WEBVTT

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This unit is about the equation
of a straight line.

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The equation of a straight line
can take different forms

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depending upon the information
that we know about the line.

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Let's start by a specific
example. Suppose we've got some.

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Points. Labeled by their X&Y
coordinates. So suppose we have

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a point where X is not why is 2?

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X is one. Why is 3X is 2?
Why is 4 and access three? Why

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is 5? Let's see what these
points look like when we put

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them on a graph.

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The first point, not 2.

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Will be here.

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An X coordinate of zero
and a Y coordinate 2.

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The second point 1 three X
coordinate of one Y coordinate

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of three. And so on 2 four.

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That will be here.

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And three 5.

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That will be there.

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See, we've got four points and
very conveniently we can put a

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straight line through them.
Notice that in every case, the Y

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value is always two more than
the X value, so if we add on two

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to zero, we get two. If we add
on 2 to one, we get three, and

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so on. the Y value is always the
X value plus two, so this gives

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us the equation of the line the

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Y value. Is always the X
value +2.

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Now there are lots and lots of
other points on this line, not

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just the four that we've
plotted, but any point that we

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choose on the line will have
this same relationship between

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Y&X. the Y value will always be
it's X value plus two, so that

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is the equation of the line, and
very often we'll label the line

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with the equation by writing it
alongside like that.

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Let's look at some more straight

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line graphs. Let's suppose
we start with the

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equation Y equals X
or drop a table

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of values and plot

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some points. Again,
let's start with some

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X values. Suppose the
X values run from

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012 up to three.

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What will the Y value be if the
equation is simply Y equals X?

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Well, in this case it's a very
simple case. the Y value is

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always equal to the X value. So
very simply we can complete the

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table. the Y value is always the
same as the X value.

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Let's plot these points on the

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graph. Access note why is not.

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Is the point of the origin.

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X is one. Why is one?

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Will be here.

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And similarly 2233.

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Will be there. And there.

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So we have a straight line.

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Passing through the origin.

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Let's ask ourselves a little bit
about the gradient of this line.

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Remember to find the gradient of
the line we take, say two points

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on it. Let's suppose we take
this point and this point, and

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we calculate the change in Y
divided by the change in X. As

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we move from one point to the

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next. Well, as we move from here
to here, why changes from one to

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three? So the change in Y is 3 -

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1. And the change in X will
exchange is from one to three,

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so the change in X is also 3 -

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1. 3 - 1 is two 3
- 1 is 2.

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So the gradient of this line is

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one. Want to write that
alongside here? Let's call

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it M1. This is the first
line of several lines I'm

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going to draw. An M1 is
one. The gradient is one.

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And also write the equation
of the line alongside as

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well. So the equation of
this line is why is X?

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Let's put another straight line
on the same graph and this time.

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Let's suppose we choose the
equation Y equals 2X.

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Let's see what the Y
coordinates will be this

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time. Well, the Y coordinate
is always two times the X

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coordinate, so if the X
coordinate is 0, the Y

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coordinate will be 2 * 0,
which is still 0.

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When X is one, why will be 2 * 1

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which is 2? When X is 2, Y is 2
* 2, which is 4.

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I'm an ex is 3. Why is 2 * 3

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which is 6? Let's put these on
as well. We've got 00, which is

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the origin again.

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When X is one way is 2. That's

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this point here. When X is 2,
why is 4?

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At this point here, I'm going to
access three wise 6.

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She's up there and again we have
a straight line graph and again.

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This line passes through
the origin.

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Right, so let's write its
equation alongside. Why is 2X?

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And let's just think for a
minute about the gradient of

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this line. Let's take two
points. Let's suppose we take

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this point and this point.

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The change in Y.

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Well, why is changing from two
to four? So the changing? Why is

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4 takeaway 2 which is 2?

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The change in X will exchange is
from one to two, so the change

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in X is just 2 - 1 or one, so
the slope of this line.

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Just two.

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That's cool that M2 is the slope
of the second line, right, M22?

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OK, let's do one more. Suppose
we have another equation. And

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let's suppose this time the
equation is Y equals 3X. So the

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Y value is always three times
the X value. We can put these in

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straightaway 3 notes and not
314-3326 and three threes and 9.

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And we can plot these on
the same graph.

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Again, 00 so the graph is going
to pass through the origin.

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Two, when X is one, why is
3 so when X is one? Why is

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3 gives me this point?

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When X is 2, why is now 6? So
I've got a point up here and

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that's sufficient to to draw in
the straight line and again with

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a straight line passing through
the origin is a steeper line

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this time. And it's equation is

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Y is 3X. So we've got 3 lines
drawn. Now, why is XY is 2, XY

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is 3X and all these lines pass

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through the origin? Let's just
get the gradient of this line or

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the gradient of this line.
Again. Let's take two points on

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it. The change in Y going from
this point to this point. Well,

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why is changing from 3 up to
six? So the change in Y is 6?

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Subtract 3 or three.

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And the change in XLX is
changing from one to two.

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So the change in access 2 - 1,
which is just one.

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So the gradient this time is 3.
Let's label that and three.

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Now this is no coincidence.
You'll notice that in every

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case, the gradient in this case
3 is the same as the number that

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is multiplying the X in the
equation. Same is true here. The

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gradient is 2, which is the
number. Multiplying the X in the

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equation. And again here. Why is
X the number? Multiplying X is

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one and the gradient is one.

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Now we deduce from this a
general result that whenever we

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have an equation of the form Y
equals MX. What this represents

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is a straight line.

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It's a line which is passing
through the origin.

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And it's gradient is
M. The number multiplying

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the X is the

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gradient. That's a very
important result, it's well

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worth remembering that whenever
you see why is a constant M

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Times X will be a straight line
will be passing through the

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origin and the gradient will be
the number that's multiplying X.

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Let's have a
look at some

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other equations of

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straight lines. Let's
have a look at Y equals 2X

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plus one. Very similar to the
one we had before, but now I've

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added on a number at the end
here. Let's choose some X and

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some Y values. When
X is

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0. Why will be
2 * 0 which is 0 plus one? So

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when X is 0 while B1.

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When X is one 2, one or two plus
one gives you 3.

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And when X is 222 to four and
one is 5, so with those three

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points we can plot a graph when
X is not. Why is one?

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It's there.

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When X is one, why is 3? So we
come up to here.

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I'm in access 2. Why is 5 which
takes us up to there?

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And there's my straight line
graph through those points.

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Not label it. Y equals 2X
plus one.

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Let's have another one. Suppose
we have Y equals 2 X +4.

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Let's see what happens this

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time. Let's suppose we
start with a negative

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X value.

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Access minus one. What will the
Y value be? Effects is minus,

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one will get two times minus one

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is minus 2. And 4 - 2
is 2.

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Let's choose X to be 0 when X is
zero, will get 2 zeros as O plus

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454. So ex is one we get 2 ones
or 2 + 4 is 6.

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Let's put those points.

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So if X is minus one, why is 2?

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If X is zero, why is 4?

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And effects is one. Why is 6,
which is a point of the.

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That's the line Y equals 2X
plus four, and you'll notice

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from looking at it that the
two lines that we have now

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drawn a parallel, and that's
precisely because they've got

00:11:37.438 --> 00:11:40.546
the same gradient. The number
multiplying X.

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Let's look at one more. Why is
2X minus one?

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Again, let's have some X values.

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And some Y values supposing X

00:11:53.158 --> 00:11:57.992
is 0. Well, if X is zero and why
is 2X minus one?

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The Y value will be 2 notes and
not subtract. 1 is minus one.

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If X is, one will get 2 ones, or
two. Subtract 1 is plus one.

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And effects is 2.

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Two tubes of 4 - 1 is 3. Again,
we've got three points. That's

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plenty points to put on the
graph. Effects is not. Why is

00:12:20.358 --> 00:12:23.984
minus one? Thanks is not
wise minus one gives me a

00:12:23.984 --> 00:12:24.782
point down here.

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If X is one.

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Why is one gives
me a point here?

00:12:32.340 --> 00:12:35.724
And if X is 2 wise, three gives
me that point there.

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And there's the straight line Y
equals 2X minus one, and again

00:12:46.242 --> 00:12:47.820
this third line.

00:12:48.540 --> 00:12:51.972
Is parallel to the previous two
lines and it's parallel because

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it's got the same gradient and
it's got the same gradient

00:12:55.404 --> 00:12:59.148
because in every case we've got
2X the number. Multiplying X is

00:12:59.148 --> 00:13:03.459
the same. So what's different
about the lines? Well, what is

00:13:03.459 --> 00:13:05.014
different is that they're all

00:13:05.014 --> 00:13:08.324
cutting. The Y axis
at a different point.

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This line is cutting the Y axis
at the point where. Why is 4?

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Note that the number 4
appears in the equation.

00:13:18.930 --> 00:13:23.960
This line. Cuts the Y axis when
Y is one, and again one appears

00:13:23.960 --> 00:13:29.040
in the equation. And again, this
line cuts the Y axis at minus

00:13:29.040 --> 00:13:33.291
one and minus one appears in the
equation, and this gives us a

00:13:33.291 --> 00:13:37.542
general rule. If we have an
equation of the form Y equals MX

00:13:37.542 --> 00:13:42.447
plus C, the number that is on
its own at the end. Here the C

00:13:42.447 --> 00:13:46.698
which was the four or the one or
the minus one, tells us

00:13:46.698 --> 00:13:48.660
whereabouts on the Y axis that

00:13:48.660 --> 00:13:51.746
the graph cuts. And we call this

00:13:51.746 --> 00:13:57.520
value. Either the for their or
the one there, or the minus one

00:13:57.520 --> 00:14:02.590
there. We call that the vertical
intercept so the value of C is

00:14:02.590 --> 00:14:03.760
the vertical intercept.

00:14:03.840 --> 00:14:09.980
So now whenever you see
an equation of the form

00:14:09.980 --> 00:14:15.506
Y equals a number times
X plus another number.

00:14:15.506 --> 00:14:19.190
So why equals MX plus C?

00:14:20.330 --> 00:14:22.205
That represents a straight line

00:14:22.205 --> 00:14:24.905
graph. Where M is the gradient

00:14:24.905 --> 00:14:29.190
of the line. And sees the
vertical intercept, which is the

00:14:29.190 --> 00:14:31.950
place where the graph crosses
the vertical axis.

00:14:33.480 --> 00:14:39.900
Now sometimes when we get the
equation of a straight line, it

00:14:39.900 --> 00:14:46.320
doesn't always appear in the
form Y equals MX plus C. Let

00:14:46.320 --> 00:14:48.995
me give you an example.

00:14:49.000 --> 00:14:53.901
Let's consider this equation 3.
Y minus two X equals 6. Now at

00:14:53.901 --> 00:14:58.802
first sight that doesn't look as
though it's in the form Y equals

00:14:58.802 --> 00:15:02.572
MX plus C which is our
recognisable form of the

00:15:02.572 --> 00:15:08.227
equation of a straight line. But
what we can do is we can do some

00:15:08.227 --> 00:15:12.751
algebraic manipulation on this
to try to write it in this form

00:15:12.751 --> 00:15:18.029
and one of the advantages of
doing that is that if we can get

00:15:18.029 --> 00:15:19.537
it into this form.

00:15:19.540 --> 00:15:23.180
We can read off what the
gradients and the vertical

00:15:23.180 --> 00:15:25.364
intercept are, so let's work on

00:15:25.364 --> 00:15:30.408
this. I'll start by adding 2X
to both sides.

00:15:30.410 --> 00:15:35.366
To remove this minus 2X from
here. So if we add 2X to both

00:15:35.366 --> 00:15:37.844
sides will get 2X plus six on

00:15:37.844 --> 00:15:43.416
the right. And now if I divide
both sides by three, I'll get Y

00:15:43.416 --> 00:15:45.922
on its own, which is what I'm

00:15:45.922 --> 00:15:51.690
looking for. Dividing 2X by
three gives me why is 2/3 of

00:15:51.690 --> 00:15:57.234
X? And if I divide 6 by
three, I'll get 2.

00:15:57.240 --> 00:16:02.378
Now this is a much more familiar
form. This is of the form Y

00:16:02.378 --> 00:16:07.516
equals MX plus. See where we can
read off the gradient M is 2/3

00:16:07.516 --> 00:16:09.351
and the vertical intercept see

00:16:09.351 --> 00:16:13.645
is 2. So be aware that sometimes
an equation that you see might

00:16:13.645 --> 00:16:16.340
not at first sight look as
though it's a straight line

00:16:16.340 --> 00:16:19.770
equation, but by doing some work
on it you can get it into a

00:16:19.770 --> 00:16:25.600
recognizable form. About
another

00:16:25.600 --> 00:16:31.022
example. Suppose we're given
some information about a

00:16:31.022 --> 00:16:36.690
straight line graph, and we want
to try and find out what the

00:16:36.690 --> 00:16:41.050
equation is. So, for example,
suppose we're told that a

00:16:41.050 --> 00:16:45.846
straight line has gradient, a
fifth and were told also that

00:16:45.846 --> 00:16:48.026
it's vertical intercept. See is

00:16:48.026 --> 00:16:51.350
one. Let's see if we can
write down the equation.

00:16:52.500 --> 00:16:56.208
Well, we know that a
straight line has equation

00:16:56.208 --> 00:16:58.268
Y equals MX plus C.

00:16:59.620 --> 00:17:05.220
So we can substitute are known
values in M is going to be 1/5.

00:17:05.220 --> 00:17:11.220
See is going to be one. So our
equation is Y equals 1/5 of X

00:17:11.220 --> 00:17:14.020
plus one Y equals MX plus C.

00:17:14.770 --> 00:17:18.718
Now we might not always choose
to leave it in that form, so let

00:17:18.718 --> 00:17:22.102
me just show you how else we
might write it. There's a

00:17:22.102 --> 00:17:25.204
fraction here of the 5th, and if
we multiply everything through

00:17:25.204 --> 00:17:28.306
by 5, we can remove this
fraction. So let's multiply both

00:17:28.306 --> 00:17:31.972
sides by 5, will get 5 Y the
files or cancel. When we

00:17:31.972 --> 00:17:33.946
multiply by 5 here just to leave

00:17:33.946 --> 00:17:36.810
X. And five ones of five.

00:17:37.620 --> 00:17:42.690
So this form is equivalent to
this form, but just written in a

00:17:42.690 --> 00:17:46.615
different way. We could
rearrange it again just by

00:17:46.615 --> 00:17:50.520
bringing everything over to the
left hand side, so we might

00:17:50.520 --> 00:17:55.845
write 5 Y minus X. Minus five is
not, so that is another form of

00:17:55.845 --> 00:17:59.750
the same equation and we'll see
some equations written in this

00:17:59.750 --> 00:18:01.170
form which later on.

00:18:02.410 --> 00:18:09.310
OK, let's have
a look at

00:18:09.310 --> 00:18:14.353
another example. Suppose now
we're interested in trying to

00:18:14.353 --> 00:18:17.821
find the equation of a line
which has a gradient of 1/3.

00:18:18.960 --> 00:18:22.470
And this time, instead of being
given the vertical intercept,

00:18:22.470 --> 00:18:25.980
we're going to be given some
information about a point

00:18:25.980 --> 00:18:30.192
through which the line passes.
So suppose that we know that the

00:18:30.192 --> 00:18:33.000
line passes through the points
with coordinates 12.

00:18:33.800 --> 00:18:36.374
Let's see if we can figure
out what the equation of

00:18:36.374 --> 00:18:37.076
the line is.

00:18:39.090 --> 00:18:44.508
Start with our general form Y
equals MX plus C and we put in

00:18:44.508 --> 00:18:46.443
the information that we already

00:18:46.443 --> 00:18:52.570
know. We know that the gradient
M is 1/3, so we can put that in

00:18:52.570 --> 00:18:53.665
here straight away.

00:18:53.680 --> 00:18:57.310
We don't know the vertical
intercept. We're going to have

00:18:57.310 --> 00:19:00.214
to do a bit of work to find

00:19:00.214 --> 00:19:05.263
that. But what we do know is
that the line passes through

00:19:05.263 --> 00:19:10.485
this point. What that means is
that when X is one, why has the

00:19:10.485 --> 00:19:14.588
value 2? And we can use that
information in this equation.

00:19:15.380 --> 00:19:17.382
So we're going to put, why is

00:19:17.382 --> 00:19:23.620
2IN. X is one in home, 3
third times, one is just a

00:19:23.620 --> 00:19:28.879
third. Let's see from this we
can workout what Sears.

00:19:29.440 --> 00:19:33.968
So two is the same as 6 thirds
and if we take a third off, both

00:19:33.968 --> 00:19:38.213
sides will have 5 thirds is see
so you can see we can use the

00:19:38.213 --> 00:19:39.911
information about a point on the

00:19:39.911 --> 00:19:44.450
line. To find the vertical
intercept, see so. Now we know

00:19:44.450 --> 00:19:48.180
everything about this line. We
know it's vertical intercept and

00:19:48.180 --> 00:19:53.029
we know its gradient. So the
equation of the line is why is

00:19:53.029 --> 00:19:54.521
1/3 X +5 thirds?

00:19:54.530 --> 00:20:00.905
I want to do that again, but I
want to do it for more general

00:20:00.905 --> 00:20:05.155
case where we haven't got
specific values for the gradient

00:20:05.155 --> 00:20:09.830
and we haven't got specific
values for the point. So this

00:20:09.830 --> 00:20:14.505
time, let's suppose we've got a
straight line. This gradient is

00:20:14.505 --> 00:20:20.163
M. But it passes through a
point with arbitrary coordinates

00:20:20.163 --> 00:20:21.590
X one. I want.

00:20:22.250 --> 00:20:24.900
Let's see if we can
find a formula for the

00:20:24.900 --> 00:20:25.960
equation of the line.

00:20:27.620 --> 00:20:31.676
Always go back to what we know.
We know already that any

00:20:31.676 --> 00:20:35.056
straight line has this equation.
Y is MX plus C.

00:20:35.070 --> 00:20:40.230
What do we know what we're told
the gradient is M so that we can

00:20:40.230 --> 00:20:43.430
leave alone. But we don't know
the vertical intercept. See

00:20:43.430 --> 00:20:45.040
let's see if we can find it.

00:20:46.240 --> 00:20:49.698
Use what we do now. We do know
that the line passes through

00:20:49.698 --> 00:20:54.660
this point. So that we know that
when X has the value X one.

00:20:55.530 --> 00:20:57.258
Why has the value? Why one?

00:20:57.880 --> 00:20:59.322
So I'm going to put those values

00:20:59.322 --> 00:21:06.444
in here. So why has the value?
Why one when X has the value

00:21:06.444 --> 00:21:13.118
X one? See, now we can rearrange
this to find C. So take the MX

00:21:13.118 --> 00:21:14.862
one off both sides.

00:21:15.740 --> 00:21:20.444
That will give me the value for
C and this value for see that

00:21:20.444 --> 00:21:24.476
we have found, which you
realize now is made up of. Only

00:21:24.476 --> 00:21:29.852
the things we knew. We knew the
M we knew the X one and Y one.

00:21:29.852 --> 00:21:34.556
So in fact we know this value.
Now we put this value back into

00:21:34.556 --> 00:21:38.924
the general equation so will
have Y equals MX plus C and see

00:21:38.924 --> 00:21:40.268
now is all this.

00:21:41.610 --> 00:21:45.786
And that is the equation of a
line with gradient M passing

00:21:45.786 --> 00:21:50.310
through X one Y one. We don't
normally leave it in this form.

00:21:50.310 --> 00:21:53.790
We write it in a slightly
different way. It's usually

00:21:53.790 --> 00:21:57.966
written like this. We subtract
why one off both sides to give

00:21:57.966 --> 00:22:00.750
us Y, minus Y one, and that will

00:22:00.750 --> 00:22:06.632
disappear. And we factorize the
MX and the NX one by taking out

00:22:06.632 --> 00:22:12.386
the common factor of M will be
left with X and minus X one.

00:22:13.290 --> 00:22:18.361
And that is an important result,
because this formula gives us

00:22:18.361 --> 00:22:20.666
the equation of a line.

00:22:21.490 --> 00:22:25.270
With gradient M and which passes
through a point where the X

00:22:25.270 --> 00:22:28.735
coordinate is X one and the Y
coordinate is why one?

00:22:29.330 --> 00:22:36.476
Let's look at
a specific example.

00:22:36.650 --> 00:22:41.610
Suppose we're interested in a
straight line where the gradient

00:22:41.610 --> 00:22:46.570
is minus 2, and it passes
through the point with

00:22:46.570 --> 00:22:48.554
coordinates minus 3 two.

00:22:50.420 --> 00:22:55.556
We know the general form of a
straight line, it's why minus

00:22:55.556 --> 00:23:00.692
why one is MX minus X one?
That's our general results and

00:23:00.692 --> 00:23:05.828
all we need to do is put this
information into this formula.

00:23:06.690 --> 00:23:11.079
Why one is the Y coordinate
coordinate of the known point

00:23:11.079 --> 00:23:12.276
which is 2?

00:23:12.450 --> 00:23:14.445
M is the gradient which is minus

00:23:14.445 --> 00:23:20.002
2. X minus X one is the X
coordinates of the known point,

00:23:20.002 --> 00:23:21.466
which is minus 3.

00:23:22.020 --> 00:23:27.774
So tidying this up, we've got Y
minus two is going to be minus

00:23:27.774 --> 00:23:33.117
2X plus three, and if we remove
the brackets, Y minus two is

00:23:33.117 --> 00:23:35.172
minus 2 X minus 6.

00:23:36.380 --> 00:23:40.790
And finally, if we add two to
both sides, we shall get why is

00:23:40.790 --> 00:23:44.570
minus 2 X minus four, and that's
the equation of the straight

00:23:44.570 --> 00:23:47.720
line with gradient minus 2
passing through this point. And

00:23:47.720 --> 00:23:51.815
there's always a check you can
apply because we can look at the

00:23:51.815 --> 00:23:55.595
final equation we've got and we
can observe from here that the

00:23:55.595 --> 00:23:57.170
gradient is indeed minus 2.

00:23:57.800 --> 00:24:01.706
And we can also pop in an X
value of minus three into here.

00:24:02.420 --> 00:24:06.932
Minus two times minus three is
plus six and six takeaway four

00:24:06.932 --> 00:24:10.692
is 2 and that's the
corresponding why value? So this

00:24:10.692 --> 00:24:12.948
built in checks that you can

00:24:12.948 --> 00:24:15.894
apply. Let's have a look at
another slightly different

00:24:15.894 --> 00:24:19.710
example, and in this example
I'm not going to give you the

00:24:19.710 --> 00:24:22.572
gradient of the line.
Instead, we're going to have

00:24:22.572 --> 00:24:26.070
two points on the line. So
let's suppose are two points

00:24:26.070 --> 00:24:30.204
are minus 1, two and two 4,
so we don't know the gradient

00:24:30.204 --> 00:24:33.066
and we don't know the
vertical intercept. We just

00:24:33.066 --> 00:24:36.882
know two points on the line,
and we've got to try to

00:24:36.882 --> 00:24:39.426
determine what the equation
of the line is.

00:24:40.660 --> 00:24:42.452
Now let's see how we can do

00:24:42.452 --> 00:24:47.202
this. One thing we can do is we
can calculate the gradient of

00:24:47.202 --> 00:24:50.694
the line because we know how to
calculate the gradient of the

00:24:50.694 --> 00:24:51.858
line joining two points.

00:24:52.380 --> 00:24:57.084
So let's do that. First of all,
the gradient of the line will be

00:24:57.084 --> 00:25:00.780
the difference in the Y
coordinates, which is 4 - 2.

00:25:01.200 --> 00:25:06.864
Divided by the difference in the
X coordinates, which is 2 minus

00:25:06.864 --> 00:25:13.710
minus one. 4 - 2 is 2 and 2
minus minus one is 2 + 1 which

00:25:13.710 --> 00:25:17.590
is 3. So the gradient of this
line is 2/3.

00:25:18.660 --> 00:25:23.054
Now we know the gradients and we
know at least one point through

00:25:23.054 --> 00:25:27.448
which the line passes. Because
we know two. So we can use the

00:25:27.448 --> 00:25:32.180
previous formula Y minus Y one
is MX minus X one. So that's the

00:25:32.180 --> 00:25:33.194
formula we use.

00:25:33.930 --> 00:25:37.725
Why minus why? One doesn't
matter which of the two points

00:25:37.725 --> 00:25:39.795
we take? Let's take the .24.

00:25:40.780 --> 00:25:44.350
So the Y coordinate is 4.

00:25:44.450 --> 00:25:47.650
M we found is 2/3.

00:25:47.650 --> 00:25:52.291
X minus the X coordinate, which
is 2. So that's our equation of

00:25:52.291 --> 00:25:57.646
the line and if we wanted to do
we can tidy this up a little

00:25:57.646 --> 00:26:03.358
bit. Y minus four is 2/3 of X
minus 4 thirds, and if we add 4

00:26:03.358 --> 00:26:07.999
to both sides, we can write this
as Y is 2X over 3.

00:26:08.630 --> 00:26:12.557
And we've got minus 4 thirds
here already, and we're bringing

00:26:12.557 --> 00:26:17.198
over four will be adding four.
So finally will have two X over

00:26:17.198 --> 00:26:23.266
3. And four is the same as 12
thirds, 12 thirds. Subtract 4

00:26:23.266 --> 00:26:28.029
thirds is 8 thirds. So that's
the equation of the line.

00:26:28.610 --> 00:26:34.231
I want to do that same
argument when were given two

00:26:34.231 --> 00:26:37.808
arbitrary points instead of
two specific points.

00:26:39.030 --> 00:26:43.879
So suppose we have the point a
coordinates X one and Y 1.

00:26:44.840 --> 00:26:48.284
And be with coordinates X2Y2
and. Let's see if we can figure

00:26:48.284 --> 00:26:51.441
out what the equation of the
line is joining these two

00:26:51.441 --> 00:26:55.459
points, and I think in this case
a graph is going to help us.

00:26:56.240 --> 00:26:58.280
Just have a quick sketch.

00:26:59.040 --> 00:27:05.070
So I've got
a point A.

00:27:05.760 --> 00:27:08.468
Coordinates X One Y1.

00:27:09.410 --> 00:27:13.690
And another point B
coordinates X2Y2 and were

00:27:13.690 --> 00:27:19.040
interested in the equation of
this line which is joining

00:27:19.040 --> 00:27:19.575
them.

00:27:21.070 --> 00:27:26.009
Suppose we pick an arbitrary
point on the line anywhere along

00:27:26.009 --> 00:27:30.499
the line at all, and let's call
that point P.

00:27:31.090 --> 00:27:34.570
P is an arbitrary point, and
let's suppose it's coordinates

00:27:34.570 --> 00:27:37.702
are X&Y. For any arbitrary X&Y
on the line.

00:27:38.610 --> 00:27:43.935
And what we do know is that the
gradient of AP is the same as

00:27:43.935 --> 00:27:45.710
the gradient of a bee.

00:27:46.370 --> 00:27:49.674
Let me write that
down the gradient.

00:27:50.730 --> 00:27:57.160
Of AP. Is
equal to the gradient.

00:27:57.310 --> 00:28:01.189
Of a B.

00:28:01.190 --> 00:28:04.850
Let's see what that means. Well,
the gradient of AP.

00:28:06.400 --> 00:28:08.332
Is just the difference in the Y

00:28:08.332 --> 00:28:12.841
coordinates. Is Y minus
Y one over the

00:28:12.841 --> 00:28:16.625
difference in the X
coordinates, which is X

00:28:16.625 --> 00:28:18.044
minus X one?

00:28:19.130 --> 00:28:23.123
So that's the gradient of this
line segment between amp and

00:28:23.123 --> 00:28:27.479
that's got to equal the gradient
of the line segment between A&B.

00:28:28.060 --> 00:28:31.723
And once the gradient of the
line segment between A&B, well,

00:28:31.723 --> 00:28:35.386
it's again. It's the difference
of the Y coordinates, which now

00:28:35.386 --> 00:28:37.384
is Y 2 minus Y 1.

00:28:37.390 --> 00:28:42.898
Divided by the difference in the
X coordinates which is X2 minus

00:28:42.898 --> 00:28:47.488
X one. So that is a formula
which will tell us.

00:28:48.160 --> 00:28:50.870
The equation of a line passing
through two arbitrary points.

00:28:50.870 --> 00:28:54.122
Now we don't usually leave it in
that form. It's normally written

00:28:54.122 --> 00:28:57.103
in a slightly different form,
and it's normally written in a

00:28:57.103 --> 00:29:01.168
form so that all the wise appear
on one side and all the ex is

00:29:01.168 --> 00:29:04.420
appear on another side, and we
can do that by dividing both

00:29:04.420 --> 00:29:06.317
sides by Y 2 minus Y 1.

00:29:07.010 --> 00:29:14.056
And multiplying both sides by X
minus X One which moves that up

00:29:14.056 --> 00:29:21.090
to here. And that's the form
which is normally quoted as the

00:29:21.090 --> 00:29:26.400
equation of a line passing
through two arbitrary points.

00:29:28.850 --> 00:29:32.126
Let's use that in an example.

00:29:32.690 --> 00:29:38.996
Suppose we have
two points A.

00:29:39.000 --> 00:29:44.350
Coordinates one and minus two
and B which has coordinates

00:29:44.350 --> 00:29:46.490
minus three and not.

00:29:47.220 --> 00:29:52.576
Let me write down the formula
again. Why minus why one over Y

00:29:52.576 --> 00:29:58.344
2 minus Y one is X Minus X one
over X2 minus X one?

00:29:59.010 --> 00:30:02.429
With pop everything we know into
the formula and see what we get.

00:30:03.200 --> 00:30:10.340
So we want Y minus Y1Y one is
the first of the Y values which

00:30:10.340 --> 00:30:11.768
is minus 2.

00:30:12.000 --> 00:30:16.940
Why 2 minus? Why one is the
difference of the Y values that

00:30:16.940 --> 00:30:20.400
zero? Minus minus 2.

00:30:20.910 --> 00:30:27.126
Equals X minus X one is the
first of the X values, which is

00:30:27.126 --> 00:30:32.898
one and X2 minus X. One is the
difference of the X values.

00:30:32.898 --> 00:30:35.118
That's minus three, subtract 1.

00:30:35.980 --> 00:30:41.164
And just to tidy this up on the
top line here will get Y +2 on

00:30:41.164 --> 00:30:43.108
the bottom line. Here will get

00:30:43.108 --> 00:30:47.317
+2. X minus one there on
the right at the top and

00:30:47.317 --> 00:30:49.396
minus 3 - 1 is minus 4.

00:30:51.870 --> 00:30:56.112
Again, we can tie this up a
little bit more to will go into

00:30:56.112 --> 00:30:57.627
minus 4 - 2 times.

00:30:58.190 --> 00:31:02.370
And if we multiply everything
through by minus, two will get

00:31:02.370 --> 00:31:07.690
minus two Y minus four equals X
minus one, and we can write this

00:31:07.690 --> 00:31:12.250
in lots of different ways. For
example, we could write this as

00:31:12.250 --> 00:31:14.150
minus two Y minus X.

00:31:15.050 --> 00:31:18.690
And we could add 1 to both
sides to give minus 3 zero.

00:31:18.690 --> 00:31:21.210
That's one way we could leave
the final answer.

00:31:22.430 --> 00:31:26.252
Another way we could leave it as
we could rearrange it to get Y

00:31:26.252 --> 00:31:29.528
equals something. So if I do
that, I'll have minus two Y

00:31:29.528 --> 00:31:30.893
equals X and if we.

00:31:31.670 --> 00:31:37.091
Add 4 to both sides will get
plus three there, and if we

00:31:37.091 --> 00:31:41.678
divide everything by minus two
will get minus 1/2 X minus

00:31:41.678 --> 00:31:45.848
three over 2. So all of these
forms are equivalent.

00:31:47.480 --> 00:31:53.852
Now, that's not quite the whole
story. The most general form of

00:31:53.852 --> 00:31:58.100
equation of a straight line
looks like this.

00:31:58.430 --> 00:32:03.510
And earlier on in this unit,
we've seen some equations

00:32:03.510 --> 00:32:09.098
written in this form. Let's look
at some specific cases. Suppose

00:32:09.098 --> 00:32:14.178
that a this number here turns
out to be 0.

00:32:14.740 --> 00:32:16.468
What will that mean if a is 0?

00:32:17.000 --> 00:32:20.180
But if a is zero, we can
rearrange this and write BY.

00:32:20.920 --> 00:32:22.930
Equals minus C.

00:32:24.010 --> 00:32:27.520
Why is minus C Overby?

00:32:29.270 --> 00:32:32.942
And what does this mean?
Remember the A and the beat and

00:32:32.942 --> 00:32:36.614
the CIA just numbers their
constants. So when a is zero, we

00:32:36.614 --> 00:32:40.592
find that this number on the
right here minus C over B is

00:32:40.592 --> 00:32:44.876
just a constant. So what this is
saying is that Y is a constant.

00:32:45.720 --> 00:32:48.426
Now align where why is constant.

00:32:49.000 --> 00:32:53.810
Must be. A horizontal line,
because why doesn't change

00:32:53.810 --> 00:32:57.230
the value of Y is always
minus C Overby.

00:32:59.080 --> 00:33:04.218
So if you have an equation of
this form where a is zero that

00:33:04.218 --> 00:33:05.319
represents horizontal lines.

00:33:06.530 --> 00:33:09.578
What about if be with zero?

00:33:09.910 --> 00:33:15.370
We're putting B is 0 in here,
will get the AX Plus Co.

00:33:16.060 --> 00:33:20.980
And if we rearrange, this will
get AX equals minus C and

00:33:20.980 --> 00:33:24.670
dividing through by AX is minus
C over A.

00:33:25.990 --> 00:33:30.566
Again, a encia constants so this
time what this is saying is that

00:33:30.566 --> 00:33:31.974
X is a constant.

00:33:32.740 --> 00:33:35.575
Now lines were X is a constant.

00:33:36.240 --> 00:33:40.959
Must look like this. They are
vertical lines because the X

00:33:40.959 --> 00:33:42.246
value doesn't change.

00:33:42.250 --> 00:33:46.362
So this general case
includes both vertical lines

00:33:46.362 --> 00:33:47.904
and horizontal lines.

00:33:49.030 --> 00:33:52.660
So remember, the most general
form will appear like that.

00:33:54.200 --> 00:33:57.130
Provided that be isn't zero,
you can always write the

00:33:57.130 --> 00:34:00.646
equation in the more familiar
form Y equals MX plus C, but

00:34:00.646 --> 00:34:04.455
in the case in which B is 0,
you get this specific case

00:34:04.455 --> 00:34:05.920
where you've got vertical
lines.