:03:16.240
So rise over run in this
example right over here

00:03:16.240 --> 00:03:19.880
is going to be 2
over negative 6,

00:03:19.880 --> 00:03:23.229
which is the same
thing as negative 1/3.

00:03:23.229 --> 00:03:24.770
And you could verify
it for yourself.

00:03:24.770 --> 00:03:26.228
Take any of these
two points, start

00:03:26.228 --> 00:03:28.150
at one of these two
points, and figure out

00:03:28.150 --> 00:03:30.870
what is the run to
get to the next point,

00:03:30.870 --> 00:03:33.130
and then what is the rise
to get the next point.

00:03:33.130 --> 00:03:35.149
And for any line, the
slope won't change.

00:03:35.149 --> 00:03:35.940
Let me do it again.

00:03:35.940 --> 00:03:39.320
Over here, we had to move in the
positive 3 direction, so that

00:03:39.320 --> 00:03:41.370
is our run.

00:03:41.370 --> 00:03:44.420
So this right here
is positive 3.

00:03:44.420 --> 00:03:45.100
That's our run.

00:03:45.100 --> 00:03:46.377
But what's our rise?

00:03:46.377 --> 00:03:48.710
Well, we actually went down,
so we have a negative rise.

00:03:48.710 --> 00:03:50.270
Our rise is negative 1.

00:03:50.270 --> 00:03:52.420
So we have negative
1 as our rise.

00:03:52.420 --> 00:03:53.490
We went down.

00:03:53.490 --> 00:03:55.340
And our run was positive 3.

00:03:55.340 --> 00:03:58.345
So our slope here
is negative 1/3.


Slope-intercept equation from slope and point

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:04.420
A line has a slope of negative
3/4 and goes through the point

00:00:04.420 --> 00:00:05.950
0 comma 8.

00:00:05.950 --> 00:00:12.060
What is the equation of this
line in slope-intercept form?

00:00:12.060 --> 00:00:15.570
So any line can be represented
in slope-intercept form, is

00:00:15.570 --> 00:00:22.680
y is equal to mx plus b,
where this m right over here,

00:00:22.680 --> 00:00:25.530
that is of the
slope of the line.

00:00:25.530 --> 00:00:31.870
And this b over here, this is
the y-intercept of the line.

00:00:31.870 --> 00:00:34.500
Let me draw a quick
line here just so

00:00:34.500 --> 00:00:36.790
that we can visualize
that a little bit.

00:00:36.790 --> 00:00:39.580
So that is my y-axis.

00:00:39.580 --> 00:00:41.826
And then that is my x-axis.

00:00:44.312 --> 00:00:45.270
And let me draw a line.

00:00:45.270 --> 00:00:47.630
And since our line here
has a negative slope,

00:00:47.630 --> 00:00:49.560
I'll draw a downward
sloping line.

00:00:49.560 --> 00:00:51.995
So let's say our line
looks something like that.

00:00:55.720 --> 00:00:57.890
So hopefully, we're a little
familiar with the slope

00:00:57.890 --> 00:00:58.390
already.

00:00:58.390 --> 00:00:59.931
The slope essentially
tells us, look,

00:00:59.931 --> 00:01:02.960
start at some point
on the line, and go

00:01:02.960 --> 00:01:05.200
to some other point
of the line, measure

00:01:05.200 --> 00:01:08.846
how much you had to move in the
x direction, that is your run,

00:01:08.846 --> 00:01:10.220
and then measure
how much you had

00:01:10.220 --> 00:01:13.440
to move in the y direction,
that is your rise.

00:01:13.440 --> 00:01:18.870
And our slope is equal
to rise over run.

00:01:18.870 --> 00:01:21.190
And you can see over here,
we'd be downward sloping.

00:01:21.190 --> 00:01:23.820
Because if you move in
the positive x direction,

00:01:23.820 --> 00:01:25.510
we have to go down.

00:01:25.510 --> 00:01:28.052
If our run is positive,
our rise here is negative.

00:01:28.052 --> 00:01:29.885
So this would be a
negative over a positive,

00:01:29.885 --> 00:01:31.400
it would give you
a negative number.

00:01:31.400 --> 00:01:33.870
That makes sense, because
we're downward sloping.

00:01:33.870 --> 00:01:37.100
The more we go down
in this situation,

00:01:37.100 --> 00:01:38.950
for every step we
move to the right,

00:01:38.950 --> 00:01:40.490
the more downward
sloping will be,

00:01:40.490 --> 00:01:44.190
the more of a negative
slope we'll have.

00:01:44.190 --> 00:01:46.230
So that's slope right over here.

00:01:46.230 --> 00:01:51.220
The y-intercept just tells us
where we intercept the y-axis.

00:01:51.220 --> 00:01:53.640
So the y-intercept, this
point right over here,

00:01:53.640 --> 00:01:57.690
this is where the line
intersects with the y-axis.

00:01:57.690 --> 00:02:00.830
This will be the
point 0 comma b.

00:02:00.830 --> 00:02:03.310
And this actually just falls
straight out of this equation.

00:02:03.310 --> 00:02:05.730
When x is equal to
0-- so let's evaluate

00:02:05.730 --> 00:02:07.920
this equation, when
x is equal to 0.

00:02:07.920 --> 00:02:13.200
y will be equal to
m times 0 plus b.

00:02:13.200 --> 00:02:15.770
Well, anything times 0 is 0.

00:02:15.770 --> 00:02:18.560
So y is equal to 0
plus b, or y will

00:02:18.560 --> 00:02:21.860
be equal to b, when
x is equal to 0.

00:02:27.370 --> 00:02:30.660
So this is the point 0 comma b.

00:02:30.660 --> 00:02:33.860
Now, they tell us what
the slope of this line is.

00:02:33.860 --> 00:02:37.930
They tell us a line has
a slope of negative 3/4.

00:02:37.930 --> 00:02:41.112
So we know that our
slope is negative 3/4,

00:02:41.112 --> 00:02:43.320
and they tell us that the
line goes through the point

00:02:43.320 --> 00:02:45.900
0 comma 8.

00:02:45.900 --> 00:02:49.100
They tell us we go through the--
Let me just, in a new color.

00:02:49.100 --> 00:02:51.950
I've already used orange,
let me use this green color.

00:02:51.950 --> 00:02:56.010
They tell us what we go
through the point 0 comma 8.

00:02:56.010 --> 00:02:58.160
Notice, x is 0.

00:02:58.160 --> 00:02:59.750
So we're on the y-axis.

00:02:59.750 --> 00:03:02.510
When x is 0, we're
on the y-axis.

00:03:02.510 --> 00:03:04.720
So this is our y-intercept.

00:03:04.720 --> 00:03:10.060
So b, we could say-- we could
do a couple-- our y-intercept is

00:03:10.060 --> 00:03:14.630
the point 0 comma 8, or we
could say that b-- Remember,

00:03:14.630 --> 00:03:16.240
it's also 0 comma b.

00:03:16.240 --> 00:03:18.710
We could say b is equal to 8.

00:03:18.710 --> 00:03:21.860
So we know m is equal to
negative 3/4, b is equal to 8,

00:03:21.860 --> 00:03:24.580
so we can write the
equation of this line

00:03:24.580 --> 00:03:26.200
in slope-intercept form.

00:03:26.200 --> 00:03:39.540
It's y is equal to negative
3/4 times x plus b, plus 8.

00:03:39.540 --> 00:03:42.780
And we are done.


Slope-intercept equation from two solutions example

WEBVTT
Kind: captions
Language: en

00:00:00.333 --> 00:00:03.800
A line goes through the points (-1, 6)

00:00:03.800 --> 00:00:08.333
and (5, -4). What is the equation of the line?

00:00:08.333 --> 00:00:12.133
Let's just try to visualize this. So that is my

00:00:12.133 --> 00:00:14.800
x axis. And you don't have to draw it

00:00:14.800 --> 00:00:17.200
to do this problem but it always help to visualize

00:00:17.200 --> 00:00:23.733
That is my y axis. And the first point is (-1,6)

00:00:23.733 --> 00:00:29.400
So (-1, 6). So negative 1 coma, 1, 2, 3, 4 ,5 6.

00:00:29.400 --> 00:00:33.400
So it's this point, rigth over there, it's (-1, 6).

00:00:33.400 --> 00:00:38.400
And the other point is (5, -4). So 1, 2, 3, 4, 5.

00:00:38.400 --> 00:00:42.133
And we go down 4, So 1, 2, 3, 4

00:00:42.133 --> 00:00:46.400
So it's right over there. So the line connects them

00:00:46.400 --> 00:00:49.000
will looks something like this. Line will

00:00:49.000 --> 00:00:52.333
draw a rough approximation.

00:00:52.333 --> 00:00:55.467
I can draw a straighter than that. I will draw a dotted line maybe

00:00:55.467 --> 00:00:57.533
Easier do dotted line.

00:00:57.533 --> 00:00:59.733
So the line will looks something like that.

00:00:59.733 --> 00:01:02.600
So let's find its equation. So good place to start

00:01:02.600 --> 00:01:04.000
is we can find its slope.

00:01:04.000 --> 00:01:06.000
Remember, we want, we can find the equation

00:01:06.000 --> 00:01:08.801
y is equal to mx plus b.

00:01:08.801 --> 00:01:12.733
This is the slope-intercept form where m is the slope and b is the y-intercept.

00:01:12.733 --> 00:01:16.467
We can first try to solve for m. We can find the slope of this line.

00:01:16.467 --> 00:01:21.133
So m, or the slope is the change in y over the change in x.

00:01:21.133 --> 00:01:26.216
Or, we can view it as the y value of our end point

00:01:26.216 --> 00:01:29.467
minus the y value of our starting point over

00:01:29.467 --> 00:01:33.467
the x-value of our end point minus the x-value of our starting point.

00:01:33.467 --> 00:01:36.000
Let me make that clear. So this is equal to

00:01:36.000 --> 00:01:41.267
change in y over change in x wich is the same thing as

00:01:41.267 --> 00:01:45.867
rise over run wich is the same thing as the y-value of

00:01:45.867 --> 00:01:49.267
your ending point minus the y-value of your starting point.

00:01:49.267 --> 00:01:53.400
This is the same exact thing as change in y and that over

00:01:53.400 --> 00:01:57.200
the x value of your ending point minus the x-value of your starting point

00:01:57.200 --> 00:01:59.933
This is the exact same thing as change in x.

00:01:59.933 --> 00:02:02.533
And you just have to pick one of these as the starting point

00:02:02.533 --> 00:02:05.411
and one as the ending point. So let's just make this over here

00:02:05.411 --> 00:02:07.467
our starting point and make that our ending point.

00:02:07.467 --> 00:02:10.400
So what is our change in y?

00:02:10.400 --> 00:02:14.049
So our change in y, to go we started at y

00:02:14.049 --> 00:02:18.000
is equal to six, we started at y is equal to 6.

00:02:18.000 --> 00:02:21.133
And we go down all the way to y is equal to negative 4

00:02:21.133 --> 00:02:24.733
So this is rigth here, that is our change in y

00:02:24.733 --> 00:02:26.634
You can look at the graph and say, oh,

00:02:26.634 --> 00:02:29.978
if I start at 6 and I go to negative 4 I went down 10.

00:02:29.978 --> 00:02:33.267
or if you just want to use this formula here it will give you the same thing

00:02:33.267 --> 00:02:37.315
We finished at negative 4, we finished at negative 4

00:02:37.315 --> 00:02:40.380
and from that we want to subtract, we want to subtract 6.

00:02:40.380 --> 00:02:47.000
This right here is y2, our ending y and this is our beginning y

00:02:47.000 --> 00:02:53.067
This is y1. So y2, negative 4 minus y1, 6.

00:02:53.067 --> 00:02:55.933
or negative 4 minus 6.

00:02:55.933 --> 00:02:57.800
That is equal to negative 10.

00:02:57.800 --> 00:03:00.303
And all it does is tell us the change in y

00:03:00.303 --> 00:03:01.667
you go from this point to that point

00:03:01.667 --> 00:03:05.000
We have to go down, our rise is negative

00:03:05.000 --> 00:03:08.012
we have to go down 10. That's where the negative 10 comes from.

00:03:08.012 --> 00:03:10.867
Now we just have to find our change in x.

00:03:10.867 --> 00:03:13.133
So we can look at this graph over here.

00:03:13.133 --> 00:03:16.400
We started at x is equal to negative 1 and we go all the way

00:03:16.400 --> 00:03:19.400
to x is equal to 5. So we started at x is equal to negative 1,

00:03:19.400 --> 00:03:22.533
and we go all the way to x is equal to 5.

00:03:22.533 --> 00:03:26.200
So it takes us one to go to zero and then five more.

00:03:26.200 --> 00:03:29.049
So are change in x is 6.

00:03:29.049 --> 00:03:31.200
You can look at that visually there or

00:03:31.200 --> 00:03:33.400
you can use this formula same exact idea,

00:03:33.400 --> 00:03:37.667
our ending x-value, our ending x-value is 5

00:03:37.667 --> 00:03:41.634
and our starting x-value is negative 1.

00:03:41.634 --> 00:03:46.533
5 minus negative 1. 5 minus negative 1 is the same thing as

00:03:46.533 --> 00:03:52.037
5 plus 1. So it is 6. So our slope here is negative 10 over 6.

00:03:52.037 --> 00:03:55.400
wich is the exact same thing as negative 5 thirds.

00:03:55.400 --> 00:04:02.067
as negative 5 over 3 I divided the numerator and the denominator by 2.

00:04:02.067 --> 00:04:05.733
So we now know our equation will be

00:04:05.733 --> 00:04:10.067
y is equal to negative 5 thirds, that's our slope,

00:04:10.067 --> 00:04:16.333
x plus b. So we still need to solve for y-intercept to get our equation.

00:04:16.333 --> 00:04:20.333
And to do that, we can use the information that we know

00:04:20.333 --> 00:04:23.384
in fact we have several points of information

00:04:23.384 --> 00:04:27.133
We can use the fact that the line goes through the point (-1,6)

00:04:27.133 --> 00:04:29.067
you could use the other point as well.

00:04:29.067 --> 00:04:31.200
We know that when is equal to negative 1,

00:04:31.200 --> 00:04:36.933
So y is eqaul to 6. So y is equal to six when x is equal to negative 1

00:04:36.933 --> 00:04:41.800
So negative 5 thirds times x, when x is equal to negative 1

00:04:41.800 --> 00:04:46.200
y is equal to 6. So we literally just substitute this x and y value

00:04:46.200 --> 00:04:49.867
back into this and know we can solve for b.

00:04:49.867 --> 00:04:52.733
So let's see, this negative 1 times negative 5 thirds.

00:04:52.733 --> 00:04:58.133
So we have 6 is equal to positive five thirds plus b.

00:04:58.133 --> 00:05:01.667
And now we can subtract 5 thirds from both sides of this equation.

00:05:01.667 --> 00:05:04.733
so we have subtracted the left hand side.

00:05:04.733 --> 00:05:07.400
From the left handside and subtracted from the rigth handside

00:05:07.400 --> 00:05:11.800
And then we get, what's 6 minus 5 thirds.

00:05:11.800 --> 00:05:14.561
So that's going to be, let me do it over here

00:05:14.561 --> 00:05:17.867
We take a common denominator. So 6 is the same thing as

00:05:17.867 --> 00:05:23.291
Let's do it over here. So 6 minus 5 over 3 is the same thing as

00:05:23.291 --> 00:05:27.867
6 is the same thing as 18 over 3 minus 5 over 3

00:05:27.867 --> 00:05:36.933
6 is 18 over 3. And this is just 13 over 3. And this is just 13 over 3.

00:05:36.933 --> 00:05:39.533
And then of course, these cancel out. So we get

00:05:39.533 --> 00:05:43.400
b is equal to 13 thirds. So we are done. We know

00:05:43.400 --> 00:05:45.867
We know the slope and we know the y-intercept.

00:05:45.867 --> 00:05:49.000
The equation of our line is y is equal to

00:05:49.000 --> 00:05:55.200
negative 5 thirds x plus our y-intercept which is 13

00:05:55.200 --> 00:05:59.329
which is 13 over 3. And we can write these as mixed numbers.

00:05:59.329 --> 00:06:02.733
if it's easier to visualize. 13 over 3 is four and 1 thirds.

00:06:02.733 --> 00:06:05.267
So this y-intercept right over here.

00:06:05.267 --> 00:06:09.800
this y-intercept right over here. That's 0 coma 13 over 3 or

00:06:09.800 --> 00:06:15.200
0 coma 4 and 1 thirds. And even with my very roughly drawn diagram

00:06:15.200 --> 00:06:19.467
it those looks like this. And the slope negative 5 thirds

00:06:19.467 --> 00:06:22.177
that's the same thing as negative 1 and 2 thirds.

00:06:22.177 --> 00:06:25.067
You can see here the slope is downward because the slope is negative.

00:06:25.067 --> 00:06:27.667
It's a little bit steeper than a slope of 1.

00:06:27.667 --> 00:06:32.333
It's not quite a negative 2. It's negative 1 and 2 thirds.

00:06:32.333 --> 00:06:35.133
if you write this as a negative, as a mixed number.

00:06:35.482 --> 99:59:59.999
So, hopefully, you found that entertaining.


Graphing inequalities 2

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:06.885
Graph the inequality y minus
4x is less than negative 3.

00:00:06.885 --> 00:00:08.260
So the first thing
we could do is

00:00:08.260 --> 00:00:10.990
we could kind of put
this in mx plus b form,

00:00:10.990 --> 00:00:13.950
or slope-intercept form,
but as an inequality.

00:00:13.950 --> 00:00:20.420
So we're starting with y minus
4x is less than negative 3.

00:00:20.420 --> 00:00:24.650
We can add 4x to both
sides of this inequality.

00:00:24.650 --> 00:00:27.670
So let's add 4x to both
sides of this inequality,

00:00:27.670 --> 00:00:30.030
and then we'll just have
a y on the left-hand side.

00:00:30.030 --> 00:00:31.040
These guys cancel out.

00:00:31.040 --> 00:00:35.410
So you have y is
less than 4x minus 3.

00:00:35.410 --> 00:00:37.890
We could have had
negative 3 plus 4x,

00:00:37.890 --> 00:00:40.320
but we want to write the 4x
first just because that's

00:00:40.320 --> 00:00:42.490
a form that we're
more familiar with.

00:00:42.490 --> 00:00:46.620
So it's less than 4x minus 3.

00:00:46.620 --> 00:00:49.160
And now we can
attempt to graph it.

00:00:49.160 --> 00:00:54.210
But before I graph it, I want
to be a little bit careful here.

00:00:54.210 --> 00:00:56.610
So let me draw our axes.

00:00:56.610 --> 00:01:03.790
So this is the x-axis,
and is that is the y-axis.

00:01:03.790 --> 00:01:05.740
And we want to be
careful, because this

00:01:05.740 --> 00:01:09.540
says y is less than 4x
minus 3, not less than

00:01:09.540 --> 00:01:14.220
or equal to 4x minus 3, or
not y is equal to 4x minus 3.

00:01:14.220 --> 00:01:16.760
So what we want to do is kind
of create a boundary at y

00:01:16.760 --> 00:01:21.110
is equal to 4x minus 3, and
the solution to this inequality

00:01:21.110 --> 00:01:24.110
will be all of the area below
that, all of the y values

00:01:24.110 --> 00:01:25.640
less than that.

00:01:25.640 --> 00:01:26.650
So let's try to do it.

00:01:26.650 --> 00:01:31.140
So the boundary line would look
like-- so let me write it over

00:01:31.140 --> 00:01:39.470
here-- so we have a boundary
at y is equal to 4x minus 3.

00:01:39.470 --> 00:01:41.120
Notice this isn't
part of the solution.

00:01:41.120 --> 00:01:42.369
This isn't less than or equal.

00:01:42.369 --> 00:01:43.580
It's just less than.

00:01:43.580 --> 00:01:45.990
But this will at
least help us draw,

00:01:45.990 --> 00:01:47.405
essentially, the boundary.

00:01:47.405 --> 00:01:48.530
So we could do it two ways.

00:01:48.530 --> 00:01:50.240
If you know slope
and y-intercept,

00:01:50.240 --> 00:01:53.110
you know that 4 is our
slope and that negative 3

00:01:53.110 --> 00:01:54.400
is our y-intercept.

00:01:54.400 --> 00:01:56.150
Or you can literally
just take two points,

00:01:56.150 --> 00:01:58.490
and that'll help you
define a line here.

00:01:58.490 --> 00:02:05.280
So you could say, well, when
x is equal to 0, what is y?

00:02:05.280 --> 00:02:08.340
You get 4 times 0 minus 3, you
get y is equal to negative 3.

00:02:08.340 --> 00:02:10.710
And we knew that because
it was the y-intercept.

00:02:10.710 --> 00:02:16.080
So you have 0, and then you
have 1, 2, 0, and negative 3.

00:02:16.080 --> 00:02:19.740
And then you have
the point, let's say,

00:02:19.740 --> 00:02:22.490
when x is equal
to-- I don't know--

00:02:22.490 --> 00:02:24.920
let's say when x is equal to 2.

00:02:24.920 --> 00:02:26.800
When x is equal to 2, what is y?

00:02:26.800 --> 00:02:30.150
We have 4 times
2 is 8 minus 3, y

00:02:30.150 --> 00:02:32.540
is then going to be equal to 5.

00:02:32.540 --> 00:02:39.350
So then you go 1, 2, and
you go 1, 2, 3, 4, 5.

00:02:39.350 --> 00:02:41.630
And so you have that
point there as well.

00:02:41.630 --> 00:02:45.676
And then we can just
connect the dots.

00:02:45.676 --> 00:02:47.550
Or you could say, look,
there's a slope of 4.

00:02:47.550 --> 00:02:50.230
So every time we move
over 1, every time we

00:02:50.230 --> 00:02:55.020
move 1 in the x direction, we
move up 4 in the y direction.

00:02:55.020 --> 00:02:56.270
So we could draw it like that.

00:02:56.270 --> 00:02:58.504
So the line will look
something like this.

00:02:58.504 --> 00:03:00.420
And I'm just going to
draw it in a dotted line

00:03:00.420 --> 00:03:05.027
because, remember, this
isn't part of the solution.

00:03:05.027 --> 00:03:06.860
Actually, let me draw
it a little bit neater

00:03:06.860 --> 00:03:09.130
because that point should
be right about there,

00:03:09.130 --> 00:03:11.740
and this point should
be right about there.

00:03:11.740 --> 00:03:14.770
And then this boundary line I'm
going to draw as a dotted line.

00:03:14.770 --> 00:03:19.875
So it's going to look
something like that.

00:03:19.875 --> 00:03:21.750
I draw it a dotted line
to show that it's not

00:03:21.750 --> 00:03:23.080
part of the solution.

00:03:23.080 --> 00:03:25.610
Our solution has the
y's less than that.

00:03:25.610 --> 00:03:31.140
So for any x, so you pick an x
here, if you took 4x minus 3,

00:03:31.140 --> 00:03:32.871
you're going to
end up on the line.

00:03:32.871 --> 00:03:35.120
But we don't want the y's
that are equal to that line.

00:03:35.120 --> 00:03:36.830
We want for that
particular x, the y's

00:03:36.830 --> 00:03:38.480
that are less than the line.

00:03:38.480 --> 00:03:40.780
So it's going to be all
of this area over here.

00:03:40.780 --> 00:03:44.010
We're less than the line, and
we're not including the line,

00:03:44.010 --> 00:03:45.960
and that's why I put
a dotted line here.

00:03:45.960 --> 00:03:47.880
You can also try values out.

00:03:47.880 --> 00:03:51.460
You can say, well, this line
is dividing our coordinate axes

00:03:51.460 --> 00:03:53.550
into, essentially,
the region above it

00:03:53.550 --> 00:03:55.890
and the region below it,
and you can test it out.

00:03:55.890 --> 00:03:57.431
Let's take something
that's above it.

00:03:57.431 --> 00:04:02.250
Let's take the
point 0, 0 and see

00:04:02.250 --> 00:04:04.650
if that satisfies
our inequality.

00:04:04.650 --> 00:04:10.110
If we have y is 0 is
less than 0 minus 3,

00:04:10.110 --> 00:04:13.540
or we get 0 is less
than negative 3.

00:04:13.540 --> 00:04:15.490
This is definitely not the case.

00:04:15.490 --> 00:04:16.510
This is not true.

00:04:19.160 --> 00:04:22.370
And it makes sense
because that 0, 0 is not

00:04:22.370 --> 00:04:24.040
part of the solution.

00:04:24.040 --> 00:04:27.440
Now, we could go on the other
side of our boundary line.

00:04:27.440 --> 00:04:29.190
And we could take the
point, I don't know,

00:04:29.190 --> 00:04:31.805
let's take the point 3 comma 0.

00:04:31.805 --> 00:04:34.180
So let's say that this is the
point-- well, that's right.

00:04:34.180 --> 00:04:35.330
There's a point 2 comma 0.

00:04:35.330 --> 00:04:38.170
Let's take the point 3
comma 0 right over here.

00:04:38.170 --> 00:04:40.490
This should work because
it's in the region less than.

00:04:40.490 --> 00:04:42.760
But let's verify
it for ourselves.

00:04:42.760 --> 00:04:46.490
So we have y is 0.

00:04:46.490 --> 00:04:51.576
0 is less than 4
times 3 minus 3.

00:04:51.576 --> 00:04:54.865
0 is less than 12 minus 3.

00:04:54.865 --> 00:04:58.190
0 Is less than 9, which
is definitely true.

00:04:58.190 --> 00:05:01.100
So that point does
satisfy the inequality.

00:05:01.100 --> 00:05:03.090
So in general, you
want to kind of look

00:05:03.090 --> 00:05:05.150
at this as an equal to
draw the boundary line.

00:05:05.150 --> 00:05:05.791
We did it.

00:05:05.791 --> 00:05:07.540
But we drew it as a
dotted line because we

00:05:07.540 --> 00:05:08.780
don't want to include
it because this

00:05:08.780 --> 00:05:09.946
isn't less than or equal to.

00:05:09.946 --> 00:05:11.240
It's just less than.

00:05:11.240 --> 00:05:13.040
And then our solution
to the inequality

00:05:13.040 --> 00:05:15.270
will be the region
below it, all the y's

00:05:15.270 --> 00:05:19.191
less than the line
for x minus 3.


Finding a linear equation given a point and slope

WEBVTT
Kind: captions
Language: en

00:00:01.000 --> 00:00:03.840
A line has a slope of 7
and goes through the point

00:00:03.840 --> 00:00:06.410
negative 4, negative 11.

00:00:06.410 --> 00:00:10.860
What is the equation of this
line in slope-intercept form?

00:00:10.860 --> 00:00:13.570
So the equation of any line
in slope-intercept form

00:00:13.570 --> 00:00:21.330
is y is equal to mx plus
b, where m is the slope

00:00:21.330 --> 00:00:24.740
and b is the y-intercept.

00:00:24.740 --> 00:00:27.600
Now, in this problem right
here, they tell us the slope.

00:00:27.600 --> 00:00:31.140
They tell us that a
line has a slope of 7.

00:00:31.140 --> 00:00:35.360
So we know right from the
get go that m is equal to 7.

00:00:35.360 --> 00:00:38.510
So we know the equation of this
line in slope-intercept form

00:00:38.510 --> 00:00:43.457
is going to look like y is
equal to, we know m is 7,

00:00:43.457 --> 00:00:49.200
so 7x plus-- let me
make that x a little bit

00:00:49.200 --> 00:00:53.960
neater-- 7x plus b.

00:00:53.960 --> 00:00:56.420
And now what we need to do,
we need to figure out b,

00:00:56.420 --> 00:00:58.860
and they give us one more
piece of information.

00:00:58.860 --> 00:01:03.210
They say that the line goes
through the point negative 4

00:01:03.210 --> 00:01:05.560
comma negative 11.

00:01:05.560 --> 00:01:10.110
So that tells us that when
x is equal to negative 4,

00:01:10.110 --> 00:01:14.010
then y is equal to negative 11.

00:01:14.010 --> 00:01:17.110
So we can use this
information in what we have

00:01:17.110 --> 00:01:18.750
or the part of our
equation that we've

00:01:18.750 --> 00:01:20.450
been able to figure out so far.

00:01:20.450 --> 00:01:23.990
We know that when x is
equal to negative 4,

00:01:23.990 --> 00:01:27.520
y is going to be
equal to negative 11.

00:01:27.520 --> 00:01:31.000
So what b do we need
to make that happen?

00:01:31.000 --> 00:01:31.830
Let's try it out.

00:01:31.830 --> 00:01:36.080
So y is negative 11 when
x is equal to negative 4.

00:01:36.080 --> 00:01:41.650
So negative 11 is equal to
7 times x-- and in this case

00:01:41.650 --> 00:01:44.420
x is negative 4-- plus b.

00:01:47.680 --> 00:01:49.580
And now we can just solve for b.

00:01:49.580 --> 00:01:53.010
A b that makes this
equation, or that

00:01:53.010 --> 00:01:56.030
satisfies the constraint that
when x is equal to negative 4,

00:01:56.030 --> 00:01:57.790
y is equal to negative 11.

00:01:57.790 --> 00:02:01.890
So let's see, we get negative 11
is equal to 7 times negative 4

00:02:01.890 --> 00:02:06.340
is negative 28
plus b, and now we

00:02:06.340 --> 00:02:09.550
can add a 28 to both
sides of this equation.

00:02:09.550 --> 00:02:10.800
So let's add a 28.

00:02:10.800 --> 00:02:15.000
I'm just trying to isolate
the b on the right-hand side.

00:02:15.000 --> 00:02:18.530
And so on the left-hand
side, negative 11 plus 28,

00:02:18.530 --> 00:02:21.090
that is just positive 17.

00:02:21.090 --> 00:02:23.250
These guys cancel
out on purpose.

00:02:23.250 --> 00:02:25.320
And I just have a b on
the right-hand side.

00:02:25.320 --> 00:02:28.070
So I get b is equal to 17.

00:02:28.070 --> 00:02:30.020
Let me write it in green.

00:02:30.020 --> 00:02:31.690
That's not green.

00:02:31.690 --> 00:02:37.110
We get b is equal to 17.

00:02:37.110 --> 00:02:40.890
So we know m is 7, they told
us that right at the beginning.

00:02:40.890 --> 00:02:42.460
And now we know b is 17.

00:02:42.460 --> 00:02:45.500
So the equation of
our line is y is

00:02:45.500 --> 00:02:48.850
equal to 7x, that's our slope.

00:02:48.850 --> 00:02:57.304
7 times x plus b,
and b here is 17.

00:02:57.304 --> 00:02:59.970
And if we wanted to graph it, it
would look something like this.

00:02:59.970 --> 00:03:03.020
I'll just do a real rough graph.

00:03:03.020 --> 00:03:06.830
So if we wanted to graph
this line, that's my x-axis,

00:03:06.830 --> 00:03:11.030
and this is my y-axis.

00:03:11.030 --> 00:03:12.690
The y-intercept is 17.

00:03:12.690 --> 00:03:18.230
So that means that the
point 0, 17 is on this line.

00:03:18.230 --> 00:03:22.690
So this point right over
here is going to be 0, 17.

00:03:22.690 --> 00:03:24.550
And our slope is 7.

00:03:24.550 --> 00:03:27.725
So that means if we move to the
right one, we move up t seven.

00:03:27.725 --> 00:03:28.600
So it's a high slope.

00:03:28.600 --> 00:03:31.580
So if we move to the right
one, we move up seven.

00:03:31.580 --> 00:03:34.540
Or if we move back one,
will move down seven.

00:03:34.540 --> 00:03:36.380
So we'll move down
seven, so the line

00:03:36.380 --> 00:03:37.680
will look roughly like this.

00:03:37.680 --> 00:03:40.202
Obviously, haven't
done it very exactly,

00:03:40.202 --> 00:03:41.660
but our line is
going to look like.

00:03:41.660 --> 00:03:45.090
That's going to be a pretty
steep upward-sloping line.

00:03:45.090 --> 00:03:46.902
It has a very high
slope, slope of seven.

00:03:46.902 --> 00:03:48.360
If you move one in
the x direction,

00:03:48.360 --> 00:03:49.920
you have to move up seven.

00:03:49.920 --> 00:03:53.410
And its y-intercept
is at y is 17.

00:03:53.410 --> 00:03:56.203
When x is 0, y is 17.