0:00:00.450,0:00:03.570
In this video I'm going to do a[br]bunch of examples of finding

0:00:03.570,0:00:07.170
the equations of lines in[br]slope-intercept form.

0:00:07.170,0:00:09.610
Just as a bit of a review, that[br]means equations of lines

0:00:09.610,0:00:17.050
in the form of y is equal to mx[br]plus b where m is the slope

0:00:17.050,0:00:21.200
and b is the y-intercept.

0:00:21.200,0:00:24.870
So let's just do a bunch of[br]these problems. So here they

0:00:24.870,0:00:28.900
tell us that a line has a slope[br]of negative 5, so m is

0:00:28.900,0:00:30.740
equal to negative 5.

0:00:30.740,0:00:34.290
And it has a y-intercept of 6.

0:00:34.290,0:00:36.300
So b is equal to 6.

0:00:36.300,0:00:37.985
So this is pretty[br]straightforward.

0:00:37.985,0:00:41.530
The equation of this line[br]is y is equal to

0:00:41.530,0:00:47.550
negative 5x plus 6.

0:00:47.550,0:00:49.570
That wasn't too bad.

0:00:49.570,0:00:51.570
Let's do this next[br]one over here.

0:00:51.570,0:00:54.300
The line has a slope of negative[br]1 and contains the

0:00:54.300,0:00:57.320
point 4/5 comma 0.

0:00:57.320,0:01:00.600
So they're telling us the slope,[br]slope of negative 1.

0:01:00.600,0:01:05.230
So we know that m is equal to[br]negative 1, but we're not 100%

0:01:05.230,0:01:09.190
sure about where the y-intercept[br]is just yet.

0:01:09.190,0:01:12.510
So we know that this equation[br]is going to be of the form y

0:01:12.510,0:01:19.300
is equal to the slope negative[br]1x plus b, where b is the

0:01:19.300,0:01:20.460
y-intercept.

0:01:20.460,0:01:23.650
Now, we can use this coordinate[br]information, the

0:01:23.650,0:01:25.870
fact that it contains this[br]point, we can use that

0:01:25.870,0:01:28.590
information to solve for b.

0:01:28.590,0:01:31.530
The fact that the line contains[br]this point means that

0:01:31.530,0:01:37.690
the value x is equal to 4/5, y[br]is equal to 0 must satisfy

0:01:37.690,0:01:38.265
this equation.

0:01:38.265,0:01:43.120
So let's substitute those in.[br]y is equal to 0 when x is

0:01:43.120,0:01:44.090
equal to 4/5.

0:01:44.090,0:01:50.170
So 0 is equal to negative[br]1 times 4/5 plus b.

0:01:50.170,0:01:52.810
I'll scroll down a little bit.

0:01:52.810,0:01:58.110
So let's see, we get a 0 is[br]equal to negative 4/5 plus b.

0:01:58.110,0:02:02.040
We can add 4/5 to both sides[br]of this equation.

0:02:02.040,0:02:04.250
So we get add a 4/5 there.

0:02:04.250,0:02:07.320
We could add a 4/5 to[br]that side as well.

0:02:07.320,0:02:10.100
The whole reason I did that is[br]so that cancels out with that.

0:02:10.100,0:02:12.130
You get b is equal to 4/5.

0:02:16.250,0:02:19.180
So we now have the equation[br]of the line.

0:02:19.180,0:02:23.040
y is equal to negative 1 times[br]x, which we write as negative

0:02:23.040,0:02:32.500
x, plus b, which is 4/5,[br]just like that.

0:02:32.500,0:02:34.480
Now we have this one.

0:02:34.480,0:02:39.580
The line contains the point[br]2 comma 6 and 5 comma 0.

0:02:39.580,0:02:42.540
So they haven't given us the[br]slope or the y-intercept

0:02:42.540,0:02:43.030
explicitly.

0:02:43.030,0:02:45.350
But we could figure out both[br]of them from these

0:02:45.350,0:02:45.650
coordinates.

0:02:45.650,0:02:48.270
So the first thing we can do[br]is figure out the slope.

0:02:48.270,0:02:53.750
So we know that the slope m is[br]equal to change in y over

0:02:53.750,0:02:58.100
change in x, which is equal to--[br]What is the change in y?

0:02:58.100,0:02:59.490
Let's start with this[br]one right here.

0:02:59.490,0:03:00.985
So we do 6 minus 0.

0:03:04.210,0:03:05.070
Let me do it this way.

0:03:05.070,0:03:10.410
So that's a 6-- I want to make[br]it color-coded-- minus 0.

0:03:10.410,0:03:14.340
So 6 minus 0, that's[br]our change in y.

0:03:14.340,0:03:24.190
Our change in x is 2[br]minus 2 minus 5.

0:03:24.190,0:03:26.320
The reason why I color-coded[br]it is I wanted to show you

0:03:26.320,0:03:30.890
when I used this y term first,[br]I used the 6 up here, that I

0:03:30.890,0:03:33.380
have to use this x term[br]first as well.

0:03:33.380,0:03:36.730
So I wanted to show you, this[br]is the coordinate 2 comma 6.

0:03:36.730,0:03:38.590
This is the coordinate[br]5 comma 0.

0:03:38.590,0:03:41.650
I couldn't have swapped[br]the 2 and the 5 then.

0:03:41.650,0:03:45.030
Then I would have gotten the[br]negative of the answer.

0:03:45.030,0:03:46.080
But what do we get here?

0:03:46.080,0:03:51.210
This is equal to[br]6 minus 0 is 6.

0:03:51.210,0:03:54.770
2 minus 5 is negative 3.

0:03:54.770,0:03:58.910
So this becomes negative 6[br]over 3, which is the same

0:03:58.910,0:04:01.310
thing as negative 2.

0:04:01.310,0:04:02.250
So that's our slope.

0:04:02.250,0:04:06.920
So, so far we know that the line[br]must be, y is equal to

0:04:06.920,0:04:12.580
the slope-- I'll do that in[br]orange-- negative 2 times x

0:04:12.580,0:04:15.160
plus our y-intercept.

0:04:15.160,0:04:17.779
Now we can do exactly what we[br]did in the last problem.

0:04:17.779,0:04:20.579
We can use one of these[br]points to solve for b.

0:04:20.579,0:04:22.029
We can use either one.

0:04:22.029,0:04:25.920
Both of these are on the line,[br]so both of these must satisfy

0:04:25.920,0:04:26.900
this equation.

0:04:26.900,0:04:29.800
I'll use the 5 comma 0 because[br]it's always nice when

0:04:29.800,0:04:31.020
you have a 0 there.

0:04:31.020,0:04:32.820
The math is a little[br]bit easier.

0:04:32.820,0:04:34.510
So let's put the 5[br]comma 0 there.

0:04:34.510,0:04:38.900
So y is equal to 0 when[br]x is equal to 5.

0:04:38.900,0:04:43.820
So y is equal to 0 when you have[br]negative 2 times 5, when

0:04:43.820,0:04:47.700
x is equal to 5 plus b.

0:04:47.700,0:04:52.650
So you get 0 is equal[br]to -10 plus b.

0:04:52.650,0:04:57.820
If you add 10 to both sides of[br]this equation, let's add 10 to

0:04:57.820,0:05:00.680
both sides, these[br]two cancel out.

0:05:00.680,0:05:03.970
You get b is equal to[br]10 plus 0 or 10.

0:05:03.970,0:05:06.420
So you get b is equal to 10.

0:05:06.420,0:05:07.935
Now we know the equation[br]for the line.

0:05:07.935,0:05:14.110
The equation is y-- let me do it[br]in a new color-- y is equal

0:05:14.110,0:05:22.280
to negative 2x plus b plus 10.

0:05:22.280,0:05:23.470
We are done.

0:05:23.470,0:05:24.720
Let's do another one of these.

0:05:28.180,0:05:31.270
All right, the line contains[br]the points 3 comma 5 and

0:05:31.270,0:05:32.890
negative 3 comma 0.

0:05:32.890,0:05:36.380
Just like the last problem, we[br]start by figuring out the

0:05:36.380,0:05:40.380
slope, which we will call m.

0:05:40.380,0:05:44.830
It's the same thing as the rise[br]over the run, which is

0:05:44.830,0:05:48.190
the same thing as the change[br]in y over the change in x.

0:05:48.190,0:05:50.070
If you were doing this for your[br]homework, you wouldn't

0:05:50.070,0:05:50.870
have to write all this.

0:05:50.870,0:05:52.920
I just want to make sure that[br]you understand that these are

0:05:52.920,0:05:55.150
all the same things.

0:05:55.150,0:05:58.520
Then what is our change in[br]y over our change in x?

0:05:58.520,0:06:02.280
This is equal to, let's start[br]with the side first. It's just

0:06:02.280,0:06:03.980
to show you I could pick[br]either of these points.

0:06:03.980,0:06:14.050
So let's say it's 0 minus[br]5 just like that.

0:06:14.050,0:06:17.000
So I'm using this coordinate[br]first. I'm kind of viewing it

0:06:17.000,0:06:19.770
as the endpoint.

0:06:19.770,0:06:22.420
Remember when I first learned[br]this, I would always be

0:06:22.420,0:06:24.160
tempted to do the x[br]in the numerator.

0:06:24.160,0:06:25.990
No, you use the y's[br]in the numerator.

0:06:25.990,0:06:28.470
So that's the second[br]of the coordinates.

0:06:28.470,0:06:38.435
That is going to be over[br]negative 3 minus 3.

0:06:41.250,0:06:44.370
This is the coordinate[br]negative 3, 0.

0:06:44.370,0:06:46.420
This is the coordinate 3, 5.

0:06:46.420,0:06:47.980
We're subtracting that.

0:06:47.980,0:06:49.310
So what are we going to get?

0:06:49.310,0:06:52.570
This is going to be equal to--[br]I'll do it in a neutral

0:06:52.570,0:06:56.210
color-- this is going to be[br]equal to the numerator is

0:06:56.210,0:07:02.010
negative 5 over negative 3[br]minus 3 is negative 6.

0:07:02.010,0:07:03.650
So the negatives cancel out.

0:07:03.650,0:07:05.930
You get 5/6.

0:07:05.930,0:07:08.700
So we know that the equation is[br]going to be of the form y

0:07:08.700,0:07:15.560
is equal to 5/6 x plus b.

0:07:15.560,0:07:18.600
Now we can substitute one of[br]these coordinates in for b.

0:07:18.600,0:07:19.440
So let's do.

0:07:19.440,0:07:21.310
I always like to use the one[br]that has the 0 in it.

0:07:21.310,0:07:33.270
So y is a zero when x is[br]negative 3 plus b.

0:07:33.270,0:07:37.810
So all I did is I substituted[br]negative 3 for x, 0 for y.

0:07:37.810,0:07:40.860
I know I can do that because[br]this is on the line.

0:07:40.860,0:07:44.040
This must satisfy the equation[br]of the line.

0:07:44.040,0:07:45.600
Let's solve for b.

0:07:45.600,0:07:49.990
So we get zero is equal to, well[br]if we divide negative 3

0:07:49.990,0:07:51.830
by 3, that becomes a 1.

0:07:51.830,0:07:54.890
If you divide 6 by 3,[br]that becomes a 2.

0:07:54.890,0:08:02.380
So it becomes negative[br]5/2 plus b.

0:08:02.380,0:08:05.280
We could add 5/2 to both[br]sides of the equation,

0:08:05.280,0:08:08.630
plus 5/2, plus 5/2.

0:08:08.630,0:08:10.850
I like to change my notation[br]just so you get

0:08:10.850,0:08:12.520
familiar with both.

0:08:12.520,0:08:17.800
So the equation becomes 5/2 is[br]equal to-- that's a 0-- is

0:08:17.800,0:08:19.600
equal to b.

0:08:19.600,0:08:22.090
b is 5/2.

0:08:22.090,0:08:31.940
So the equation of our line is[br]y is equal to 5/6 x plus b,

0:08:31.940,0:08:37.820
which we just figured out[br]is 5/2, plus 5/2.

0:08:37.820,0:08:38.710
We are done.

0:08:38.710,0:08:41.280
Let's do another one.

0:08:41.280,0:08:43.500
We have a graph here.

0:08:43.500,0:08:45.300
Let's figure out the equation[br]of this graph.

0:08:45.300,0:08:46.900
This is actually, on some level,[br]a little bit easier.

0:08:46.900,0:08:47.740
What's the slope?

0:08:47.740,0:08:52.250
Slope is change in y[br]over change it x.

0:08:52.250,0:08:53.310
So let's see what happens.

0:08:53.310,0:08:57.900
When we move in x, when our[br]change in x is 1, so that is

0:08:57.900,0:08:58.940
our change in x.

0:08:58.940,0:09:00.850
So change in x is 1.

0:09:00.850,0:09:04.130
I'm just deciding to change[br]my x by 1, increment by 1.

0:09:04.130,0:09:05.900
What is the change in y?

0:09:05.900,0:09:10.390
It looks like y changes[br]exactly by 4.

0:09:10.390,0:09:14.980
It looks like my delta y, my[br]change in y, is equal to 4

0:09:14.980,0:09:20.690
when my delta x is equal to 1.

0:09:20.690,0:09:24.340
So change in y over change in[br]x, change in y is 4 when

0:09:24.340,0:09:26.220
change in x is 1.

0:09:26.220,0:09:30.380
So the slope is equal to 4.

0:09:30.380,0:09:32.190
Now what's its y-intercept?

0:09:32.190,0:09:33.720
Well here we can just[br]look at the graph.

0:09:33.720,0:09:38.260
It looks like it intersects[br]y-axis at y is equal to

0:09:38.260,0:09:41.600
negative 6, or at the[br]point 0, negative 6.

0:09:41.600,0:09:44.180
So we know that b is equal[br]to negative 6.

0:09:46.950,0:09:48.875
So we know the equation[br]of the line.

0:09:48.875,0:09:56.630
The equation of the line is y is[br]equal to the slope times x

0:09:56.630,0:09:59.030
plus the y-intercept.

0:09:59.030,0:10:01.850
I should write that.

0:10:01.850,0:10:07.840
So minus 6, that is plus[br]negative 6 So that is the

0:10:07.840,0:10:09.800
equation of our line.

0:10:09.800,0:10:12.980
Let's do one more of these.

0:10:12.980,0:10:17.170
So they tell us that f of[br]1.5 is negative 3, f of

0:10:17.170,0:10:18.750
negative 1 is 2.

0:10:18.750,0:10:19.970
What is that?

0:10:19.970,0:10:23.830
Well, all this is just a fancy[br]way of telling you that the

0:10:23.830,0:10:30.530
point when x is 1.5, when you[br]put 1.5 into the function, the

0:10:30.530,0:10:33.490
function evaluates[br]as negative 3.

0:10:33.490,0:10:36.750
So this tells us that the[br]coordinate 1.5, negative 3 is

0:10:36.750,0:10:38.270
on the line.

0:10:38.270,0:10:41.960
Then this tells us that the[br]point when x is negative 1, f

0:10:41.960,0:10:44.420
of x is equal to 2.

0:10:44.420,0:10:47.540
This is just a fancy way of[br]saying that both of these two

0:10:47.540,0:10:51.400
points are on the line,[br]nothing unusual.

0:10:51.400,0:10:54.380
I think the point of this[br]problem is to get you familiar

0:10:54.380,0:10:56.870
with function notation, for you[br]to not get intimidated if

0:10:56.870,0:10:57.970
you see something like this.

0:10:57.970,0:11:01.540
If you evaluate the function[br]at 1.5, you get negative 3.

0:11:01.540,0:11:04.440
So that's the coordinate if[br]you imagine that y is

0:11:04.440,0:11:06.020
equal to f of x.

0:11:06.020,0:11:06.950
So this would be the[br]y-coordinate.

0:11:06.950,0:11:09.250
It would be equal to negative[br]3 when x is 1.5.

0:11:09.250,0:11:10.840
Anyway, I've said it[br]multiple times.

0:11:10.840,0:11:13.280
Let's figure out the[br]slope of this line.

0:11:13.280,0:11:20.020
The slope which is change in y[br]over change in x is equal to,

0:11:20.020,0:11:27.460
let's start with 2 minus this[br]guy, negative 3-- these are

0:11:27.460,0:11:32.880
the y-values-- over, all[br]of that over, negative

0:11:32.880,0:11:40.140
1 minus this guy.

0:11:40.140,0:11:43.330
Let me write it this way,[br]negative 1 minus

0:11:43.330,0:11:48.440
that guy, minus 1.5.

0:11:48.440,0:11:50.340
I do the colors because I want[br]to show you that the negative

0:11:50.340,0:11:54.060
1 and the 2 are both coming from[br]this, that's why I use

0:11:54.060,0:11:57.500
both of them first. If I used[br]these guys first, I would have

0:11:57.500,0:12:00.495
to use both the x and the y[br]first. If I use the 2 first, I

0:12:00.495,0:12:02.080
have to use the negative[br]1 first. That's why I'm

0:12:02.080,0:12:03.390
color-coding it.

0:12:03.390,0:12:08.360
So this is going to be equal[br]to 2 minus negative 3.

0:12:08.360,0:12:10.370
That's the same thing[br]as 2 plus 3.

0:12:10.370,0:12:11.620
So that is 5.

0:12:16.480,0:12:20.040
Negative 1 minus 1.5[br]is negative 2.5.

0:12:23.830,0:12:27.770
5 divided by 2.5[br]is equal to 2.

0:12:27.770,0:12:30.250
So the slope of this[br]line is negative 2.

0:12:30.250,0:12:32.130
Actually I'll take a little[br]aside to show you it doesn't

0:12:32.130,0:12:34.480
matter what order[br]I do this in.

0:12:34.480,0:12:36.180
If I use this coordinate first,[br]then I have to use that

0:12:36.180,0:12:38.140
coordinate first. Let's[br]do it the other way.

0:12:38.140,0:12:54.180
If I did it as negative 3[br]minus 2 over 1.5 minus

0:12:54.180,0:12:59.810
negative 1, this should be minus[br]the 2 over 1.5 minus the

0:12:59.810,0:13:01.060
negative 1.

0:13:03.300,0:13:04.780
This should give me[br]the same answer.

0:13:04.780,0:13:06.130
This is equal to what?

0:13:06.130,0:13:12.860
Negative 3 minus 2 is negative[br]5 over 1.5 minus negative 1.

0:13:12.860,0:13:14.520
That's 1.5 plus 1.

0:13:14.520,0:13:16.610
That's over 2.5.

0:13:16.610,0:13:18.840
So once again, this is[br]equal the negative 2.

0:13:18.840,0:13:20.340
So I just wanted to show you,[br]it doesn't matter which one

0:13:20.340,0:13:23.090
you pick as the starting or[br]the endpoint, as long as

0:13:23.090,0:13:23.980
you're consistent.

0:13:23.980,0:13:26.650
If this is the starting y,[br]this is the starting x.

0:13:26.650,0:13:28.370
If this is the finishing[br]y, this has to be

0:13:28.370,0:13:29.500
the finishing x.

0:13:29.500,0:13:33.100
But anyway, we know that the[br]slope is negative 2.

0:13:33.100,0:13:36.540
So we know the equation is y is[br]equal to negative 2x plus

0:13:36.540,0:13:39.170
some y-intercept.

0:13:39.170,0:13:40.720
Let's use one of these[br]coordinates.

0:13:40.720,0:13:43.430
I'll use this one since it[br]doesn't have a decimal in it.

0:13:43.430,0:13:47.450
So we know that y[br]is equal to 2.

0:13:47.450,0:13:52.630
So y is equal to 2 when x[br]is equal to negative 1.

0:13:55.140,0:13:57.290
Of course you have[br]your plus b.

0:13:57.290,0:14:02.710
So 2 is equal to negative 2[br]times negative 1 is 2 plus b.

0:14:02.710,0:14:06.390
If you subtract 2 from both[br]sides of this equation, minus

0:14:06.390,0:14:10.370
2, minus 2, you're subtracting[br]it from both sides of this

0:14:10.370,0:14:12.480
equation, you're going to get[br]0 on the left-hand side is

0:14:12.480,0:14:14.520
equal to b.

0:14:14.520,0:14:15.670
So b is 0.

0:14:15.670,0:14:18.430
So the equation of our[br]line is just y is

0:14:18.430,0:14:19.680
equal to negative 2x.

0:14:22.040,0:14:23.870
Actually if you wanted to write[br]it in function notation,

0:14:23.870,0:14:28.190
it would be that f of x is[br]equal to negative 2x.

0:14:28.190,0:14:30.810
I kind of just assumed that[br]y is equal to f of x.

0:14:30.810,0:14:32.420
But this is really[br]the equation.

0:14:32.420,0:14:33.990
They never mentioned y's here.

0:14:33.990,0:14:37.890
So you could just write f of x[br]is equal to 2x right here.

0:14:37.890,0:14:40.190
Each of these coordinates[br]are the coordinates

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of x and f of x.

0:14:46.960,0:14:49.960
So you could even view the[br]definition of slope as change

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in f of x over change in x.

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These are all equivalent ways[br]of viewing the same thing.