0:00:01.480,0:00:05.730
This unit is about the equation[br]of a straight line.

0:00:06.570,0:00:10.040
The equation of a straight line[br]can take different forms

0:00:10.040,0:00:13.510
depending upon the information[br]that we know about the line.

0:00:13.510,0:00:16.980
Let's start by a specific[br]example. Suppose we've got some.

0:00:17.500,0:00:22.660
Points. Labeled by their X&Y[br]coordinates. So suppose we have

0:00:22.660,0:00:25.900
a point where X is not why is 2?

0:00:26.590,0:00:33.700
X is one. Why is 3X is 2?[br]Why is 4 and access three? Why

0:00:33.700,0:00:39.388
is 5? Let's see what these[br]points look like when we put

0:00:39.388,0:00:41.284
them on a graph.

0:00:42.230,0:00:44.900
The first point, not 2.

0:00:45.680,0:00:46.379
Will be here.

0:00:47.690,0:00:50.560
An X coordinate of zero[br]and a Y coordinate 2.

0:00:51.890,0:00:56.180
The second point 1 three X[br]coordinate of one Y coordinate

0:00:56.180,0:00:59.470
of three. And so on 2 four.

0:01:00.490,0:01:01.718
That will be here.

0:01:02.830,0:01:03.868
And three 5.

0:01:05.520,0:01:07.280
That will be there.

0:01:07.920,0:01:12.648
See, we've got four points and[br]very conveniently we can put a

0:01:12.648,0:01:16.982
straight line through them.[br]Notice that in every case, the Y

0:01:16.982,0:01:22.892
value is always two more than[br]the X value, so if we add on two

0:01:22.892,0:01:29.196
to zero, we get two. If we add[br]on 2 to one, we get three, and

0:01:29.196,0:01:35.106
so on. the Y value is always the[br]X value plus two, so this gives

0:01:35.106,0:01:37.864
us the equation of the line the

0:01:37.864,0:01:41.958
Y value. Is always the X[br]value +2.

0:01:43.250,0:01:47.670
Now there are lots and lots of[br]other points on this line, not

0:01:47.670,0:01:51.410
just the four that we've[br]plotted, but any point that we

0:01:51.410,0:01:54.810
choose on the line will have[br]this same relationship between

0:01:54.810,0:01:59.570
Y&X. the Y value will always be[br]it's X value plus two, so that

0:01:59.570,0:02:03.990
is the equation of the line, and[br]very often we'll label the line

0:02:03.990,0:02:07.050
with the equation by writing it[br]alongside like that.

0:02:08.600,0:02:10.922
Let's look at some more straight

0:02:10.922,0:02:18.408
line graphs. Let's suppose[br]we start with the

0:02:18.408,0:02:26.312
equation Y equals X[br]or drop a table

0:02:26.312,0:02:30.264
of values and plot

0:02:30.264,0:02:37.260
some points. Again,[br]let's start with some

0:02:37.260,0:02:43.692
X values. Suppose the[br]X values run from

0:02:43.692,0:02:46.908
012 up to three.

0:02:47.720,0:02:51.822
What will the Y value be if the[br]equation is simply Y equals X?

0:02:51.822,0:02:55.631
Well, in this case it's a very[br]simple case. the Y value is

0:02:55.631,0:02:59.440
always equal to the X value. So[br]very simply we can complete the

0:02:59.440,0:03:02.956
table. the Y value is always the[br]same as the X value.

0:03:03.650,0:03:07.472
Let's plot these points on the

0:03:07.472,0:03:11.380
graph. Access note why is not.

0:03:11.990,0:03:13.328
Is the point of the origin.

0:03:14.770,0:03:16.660
X is one. Why is one?

0:03:17.240,0:03:18.599
Will be here.

0:03:19.140,0:03:21.168
And similarly 2233.

0:03:21.680,0:03:24.070
Will be there. And there.

0:03:24.930,0:03:27.138
So we have a straight line.

0:03:28.320,0:03:30.888
Passing through the origin.

0:03:32.400,0:03:35.856
Let's ask ourselves a little bit[br]about the gradient of this line.

0:03:36.480,0:03:40.393
Remember to find the gradient of[br]the line we take, say two points

0:03:40.393,0:03:44.005
on it. Let's suppose we take[br]this point and this point, and

0:03:44.005,0:03:47.918
we calculate the change in Y[br]divided by the change in X. As

0:03:47.918,0:03:50.025
we move from one point to the

0:03:50.025,0:03:54.823
next. Well, as we move from here[br]to here, why changes from one to

0:03:54.823,0:03:58.406
three? So the change in Y is 3 -

0:03:58.406,0:04:03.784
1. And the change in X will[br]exchange is from one to three,

0:04:03.784,0:04:07.132
so the change in X is also 3 -

0:04:07.132,0:04:12.610
1. 3 - 1 is two 3[br]- 1 is 2.

0:04:13.260,0:04:15.906
So the gradient of this line is

0:04:15.906,0:04:19.334
one. Want to write that[br]alongside here? Let's call

0:04:19.334,0:04:22.832
it M1. This is the first[br]line of several lines I'm

0:04:22.832,0:04:26.330
going to draw. An M1 is[br]one. The gradient is one.

0:04:27.520,0:04:30.140
And also write the equation[br]of the line alongside as

0:04:30.140,0:04:33.022
well. So the equation of[br]this line is why is X?

0:04:35.260,0:04:39.304
Let's put another straight line[br]on the same graph and this time.

0:04:39.304,0:04:42.337
Let's suppose we choose the[br]equation Y equals 2X.

0:04:44.190,0:04:47.133
Let's see what the Y[br]coordinates will be this

0:04:47.133,0:04:50.730
time. Well, the Y coordinate[br]is always two times the X

0:04:50.730,0:04:54.000
coordinate, so if the X[br]coordinate is 0, the Y

0:04:54.000,0:04:57.270
coordinate will be 2 * 0,[br]which is still 0.

0:04:58.780,0:05:01.510
When X is one, why will be 2 * 1

0:05:01.510,0:05:06.130
which is 2? When X is 2, Y is 2[br]* 2, which is 4.

0:05:06.870,0:05:09.190
I'm an ex is 3. Why is 2 * 3

0:05:09.190,0:05:14.951
which is 6? Let's put these on[br]as well. We've got 00, which is

0:05:14.951,0:05:16.154
the origin again.

0:05:17.220,0:05:19.796
When X is one way is 2. That's

0:05:19.796,0:05:24.284
this point here. When X is 2,[br]why is 4?

0:05:25.060,0:05:28.602
At this point here, I'm going to[br]access three wise 6.

0:05:29.140,0:05:34.678
She's up there and again we have[br]a straight line graph and again.

0:05:35.690,0:05:39.776
This line passes through[br]the origin.

0:05:40.910,0:05:45.030
Right, so let's write its[br]equation alongside. Why is 2X?

0:05:45.670,0:05:48.827
And let's just think for a[br]minute about the gradient of

0:05:48.827,0:05:51.697
this line. Let's take two[br]points. Let's suppose we take

0:05:51.697,0:05:53.132
this point and this point.

0:05:53.910,0:05:55.218
The change in Y.

0:05:56.600,0:06:00.110
Well, why is changing from two[br]to four? So the changing? Why is

0:06:00.110,0:06:01.730
4 takeaway 2 which is 2?

0:06:02.380,0:06:06.524
The change in X will exchange is[br]from one to two, so the change

0:06:06.524,0:06:10.964
in X is just 2 - 1 or one, so[br]the slope of this line.

0:06:11.520,0:06:13.300
Just two.

0:06:14.920,0:06:20.809
That's cool that M2 is the slope[br]of the second line, right, M22?

0:06:23.340,0:06:27.553
OK, let's do one more. Suppose[br]we have another equation. And

0:06:27.553,0:06:32.149
let's suppose this time the[br]equation is Y equals 3X. So the

0:06:32.149,0:06:37.511
Y value is always three times[br]the X value. We can put these in

0:06:37.511,0:06:41.724
straightaway 3 notes and not[br]314-3326 and three threes and 9.

0:06:42.280,0:06:44.017
And we can plot these on[br]the same graph.

0:06:45.130,0:06:48.898
Again, 00 so the graph is going[br]to pass through the origin.

0:06:50.440,0:06:55.330
Two, when X is one, why is[br]3 so when X is one? Why is

0:06:55.330,0:06:56.960
3 gives me this point?

0:06:58.650,0:07:03.594
When X is 2, why is now 6? So[br]I've got a point up here and

0:07:03.594,0:07:07.302
that's sufficient to to draw in[br]the straight line and again with

0:07:07.302,0:07:10.701
a straight line passing through[br]the origin is a steeper line

0:07:10.701,0:07:13.424
this time. And it's equation is

0:07:13.424,0:07:19.011
Y is 3X. So we've got 3 lines[br]drawn. Now, why is XY is 2, XY

0:07:19.011,0:07:21.090
is 3X and all these lines pass

0:07:21.090,0:07:24.984
through the origin? Let's just[br]get the gradient of this line or

0:07:24.984,0:07:28.020
the gradient of this line.[br]Again. Let's take two points on

0:07:28.020,0:07:32.402
it. The change in Y going from[br]this point to this point. Well,

0:07:32.402,0:07:36.617
why is changing from 3 up to[br]six? So the change in Y is 6?

0:07:36.617,0:07:37.741
Subtract 3 or three.

0:07:38.450,0:07:42.740
And the change in XLX is[br]changing from one to two.

0:07:43.250,0:07:47.822
So the change in access 2 - 1,[br]which is just one.

0:07:47.830,0:07:52.534
So the gradient this time is 3.[br]Let's label that and three.

0:07:53.340,0:07:56.630
Now this is no coincidence.[br]You'll notice that in every

0:07:56.630,0:08:01.236
case, the gradient in this case[br]3 is the same as the number that

0:08:01.236,0:08:05.184
is multiplying the X in the[br]equation. Same is true here. The

0:08:05.184,0:08:09.132
gradient is 2, which is the[br]number. Multiplying the X in the

0:08:09.132,0:08:13.080
equation. And again here. Why is[br]X the number? Multiplying X is

0:08:13.080,0:08:15.054
one and the gradient is one.

0:08:15.770,0:08:20.445
Now we deduce from this a[br]general result that whenever we

0:08:20.445,0:08:25.545
have an equation of the form Y[br]equals MX. What this represents

0:08:25.545,0:08:27.245
is a straight line.

0:08:28.320,0:08:34.107
It's a line which is passing[br]through the origin.

0:08:34.110,0:08:40.918
And it's gradient is[br]M. The number multiplying

0:08:40.918,0:08:44.322
the X is the

0:08:44.322,0:08:47.988
gradient. That's a very[br]important result, it's well

0:08:47.988,0:08:52.322
worth remembering that whenever[br]you see why is a constant M

0:08:52.322,0:08:57.050
Times X will be a straight line[br]will be passing through the

0:08:57.050,0:09:01.384
origin and the gradient will be[br]the number that's multiplying X.

0:09:02.590,0:09:09.232
Let's have a[br]look at some

0:09:09.232,0:09:12.553
other equations of

0:09:12.553,0:09:19.794
straight lines. Let's[br]have a look at Y equals 2X

0:09:19.794,0:09:24.432
plus one. Very similar to the[br]one we had before, but now I've

0:09:24.432,0:09:27.968
added on a number at the end[br]here. Let's choose some X and

0:09:27.968,0:09:35.316
some Y values. When[br]X is

0:09:35.316,0:09:41.914
0. Why will be[br]2 * 0 which is 0 plus one? So

0:09:41.914,0:09:43.816
when X is 0 while B1.

0:09:44.670,0:09:49.246
When X is one 2, one or two plus[br]one gives you 3.

0:09:50.250,0:09:54.765
And when X is 222 to four and[br]one is 5, so with those three

0:09:54.765,0:09:58.678
points we can plot a graph when[br]X is not. Why is one?

0:09:59.320,0:10:00.300
It's there.

0:10:01.360,0:10:05.000
When X is one, why is 3? So we[br]come up to here.

0:10:05.760,0:10:09.361
I'm in access 2. Why is 5 which[br]takes us up to there?

0:10:10.020,0:10:15.537
And there's my straight line[br]graph through those points.

0:10:17.140,0:10:22.000
Not label it. Y equals 2X[br]plus one.

0:10:24.220,0:10:29.368
Let's have another one. Suppose[br]we have Y equals 2 X +4.

0:10:30.690,0:10:32.045
Let's see what happens this

0:10:32.045,0:10:39.776
time. Let's suppose we[br]start with a negative

0:10:39.776,0:10:41.732
X value.

0:10:43.030,0:10:46.882
Access minus one. What will the[br]Y value be? Effects is minus,

0:10:46.882,0:10:49.129
one will get two times minus one

0:10:49.129,0:10:53.410
is minus 2. And 4 - 2[br]is 2.

0:10:56.040,0:11:01.735
Let's choose X to be 0 when X is[br]zero, will get 2 zeros as O plus

0:11:01.735,0:11:06.760
454. So ex is one we get 2 ones[br]or 2 + 4 is 6.

0:11:07.420,0:11:08.560
Let's put those points.

0:11:09.190,0:11:11.404
So if X is minus one, why is 2?

0:11:13.870,0:11:15.900
If X is zero, why is 4?

0:11:18.420,0:11:22.359
And effects is one. Why is 6,[br]which is a point of the.

0:11:23.230,0:11:28.114
That's the line Y equals 2X[br]plus four, and you'll notice

0:11:28.114,0:11:33.442
from looking at it that the[br]two lines that we have now

0:11:33.442,0:11:37.438
drawn a parallel, and that's[br]precisely because they've got

0:11:37.438,0:11:40.546
the same gradient. The number[br]multiplying X.

0:11:42.310,0:11:46.700
Let's look at one more. Why is[br]2X minus one?

0:11:47.250,0:11:49.170
Again, let's have some X values.

0:11:49.720,0:11:53.158
And some Y values supposing X

0:11:53.158,0:11:57.992
is 0. Well, if X is zero and why[br]is 2X minus one?

0:11:58.510,0:12:02.654
The Y value will be 2 notes and[br]not subtract. 1 is minus one.

0:12:03.860,0:12:09.380
If X is, one will get 2 ones, or[br]two. Subtract 1 is plus one.

0:12:10.490,0:12:11.850
And effects is 2.

0:12:12.350,0:12:16.662
Two tubes of 4 - 1 is 3. Again,[br]we've got three points. That's

0:12:16.662,0:12:20.358
plenty points to put on the[br]graph. Effects is not. Why is

0:12:20.358,0:12:23.984
minus one? Thanks is not[br]wise minus one gives me a

0:12:23.984,0:12:24.782
point down here.

0:12:27.040,0:12:28.100
If X is one.

0:12:28.690,0:12:30.930
Why is one gives[br]me a point here?

0:12:32.340,0:12:35.724
And if X is 2 wise, three gives[br]me that point there.

0:12:39.930,0:12:46.242
And there's the straight line Y[br]equals 2X minus one, and again

0:12:46.242,0:12:47.820
this third line.

0:12:48.540,0:12:51.972
Is parallel to the previous two[br]lines and it's parallel because

0:12:51.972,0:12:55.404
it's got the same gradient and[br]it's got the same gradient

0:12:55.404,0:12:59.148
because in every case we've got[br]2X the number. Multiplying X is

0:12:59.148,0:13:03.459
the same. So what's different[br]about the lines? Well, what is

0:13:03.459,0:13:05.014
different is that they're all

0:13:05.014,0:13:08.324
cutting. The Y axis[br]at a different point.

0:13:09.540,0:13:13.782
This line is cutting the Y axis[br]at the point where. Why is 4?

0:13:14.940,0:13:17.289
Note that the number 4[br]appears in the equation.

0:13:18.930,0:13:23.960
This line. Cuts the Y axis when[br]Y is one, and again one appears

0:13:23.960,0:13:29.040
in the equation. And again, this[br]line cuts the Y axis at minus

0:13:29.040,0:13:33.291
one and minus one appears in the[br]equation, and this gives us a

0:13:33.291,0:13:37.542
general rule. If we have an[br]equation of the form Y equals MX

0:13:37.542,0:13:42.447
plus C, the number that is on[br]its own at the end. Here the C

0:13:42.447,0:13:46.698
which was the four or the one or[br]the minus one, tells us

0:13:46.698,0:13:48.660
whereabouts on the Y axis that

0:13:48.660,0:13:51.746
the graph cuts. And we call this

0:13:51.746,0:13:57.520
value. Either the for their or[br]the one there, or the minus one

0:13:57.520,0:14:02.590
there. We call that the vertical[br]intercept so the value of C is

0:14:02.590,0:14:03.760
the vertical intercept.

0:14:03.840,0:14:09.980
So now whenever you see[br]an equation of the form

0:14:09.980,0:14:15.506
Y equals a number times[br]X plus another number.

0:14:15.506,0:14:19.190
So why equals MX plus C?

0:14:20.330,0:14:22.205
That represents a straight line

0:14:22.205,0:14:24.905
graph. Where M is the gradient

0:14:24.905,0:14:29.190
of the line. And sees the[br]vertical intercept, which is the

0:14:29.190,0:14:31.950
place where the graph crosses[br]the vertical axis.

0:14:33.480,0:14:39.900
Now sometimes when we get the[br]equation of a straight line, it

0:14:39.900,0:14:46.320
doesn't always appear in the[br]form Y equals MX plus C. Let

0:14:46.320,0:14:48.995
me give you an example.

0:14:49.000,0:14:53.901
Let's consider this equation 3.[br]Y minus two X equals 6. Now at

0:14:53.901,0:14:58.802
first sight that doesn't look as[br]though it's in the form Y equals

0:14:58.802,0:15:02.572
MX plus C which is our[br]recognisable form of the

0:15:02.572,0:15:08.227
equation of a straight line. But[br]what we can do is we can do some

0:15:08.227,0:15:12.751
algebraic manipulation on this[br]to try to write it in this form

0:15:12.751,0:15:18.029
and one of the advantages of[br]doing that is that if we can get

0:15:18.029,0:15:19.537
it into this form.

0:15:19.540,0:15:23.180
We can read off what the[br]gradients and the vertical

0:15:23.180,0:15:25.364
intercept are, so let's work on

0:15:25.364,0:15:30.408
this. I'll start by adding 2X[br]to both sides.

0:15:30.410,0:15:35.366
To remove this minus 2X from[br]here. So if we add 2X to both

0:15:35.366,0:15:37.844
sides will get 2X plus six on

0:15:37.844,0:15:43.416
the right. And now if I divide[br]both sides by three, I'll get Y

0:15:43.416,0:15:45.922
on its own, which is what I'm

0:15:45.922,0:15:51.690
looking for. Dividing 2X by[br]three gives me why is 2/3 of

0:15:51.690,0:15:57.234
X? And if I divide 6 by[br]three, I'll get 2.

0:15:57.240,0:16:02.378
Now this is a much more familiar[br]form. This is of the form Y

0:16:02.378,0:16:07.516
equals MX plus. See where we can[br]read off the gradient M is 2/3

0:16:07.516,0:16:09.351
and the vertical intercept see

0:16:09.351,0:16:13.645
is 2. So be aware that sometimes[br]an equation that you see might

0:16:13.645,0:16:16.340
not at first sight look as[br]though it's a straight line

0:16:16.340,0:16:19.770
equation, but by doing some work[br]on it you can get it into a

0:16:19.770,0:16:25.600
recognizable form. About[br]another

0:16:25.600,0:16:31.022
example. Suppose we're given[br]some information about a

0:16:31.022,0:16:36.690
straight line graph, and we want[br]to try and find out what the

0:16:36.690,0:16:41.050
equation is. So, for example,[br]suppose we're told that a

0:16:41.050,0:16:45.846
straight line has gradient, a[br]fifth and were told also that

0:16:45.846,0:16:48.026
it's vertical intercept. See is

0:16:48.026,0:16:51.350
one. Let's see if we can[br]write down the equation.

0:16:52.500,0:16:56.208
Well, we know that a[br]straight line has equation

0:16:56.208,0:16:58.268
Y equals MX plus C.

0:16:59.620,0:17:05.220
So we can substitute are known[br]values in M is going to be 1/5.

0:17:05.220,0:17:11.220
See is going to be one. So our[br]equation is Y equals 1/5 of X

0:17:11.220,0:17:14.020
plus one Y equals MX plus C.

0:17:14.770,0:17:18.718
Now we might not always choose[br]to leave it in that form, so let

0:17:18.718,0:17:22.102
me just show you how else we[br]might write it. There's a

0:17:22.102,0:17:25.204
fraction here of the 5th, and if[br]we multiply everything through

0:17:25.204,0:17:28.306
by 5, we can remove this[br]fraction. So let's multiply both

0:17:28.306,0:17:31.972
sides by 5, will get 5 Y the[br]files or cancel. When we

0:17:31.972,0:17:33.946
multiply by 5 here just to leave

0:17:33.946,0:17:36.810
X. And five ones of five.

0:17:37.620,0:17:42.690
So this form is equivalent to[br]this form, but just written in a

0:17:42.690,0:17:46.615
different way. We could[br]rearrange it again just by

0:17:46.615,0:17:50.520
bringing everything over to the[br]left hand side, so we might

0:17:50.520,0:17:55.845
write 5 Y minus X. Minus five is[br]not, so that is another form of

0:17:55.845,0:17:59.750
the same equation and we'll see[br]some equations written in this

0:17:59.750,0:18:01.170
form which later on.

0:18:02.410,0:18:09.310
OK, let's have[br]a look at

0:18:09.310,0:18:14.353
another example. Suppose now[br]we're interested in trying to

0:18:14.353,0:18:17.821
find the equation of a line[br]which has a gradient of 1/3.

0:18:18.960,0:18:22.470
And this time, instead of being[br]given the vertical intercept,

0:18:22.470,0:18:25.980
we're going to be given some[br]information about a point

0:18:25.980,0:18:30.192
through which the line passes.[br]So suppose that we know that the

0:18:30.192,0:18:33.000
line passes through the points[br]with coordinates 12.

0:18:33.800,0:18:36.374
Let's see if we can figure[br]out what the equation of

0:18:36.374,0:18:37.076
the line is.

0:18:39.090,0:18:44.508
Start with our general form Y[br]equals MX plus C and we put in

0:18:44.508,0:18:46.443
the information that we already

0:18:46.443,0:18:52.570
know. We know that the gradient[br]M is 1/3, so we can put that in

0:18:52.570,0:18:53.665
here straight away.

0:18:53.680,0:18:57.310
We don't know the vertical[br]intercept. We're going to have

0:18:57.310,0:19:00.214
to do a bit of work to find

0:19:00.214,0:19:05.263
that. But what we do know is[br]that the line passes through

0:19:05.263,0:19:10.485
this point. What that means is[br]that when X is one, why has the

0:19:10.485,0:19:14.588
value 2? And we can use that[br]information in this equation.

0:19:15.380,0:19:17.382
So we're going to put, why is

0:19:17.382,0:19:23.620
2IN. X is one in home, 3[br]third times, one is just a

0:19:23.620,0:19:28.879
third. Let's see from this we[br]can workout what Sears.

0:19:29.440,0:19:33.968
So two is the same as 6 thirds[br]and if we take a third off, both

0:19:33.968,0:19:38.213
sides will have 5 thirds is see[br]so you can see we can use the

0:19:38.213,0:19:39.911
information about a point on the

0:19:39.911,0:19:44.450
line. To find the vertical[br]intercept, see so. Now we know

0:19:44.450,0:19:48.180
everything about this line. We[br]know it's vertical intercept and

0:19:48.180,0:19:53.029
we know its gradient. So the[br]equation of the line is why is

0:19:53.029,0:19:54.521
1/3 X +5 thirds?

0:19:54.530,0:20:00.905
I want to do that again, but I[br]want to do it for more general

0:20:00.905,0:20:05.155
case where we haven't got[br]specific values for the gradient

0:20:05.155,0:20:09.830
and we haven't got specific[br]values for the point. So this

0:20:09.830,0:20:14.505
time, let's suppose we've got a[br]straight line. This gradient is

0:20:14.505,0:20:20.163
M. But it passes through a[br]point with arbitrary coordinates

0:20:20.163,0:20:21.590
X one. I want.

0:20:22.250,0:20:24.900
Let's see if we can[br]find a formula for the

0:20:24.900,0:20:25.960
equation of the line.

0:20:27.620,0:20:31.676
Always go back to what we know.[br]We know already that any

0:20:31.676,0:20:35.056
straight line has this equation.[br]Y is MX plus C.

0:20:35.070,0:20:40.230
What do we know what we're told[br]the gradient is M so that we can

0:20:40.230,0:20:43.430
leave alone. But we don't know[br]the vertical intercept. See

0:20:43.430,0:20:45.040
let's see if we can find it.

0:20:46.240,0:20:49.698
Use what we do now. We do know[br]that the line passes through

0:20:49.698,0:20:54.660
this point. So that we know that[br]when X has the value X one.

0:20:55.530,0:20:57.258
Why has the value? Why one?

0:20:57.880,0:20:59.322
So I'm going to put those values

0:20:59.322,0:21:06.444
in here. So why has the value?[br]Why one when X has the value

0:21:06.444,0:21:13.118
X one? See, now we can rearrange[br]this to find C. So take the MX

0:21:13.118,0:21:14.862
one off both sides.

0:21:15.740,0:21:20.444
That will give me the value for[br]C and this value for see that

0:21:20.444,0:21:24.476
we have found, which you[br]realize now is made up of. Only

0:21:24.476,0:21:29.852
the things we knew. We knew the[br]M we knew the X one and Y one.

0:21:29.852,0:21:34.556
So in fact we know this value.[br]Now we put this value back into

0:21:34.556,0:21:38.924
the general equation so will[br]have Y equals MX plus C and see

0:21:38.924,0:21:40.268
now is all this.

0:21:41.610,0:21:45.786
And that is the equation of a[br]line with gradient M passing

0:21:45.786,0:21:50.310
through X one Y one. We don't[br]normally leave it in this form.

0:21:50.310,0:21:53.790
We write it in a slightly[br]different way. It's usually

0:21:53.790,0:21:57.966
written like this. We subtract[br]why one off both sides to give

0:21:57.966,0:22:00.750
us Y, minus Y one, and that will

0:22:00.750,0:22:06.632
disappear. And we factorize the[br]MX and the NX one by taking out

0:22:06.632,0:22:12.386
the common factor of M will be[br]left with X and minus X one.

0:22:13.290,0:22:18.361
And that is an important result,[br]because this formula gives us

0:22:18.361,0:22:20.666
the equation of a line.

0:22:21.490,0:22:25.270
With gradient M and which passes[br]through a point where the X

0:22:25.270,0:22:28.735
coordinate is X one and the Y[br]coordinate is why one?

0:22:29.330,0:22:36.476
Let's look at[br]a specific example.

0:22:36.650,0:22:41.610
Suppose we're interested in a[br]straight line where the gradient

0:22:41.610,0:22:46.570
is minus 2, and it passes[br]through the point with

0:22:46.570,0:22:48.554
coordinates minus 3 two.

0:22:50.420,0:22:55.556
We know the general form of a[br]straight line, it's why minus

0:22:55.556,0:23:00.692
why one is MX minus X one?[br]That's our general results and

0:23:00.692,0:23:05.828
all we need to do is put this[br]information into this formula.

0:23:06.690,0:23:11.079
Why one is the Y coordinate[br]coordinate of the known point

0:23:11.079,0:23:12.276
which is 2?

0:23:12.450,0:23:14.445
M is the gradient which is minus

0:23:14.445,0:23:20.002
2. X minus X one is the X[br]coordinates of the known point,

0:23:20.002,0:23:21.466
which is minus 3.

0:23:22.020,0:23:27.774
So tidying this up, we've got Y[br]minus two is going to be minus

0:23:27.774,0:23:33.117
2X plus three, and if we remove[br]the brackets, Y minus two is

0:23:33.117,0:23:35.172
minus 2 X minus 6.

0:23:36.380,0:23:40.790
And finally, if we add two to[br]both sides, we shall get why is

0:23:40.790,0:23:44.570
minus 2 X minus four, and that's[br]the equation of the straight

0:23:44.570,0:23:47.720
line with gradient minus 2[br]passing through this point. And

0:23:47.720,0:23:51.815
there's always a check you can[br]apply because we can look at the

0:23:51.815,0:23:55.595
final equation we've got and we[br]can observe from here that the

0:23:55.595,0:23:57.170
gradient is indeed minus 2.

0:23:57.800,0:24:01.706
And we can also pop in an X[br]value of minus three into here.

0:24:02.420,0:24:06.932
Minus two times minus three is[br]plus six and six takeaway four

0:24:06.932,0:24:10.692
is 2 and that's the[br]corresponding why value? So this

0:24:10.692,0:24:12.948
built in checks that you can

0:24:12.948,0:24:15.894
apply. Let's have a look at[br]another slightly different

0:24:15.894,0:24:19.710
example, and in this example[br]I'm not going to give you the

0:24:19.710,0:24:22.572
gradient of the line.[br]Instead, we're going to have

0:24:22.572,0:24:26.070
two points on the line. So[br]let's suppose are two points

0:24:26.070,0:24:30.204
are minus 1, two and two 4,[br]so we don't know the gradient

0:24:30.204,0:24:33.066
and we don't know the[br]vertical intercept. We just

0:24:33.066,0:24:36.882
know two points on the line,[br]and we've got to try to

0:24:36.882,0:24:39.426
determine what the equation[br]of the line is.

0:24:40.660,0:24:42.452
Now let's see how we can do

0:24:42.452,0:24:47.202
this. One thing we can do is we[br]can calculate the gradient of

0:24:47.202,0:24:50.694
the line because we know how to[br]calculate the gradient of the

0:24:50.694,0:24:51.858
line joining two points.

0:24:52.380,0:24:57.084
So let's do that. First of all,[br]the gradient of the line will be

0:24:57.084,0:25:00.780
the difference in the Y[br]coordinates, which is 4 - 2.

0:25:01.200,0:25:06.864
Divided by the difference in the[br]X coordinates, which is 2 minus

0:25:06.864,0:25:13.710
minus one. 4 - 2 is 2 and 2[br]minus minus one is 2 + 1 which

0:25:13.710,0:25:17.590
is 3. So the gradient of this[br]line is 2/3.

0:25:18.660,0:25:23.054
Now we know the gradients and we[br]know at least one point through

0:25:23.054,0:25:27.448
which the line passes. Because[br]we know two. So we can use the

0:25:27.448,0:25:32.180
previous formula Y minus Y one[br]is MX minus X one. So that's the

0:25:32.180,0:25:33.194
formula we use.

0:25:33.930,0:25:37.725
Why minus why? One doesn't[br]matter which of the two points

0:25:37.725,0:25:39.795
we take? Let's take the .24.

0:25:40.780,0:25:44.350
So the Y coordinate is 4.

0:25:44.450,0:25:47.650
M we found is 2/3.

0:25:47.650,0:25:52.291
X minus the X coordinate, which[br]is 2. So that's our equation of

0:25:52.291,0:25:57.646
the line and if we wanted to do[br]we can tidy this up a little

0:25:57.646,0:26:03.358
bit. Y minus four is 2/3 of X[br]minus 4 thirds, and if we add 4

0:26:03.358,0:26:07.999
to both sides, we can write this[br]as Y is 2X over 3.

0:26:08.630,0:26:12.557
And we've got minus 4 thirds[br]here already, and we're bringing

0:26:12.557,0:26:17.198
over four will be adding four.[br]So finally will have two X over

0:26:17.198,0:26:23.266
3. And four is the same as 12[br]thirds, 12 thirds. Subtract 4

0:26:23.266,0:26:28.029
thirds is 8 thirds. So that's[br]the equation of the line.

0:26:28.610,0:26:34.231
I want to do that same[br]argument when were given two

0:26:34.231,0:26:37.808
arbitrary points instead of[br]two specific points.

0:26:39.030,0:26:43.879
So suppose we have the point a[br]coordinates X one and Y 1.

0:26:44.840,0:26:48.284
And be with coordinates X2Y2[br]and. Let's see if we can figure

0:26:48.284,0:26:51.441
out what the equation of the[br]line is joining these two

0:26:51.441,0:26:55.459
points, and I think in this case[br]a graph is going to help us.

0:26:56.240,0:26:58.280
Just have a quick sketch.

0:26:59.040,0:27:05.070
So I've got[br]a point A.

0:27:05.760,0:27:08.468
Coordinates X One Y1.

0:27:09.410,0:27:13.690
And another point B[br]coordinates X2Y2 and were

0:27:13.690,0:27:19.040
interested in the equation of[br]this line which is joining

0:27:19.040,0:27:19.575
them.

0:27:21.070,0:27:26.009
Suppose we pick an arbitrary[br]point on the line anywhere along

0:27:26.009,0:27:30.499
the line at all, and let's call[br]that point P.

0:27:31.090,0:27:34.570
P is an arbitrary point, and[br]let's suppose it's coordinates

0:27:34.570,0:27:37.702
are X&Y. For any arbitrary X&Y[br]on the line.

0:27:38.610,0:27:43.935
And what we do know is that the[br]gradient of AP is the same as

0:27:43.935,0:27:45.710
the gradient of a bee.

0:27:46.370,0:27:49.674
Let me write that[br]down the gradient.

0:27:50.730,0:27:57.160
Of AP. Is[br]equal to the gradient.

0:27:57.310,0:28:01.189
Of a B.

0:28:01.190,0:28:04.850
Let's see what that means. Well,[br]the gradient of AP.

0:28:06.400,0:28:08.332
Is just the difference in the Y

0:28:08.332,0:28:12.841
coordinates. Is Y minus[br]Y one over the

0:28:12.841,0:28:16.625
difference in the X[br]coordinates, which is X

0:28:16.625,0:28:18.044
minus X one?

0:28:19.130,0:28:23.123
So that's the gradient of this[br]line segment between amp and

0:28:23.123,0:28:27.479
that's got to equal the gradient[br]of the line segment between A&B.

0:28:28.060,0:28:31.723
And once the gradient of the[br]line segment between A&B, well,

0:28:31.723,0:28:35.386
it's again. It's the difference[br]of the Y coordinates, which now

0:28:35.386,0:28:37.384
is Y 2 minus Y 1.

0:28:37.390,0:28:42.898
Divided by the difference in the[br]X coordinates which is X2 minus

0:28:42.898,0:28:47.488
X one. So that is a formula[br]which will tell us.

0:28:48.160,0:28:50.870
The equation of a line passing[br]through two arbitrary points.

0:28:50.870,0:28:54.122
Now we don't usually leave it in[br]that form. It's normally written

0:28:54.122,0:28:57.103
in a slightly different form,[br]and it's normally written in a

0:28:57.103,0:29:01.168
form so that all the wise appear[br]on one side and all the ex is

0:29:01.168,0:29:04.420
appear on another side, and we[br]can do that by dividing both

0:29:04.420,0:29:06.317
sides by Y 2 minus Y 1.

0:29:07.010,0:29:14.056
And multiplying both sides by X[br]minus X One which moves that up

0:29:14.056,0:29:21.090
to here. And that's the form[br]which is normally quoted as the

0:29:21.090,0:29:26.400
equation of a line passing[br]through two arbitrary points.

0:29:28.850,0:29:32.126
Let's use that in an example.

0:29:32.690,0:29:38.996
Suppose we have[br]two points A.

0:29:39.000,0:29:44.350
Coordinates one and minus two[br]and B which has coordinates

0:29:44.350,0:29:46.490
minus three and not.

0:29:47.220,0:29:52.576
Let me write down the formula[br]again. Why minus why one over Y

0:29:52.576,0:29:58.344
2 minus Y one is X Minus X one[br]over X2 minus X one?

0:29:59.010,0:30:02.429
With pop everything we know into[br]the formula and see what we get.

0:30:03.200,0:30:10.340
So we want Y minus Y1Y one is[br]the first of the Y values which

0:30:10.340,0:30:11.768
is minus 2.

0:30:12.000,0:30:16.940
Why 2 minus? Why one is the[br]difference of the Y values that

0:30:16.940,0:30:20.400
zero? Minus minus 2.

0:30:20.910,0:30:27.126
Equals X minus X one is the[br]first of the X values, which is

0:30:27.126,0:30:32.898
one and X2 minus X. One is the[br]difference of the X values.

0:30:32.898,0:30:35.118
That's minus three, subtract 1.

0:30:35.980,0:30:41.164
And just to tidy this up on the[br]top line here will get Y +2 on

0:30:41.164,0:30:43.108
the bottom line. Here will get

0:30:43.108,0:30:47.317
+2. X minus one there on[br]the right at the top and

0:30:47.317,0:30:49.396
minus 3 - 1 is minus 4.

0:30:51.870,0:30:56.112
Again, we can tie this up a[br]little bit more to will go into

0:30:56.112,0:30:57.627
minus 4 - 2 times.

0:30:58.190,0:31:02.370
And if we multiply everything[br]through by minus, two will get

0:31:02.370,0:31:07.690
minus two Y minus four equals X[br]minus one, and we can write this

0:31:07.690,0:31:12.250
in lots of different ways. For[br]example, we could write this as

0:31:12.250,0:31:14.150
minus two Y minus X.

0:31:15.050,0:31:18.690
And we could add 1 to both[br]sides to give minus 3 zero.

0:31:18.690,0:31:21.210
That's one way we could leave[br]the final answer.

0:31:22.430,0:31:26.252
Another way we could leave it as[br]we could rearrange it to get Y

0:31:26.252,0:31:29.528
equals something. So if I do[br]that, I'll have minus two Y

0:31:29.528,0:31:30.893
equals X and if we.

0:31:31.670,0:31:37.091
Add 4 to both sides will get[br]plus three there, and if we

0:31:37.091,0:31:41.678
divide everything by minus two[br]will get minus 1/2 X minus

0:31:41.678,0:31:45.848
three over 2. So all of these[br]forms are equivalent.

0:31:47.480,0:31:53.852
Now, that's not quite the whole[br]story. The most general form of

0:31:53.852,0:31:58.100
equation of a straight line[br]looks like this.

0:31:58.430,0:32:03.510
And earlier on in this unit,[br]we've seen some equations

0:32:03.510,0:32:09.098
written in this form. Let's look[br]at some specific cases. Suppose

0:32:09.098,0:32:14.178
that a this number here turns[br]out to be 0.

0:32:14.740,0:32:16.468
What will that mean if a is 0?

0:32:17.000,0:32:20.180
But if a is zero, we can[br]rearrange this and write BY.

0:32:20.920,0:32:22.930
Equals minus C.

0:32:24.010,0:32:27.520
Why is minus C Overby?

0:32:29.270,0:32:32.942
And what does this mean?[br]Remember the A and the beat and

0:32:32.942,0:32:36.614
the CIA just numbers their[br]constants. So when a is zero, we

0:32:36.614,0:32:40.592
find that this number on the[br]right here minus C over B is

0:32:40.592,0:32:44.876
just a constant. So what this is[br]saying is that Y is a constant.

0:32:45.720,0:32:48.426
Now align where why is constant.

0:32:49.000,0:32:53.810
Must be. A horizontal line,[br]because why doesn't change

0:32:53.810,0:32:57.230
the value of Y is always[br]minus C Overby.

0:32:59.080,0:33:04.218
So if you have an equation of[br]this form where a is zero that

0:33:04.218,0:33:05.319
represents horizontal lines.

0:33:06.530,0:33:09.578
What about if be with zero?

0:33:09.910,0:33:15.370
We're putting B is 0 in here,[br]will get the AX Plus Co.

0:33:16.060,0:33:20.980
And if we rearrange, this will[br]get AX equals minus C and

0:33:20.980,0:33:24.670
dividing through by AX is minus[br]C over A.

0:33:25.990,0:33:30.566
Again, a encia constants so this[br]time what this is saying is that

0:33:30.566,0:33:31.974
X is a constant.

0:33:32.740,0:33:35.575
Now lines were X is a constant.

0:33:36.240,0:33:40.959
Must look like this. They are[br]vertical lines because the X

0:33:40.959,0:33:42.246
value doesn't change.

0:33:42.250,0:33:46.362
So this general case[br]includes both vertical lines

0:33:46.362,0:33:47.904
and horizontal lines.

0:33:49.030,0:33:52.660
So remember, the most general[br]form will appear like that.

0:33:54.200,0:33:57.130
Provided that be isn't zero,[br]you can always write the

0:33:57.130,0:34:00.646
equation in the more familiar[br]form Y equals MX plus C, but

0:34:00.646,0:34:04.455
in the case in which B is 0,[br]you get this specific case

0:34:04.455,0:34:05.920
where you've got vertical[br]lines.