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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

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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

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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

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minus one? Thanks is not
wise minus one gives me a

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point down here.

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If X is one.

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Why is one gives
me a point here?

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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

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this third line.

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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

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because in every case we've got
2X the number. Multiplying X is

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the same. So what's different
about the lines? Well, what is

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different is that they're all

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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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00:13:14,940 --> 00:13:17,289
Note that the number 4
appears in the equation.

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00:13:18,930 --> 00:13:23,960
This line. Cuts the Y axis when
Y is one, and again one appears

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00:13:23,960 --> 00:13:29,040
in the equation. And again, this
line cuts the Y axis at minus

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00:13:29,040 --> 00:13:33,291
one and minus one appears in the
equation, and this gives us a

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00:13:33,291 --> 00:13:37,542
general rule. If we have an
equation of the form Y equals MX

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plus C, the number that is on
its own at the end. Here the C

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which was the four or the one or
the minus one, tells us

195
00:13:46,698 --> 00:13:48,660
whereabouts on the Y axis that

196
00:13:48,660 --> 00:13:51,746
the graph cuts. And we call this

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00:13:51,746 --> 00:13:57,520
value. Either the for their or
the one there, or the minus one

198
00:13:57,520 --> 00:14:02,590
there. We call that the vertical
intercept so the value of C is

199
00:14:02,590 --> 00:14:03,760
the vertical intercept.

200
00:14:03,840 --> 00:14:09,980
So now whenever you see
an equation of the form

201
00:14:09,980 --> 00:14:15,506
Y equals a number times
X plus another number.

202
00:14:15,506 --> 00:14:19,190
So why equals MX plus C?

203
00:14:20,330 --> 00:14:22,205
That represents a straight line

204
00:14:22,205 --> 00:14:24,905
graph. Where M is the gradient

205
00:14:24,905 --> 00:14:29,190
of the line. And sees the
vertical intercept, which is the

206
00:14:29,190 --> 00:14:31,950
place where the graph crosses
the vertical axis.

207
00:14:33,480 --> 00:14:39,900
Now sometimes when we get the
equation of a straight line, it

208
00:14:39,900 --> 00:14:46,320
doesn't always appear in the
form Y equals MX plus C. Let

209
00:14:46,320 --> 00:14:48,995
me give you an example.

210
00:14:49,000 --> 00:14:53,901
Let's consider this equation 3.
Y minus two X equals 6. Now at

211
00:14:53,901 --> 00:14:58,802
first sight that doesn't look as
though it's in the form Y equals

212
00:14:58,802 --> 00:15:02,572
MX plus C which is our
recognisable form of the

213
00:15:02,572 --> 00:15:08,227
equation of a straight line. But
what we can do is we can do some

214
00:15:08,227 --> 00:15:12,751
algebraic manipulation on this
to try to write it in this form

215
00:15:12,751 --> 00:15:18,029
and one of the advantages of
doing that is that if we can get

216
00:15:18,029 --> 00:15:19,537
it into this form.

217
00:15:19,540 --> 00:15:23,180
We can read off what the
gradients and the vertical

218
00:15:23,180 --> 00:15:25,364
intercept are, so let's work on

219
00:15:25,364 --> 00:15:30,408
this. I'll start by adding 2X
to both sides.

220
00:15:30,410 --> 00:15:35,366
To remove this minus 2X from
here. So if we add 2X to both

221
00:15:35,366 --> 00:15:37,844
sides will get 2X plus six on

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00:15:37,844 --> 00:15:43,416
the right. And now if I divide
both sides by three, I'll get Y

223
00:15:43,416 --> 00:15:45,922
on its own, which is what I'm

224
00:15:45,922 --> 00:15:51,690
looking for. Dividing 2X by
three gives me why is 2/3 of

225
00:15:51,690 --> 00:15:57,234
X? And if I divide 6 by
three, I'll get 2.

226
00:15:57,240 --> 00:16:02,378
Now this is a much more familiar
form. This is of the form Y

227
00:16:02,378 --> 00:16:07,516
equals MX plus. See where we can
read off the gradient M is 2/3

228
00:16:07,516 --> 00:16:09,351
and the vertical intercept see

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is 2. So be aware that sometimes
an equation that you see might

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00:16:13,645 --> 00:16:16,340
not at first sight look as
though it's a straight line

231
00:16:16,340 --> 00:16:19,770
equation, but by doing some work
on it you can get it into a

232
00:16:19,770 --> 00:16:25,600
recognizable form. About
another

233
00:16:25,600 --> 00:16:31,022
example. Suppose we're given
some information about a

234
00:16:31,022 --> 00:16:36,690
straight line graph, and we want
to try and find out what the

235
00:16:36,690 --> 00:16:41,050
equation is. So, for example,
suppose we're told that a

236
00:16:41,050 --> 00:16:45,846
straight line has gradient, a
fifth and were told also that

237
00:16:45,846 --> 00:16:48,026
it's vertical intercept. See is

238
00:16:48,026 --> 00:16:51,350
one. Let's see if we can
write down the equation.

239
00:16:52,500 --> 00:16:56,208
Well, we know that a
straight line has equation

240
00:16:56,208 --> 00:16:58,268
Y equals MX plus C.

241
00:16:59,620 --> 00:17:05,220
So we can substitute are known
values in M is going to be 1/5.

242
00:17:05,220 --> 00:17:11,220
See is going to be one. So our
equation is Y equals 1/5 of X

243
00:17:11,220 --> 00:17:14,020
plus one Y equals MX plus C.

244
00:17:14,770 --> 00:17:18,718
Now we might not always choose
to leave it in that form, so let

245
00:17:18,718 --> 00:17:22,102
me just show you how else we
might write it. There's a

246
00:17:22,102 --> 00:17:25,204
fraction here of the 5th, and if
we multiply everything through

247
00:17:25,204 --> 00:17:28,306
by 5, we can remove this
fraction. So let's multiply both

248
00:17:28,306 --> 00:17:31,972
sides by 5, will get 5 Y the
files or cancel. When we

249
00:17:31,972 --> 00:17:33,946
multiply by 5 here just to leave

250
00:17:33,946 --> 00:17:36,810
X. And five ones of five.

251
00:17:37,620 --> 00:17:42,690
So this form is equivalent to
this form, but just written in a

252
00:17:42,690 --> 00:17:46,615
different way. We could
rearrange it again just by

253
00:17:46,615 --> 00:17:50,520
bringing everything over to the
left hand side, so we might

254
00:17:50,520 --> 00:17:55,845
write 5 Y minus X. Minus five is
not, so that is another form of

255
00:17:55,845 --> 00:17:59,750
the same equation and we'll see
some equations written in this

256
00:17:59,750 --> 00:18:01,170
form which later on.

257
00:18:02,410 --> 00:18:09,310
OK, let's have
a look at

258
00:18:09,310 --> 00:18:14,353
another example. Suppose now
we're interested in trying to

259
00:18:14,353 --> 00:18:17,821
find the equation of a line
which has a gradient of 1/3.

260
00:18:18,960 --> 00:18:22,470
And this time, instead of being
given the vertical intercept,

261
00:18:22,470 --> 00:18:25,980
we're going to be given some
information about a point

262
00:18:25,980 --> 00:18:30,192
through which the line passes.
So suppose that we know that the

263
00:18:30,192 --> 00:18:33,000
line passes through the points
with coordinates 12.

264
00:18:33,800 --> 00:18:36,374
Let's see if we can figure
out what the equation of

265
00:18:36,374 --> 00:18:37,076
the line is.

266
00:18:39,090 --> 00:18:44,508
Start with our general form Y
equals MX plus C and we put in

267
00:18:44,508 --> 00:18:46,443
the information that we already

268
00:18:46,443 --> 00:18:52,570
know. We know that the gradient
M is 1/3, so we can put that in

269
00:18:52,570 --> 00:18:53,665
here straight away.

270
00:18:53,680 --> 00:18:57,310
We don't know the vertical
intercept. We're going to have

271
00:18:57,310 --> 00:19:00,214
to do a bit of work to find

272
00:19:00,214 --> 00:19:05,263
that. But what we do know is
that the line passes through

273
00:19:05,263 --> 00:19:10,485
this point. What that means is
that when X is one, why has the

274
00:19:10,485 --> 00:19:14,588
value 2? And we can use that
information in this equation.

275
00:19:15,380 --> 00:19:17,382
So we're going to put, why is

276
00:19:17,382 --> 00:19:23,620
2IN. X is one in home, 3
third times, one is just a

277
00:19:23,620 --> 00:19:28,879
third. Let's see from this we
can workout what Sears.

278
00:19:29,440 --> 00:19:33,968
So two is the same as 6 thirds
and if we take a third off, both

279
00:19:33,968 --> 00:19:38,213
sides will have 5 thirds is see
so you can see we can use the

280
00:19:38,213 --> 00:19:39,911
information about a point on the

281
00:19:39,911 --> 00:19:44,450
line. To find the vertical
intercept, see so. Now we know

282
00:19:44,450 --> 00:19:48,180
everything about this line. We
know it's vertical intercept and

283
00:19:48,180 --> 00:19:53,029
we know its gradient. So the
equation of the line is why is

284
00:19:53,029 --> 00:19:54,521
1/3 X +5 thirds?

285
00:19:54,530 --> 00:20:00,905
I want to do that again, but I
want to do it for more general

286
00:20:00,905 --> 00:20:05,155
case where we haven't got
specific values for the gradient

287
00:20:05,155 --> 00:20:09,830
and we haven't got specific
values for the point. So this

288
00:20:09,830 --> 00:20:14,505
time, let's suppose we've got a
straight line. This gradient is

289
00:20:14,505 --> 00:20:20,163
M. But it passes through a
point with arbitrary coordinates

290
00:20:20,163 --> 00:20:21,590
X one. I want.

291
00:20:22,250 --> 00:20:24,900
Let's see if we can
find a formula for the

292
00:20:24,900 --> 00:20:25,960
equation of the line.

293
00:20:27,620 --> 00:20:31,676
Always go back to what we know.
We know already that any

294
00:20:31,676 --> 00:20:35,056
straight line has this equation.
Y is MX plus C.

295
00:20:35,070 --> 00:20:40,230
What do we know what we're told
the gradient is M so that we can

296
00:20:40,230 --> 00:20:43,430
leave alone. But we don't know
the vertical intercept. See

297
00:20:43,430 --> 00:20:45,040
let's see if we can find it.

298
00:20:46,240 --> 00:20:49,698
Use what we do now. We do know
that the line passes through

299
00:20:49,698 --> 00:20:54,660
this point. So that we know that
when X has the value X one.

300
00:20:55,530 --> 00:20:57,258
Why has the value? Why one?

301
00:20:57,880 --> 00:20:59,322
So I'm going to put those values

302
00:20:59,322 --> 00:21:06,444
in here. So why has the value?
Why one when X has the value

303
00:21:06,444 --> 00:21:13,118
X one? See, now we can rearrange
this to find C. So take the MX

304
00:21:13,118 --> 00:21:14,862
one off both sides.

305
00:21:15,740 --> 00:21:20,444
That will give me the value for
C and this value for see that

306
00:21:20,444 --> 00:21:24,476
we have found, which you
realize now is made up of. Only

307
00:21:24,476 --> 00:21:29,852
the things we knew. We knew the
M we knew the X one and Y one.

308
00:21:29,852 --> 00:21:34,556
So in fact we know this value.
Now we put this value back into

309
00:21:34,556 --> 00:21:38,924
the general equation so will
have Y equals MX plus C and see

310
00:21:38,924 --> 00:21:40,268
now is all this.

311
00:21:41,610 --> 00:21:45,786
And that is the equation of a
line with gradient M passing

312
00:21:45,786 --> 00:21:50,310
through X one Y one. We don't
normally leave it in this form.

313
00:21:50,310 --> 00:21:53,790
We write it in a slightly
different way. It's usually

314
00:21:53,790 --> 00:21:57,966
written like this. We subtract
why one off both sides to give

315
00:21:57,966 --> 00:22:00,750
us Y, minus Y one, and that will

316
00:22:00,750 --> 00:22:06,632
disappear. And we factorize the
MX and the NX one by taking out

317
00:22:06,632 --> 00:22:12,386
the common factor of M will be
left with X and minus X one.

318
00:22:13,290 --> 00:22:18,361
And that is an important result,
because this formula gives us

319
00:22:18,361 --> 00:22:20,666
the equation of a line.

320
00:22:21,490 --> 00:22:25,270
With gradient M and which passes
through a point where the X

321
00:22:25,270 --> 00:22:28,735
coordinate is X one and the Y
coordinate is why one?

322
00:22:29,330 --> 00:22:36,476
Let's look at
a specific example.

323
00:22:36,650 --> 00:22:41,610
Suppose we're interested in a
straight line where the gradient

324
00:22:41,610 --> 00:22:46,570
is minus 2, and it passes
through the point with

325
00:22:46,570 --> 00:22:48,554
coordinates minus 3 two.

326
00:22:50,420 --> 00:22:55,556
We know the general form of a
straight line, it's why minus

327
00:22:55,556 --> 00:23:00,692
why one is MX minus X one?
That's our general results and

328
00:23:00,692 --> 00:23:05,828
all we need to do is put this
information into this formula.

329
00:23:06,690 --> 00:23:11,079
Why one is the Y coordinate
coordinate of the known point

330
00:23:11,079 --> 00:23:12,276
which is 2?

331
00:23:12,450 --> 00:23:14,445
M is the gradient which is minus

332
00:23:14,445 --> 00:23:20,002
2. X minus X one is the X
coordinates of the known point,

333
00:23:20,002 --> 00:23:21,466
which is minus 3.

334
00:23:22,020 --> 00:23:27,774
So tidying this up, we've got Y
minus two is going to be minus

335
00:23:27,774 --> 00:23:33,117
2X plus three, and if we remove
the brackets, Y minus two is

336
00:23:33,117 --> 00:23:35,172
minus 2 X minus 6.

337
00:23:36,380 --> 00:23:40,790
And finally, if we add two to
both sides, we shall get why is

338
00:23:40,790 --> 00:23:44,570
minus 2 X minus four, and that's
the equation of the straight

339
00:23:44,570 --> 00:23:47,720
line with gradient minus 2
passing through this point. And

340
00:23:47,720 --> 00:23:51,815
there's always a check you can
apply because we can look at the

341
00:23:51,815 --> 00:23:55,595
final equation we've got and we
can observe from here that the

342
00:23:55,595 --> 00:23:57,170
gradient is indeed minus 2.

343
00:23:57,800 --> 00:24:01,706
And we can also pop in an X
value of minus three into here.

344
00:24:02,420 --> 00:24:06,932
Minus two times minus three is
plus six and six takeaway four

345
00:24:06,932 --> 00:24:10,692
is 2 and that's the
corresponding why value? So this

346
00:24:10,692 --> 00:24:12,948
built in checks that you can

347
00:24:12,948 --> 00:24:15,894
apply. Let's have a look at
another slightly different

348
00:24:15,894 --> 00:24:19,710
example, and in this example
I'm not going to give you the

349
00:24:19,710 --> 00:24:22,572
gradient of the line.
Instead, we're going to have

350
00:24:22,572 --> 00:24:26,070
two points on the line. So
let's suppose are two points

351
00:24:26,070 --> 00:24:30,204
are minus 1, two and two 4,
so we don't know the gradient

352
00:24:30,204 --> 00:24:33,066
and we don't know the
vertical intercept. We just

353
00:24:33,066 --> 00:24:36,882
know two points on the line,
and we've got to try to

354
00:24:36,882 --> 00:24:39,426
determine what the equation
of the line is.

355
00:24:40,660 --> 00:24:42,452
Now let's see how we can do

356
00:24:42,452 --> 00:24:47,202
this. One thing we can do is we
can calculate the gradient of

357
00:24:47,202 --> 00:24:50,694
the line because we know how to
calculate the gradient of the

358
00:24:50,694 --> 00:24:51,858
line joining two points.

359
00:24:52,380 --> 00:24:57,084
So let's do that. First of all,
the gradient of the line will be

360
00:24:57,084 --> 00:25:00,780
the difference in the Y
coordinates, which is 4 - 2.

361
00:25:01,200 --> 00:25:06,864
Divided by the difference in the
X coordinates, which is 2 minus

362
00:25:06,864 --> 00:25:13,710
minus one. 4 - 2 is 2 and 2
minus minus one is 2 + 1 which

363
00:25:13,710 --> 00:25:17,590
is 3. So the gradient of this
line is 2/3.

364
00:25:18,660 --> 00:25:23,054
Now we know the gradients and we
know at least one point through

365
00:25:23,054 --> 00:25:27,448
which the line passes. Because
we know two. So we can use the

366
00:25:27,448 --> 00:25:32,180
previous formula Y minus Y one
is MX minus X one. So that's the

367
00:25:32,180 --> 00:25:33,194
formula we use.

368
00:25:33,930 --> 00:25:37,725
Why minus why? One doesn't
matter which of the two points

369
00:25:37,725 --> 00:25:39,795
we take? Let's take the .24.

370
00:25:40,780 --> 00:25:44,350
So the Y coordinate is 4.

371
00:25:44,450 --> 00:25:47,650
M we found is 2/3.

372
00:25:47,650 --> 00:25:52,291
X minus the X coordinate, which
is 2. So that's our equation of

373
00:25:52,291 --> 00:25:57,646
the line and if we wanted to do
we can tidy this up a little

374
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

375
00:26:03,358 --> 00:26:07,999
to both sides, we can write this
as Y is 2X over 3.

376
00:26:08,630 --> 00:26:12,557
And we've got minus 4 thirds
here already, and we're bringing

377
00:26:12,557 --> 00:26:17,198
over four will be adding four.
So finally will have two X over

378
00:26:17,198 --> 00:26:23,266
3. And four is the same as 12
thirds, 12 thirds. Subtract 4

379
00:26:23,266 --> 00:26:28,029
thirds is 8 thirds. So that's
the equation of the line.

380
00:26:28,610 --> 00:26:34,231
I want to do that same
argument when were given two

381
00:26:34,231 --> 00:26:37,808
arbitrary points instead of
two specific points.

382
00:26:39,030 --> 00:26:43,879
So suppose we have the point a
coordinates X one and Y 1.

383
00:26:44,840 --> 00:26:48,284
And be with coordinates X2Y2
and. Let's see if we can figure

384
00:26:48,284 --> 00:26:51,441
out what the equation of the
line is joining these two

385
00:26:51,441 --> 00:26:55,459
points, and I think in this case
a graph is going to help us.

386
00:26:56,240 --> 00:26:58,280
Just have a quick sketch.

387
00:26:59,040 --> 00:27:05,070
So I've got
a point A.

388
00:27:05,760 --> 00:27:08,468
Coordinates X One Y1.

389
00:27:09,410 --> 00:27:13,690
And another point B
coordinates X2Y2 and were

390
00:27:13,690 --> 00:27:19,040
interested in the equation of
this line which is joining

391
00:27:19,040 --> 00:27:19,575
them.

392
00:27:21,070 --> 00:27:26,009
Suppose we pick an arbitrary
point on the line anywhere along

393
00:27:26,009 --> 00:27:30,499
the line at all, and let's call
that point P.

394
00:27:31,090 --> 00:27:34,570
P is an arbitrary point, and
let's suppose it's coordinates

395
00:27:34,570 --> 00:27:37,702
are X&Y. For any arbitrary X&Y
on the line.

396
00:27:38,610 --> 00:27:43,935
And what we do know is that the
gradient of AP is the same as

397
00:27:43,935 --> 00:27:45,710
the gradient of a bee.

398
00:27:46,370 --> 00:27:49,674
Let me write that
down the gradient.

399
00:27:50,730 --> 00:27:57,160
Of AP. Is
equal to the gradient.

400
00:27:57,310 --> 00:28:01,189
Of a B.

401
00:28:01,190 --> 00:28:04,850
Let's see what that means. Well,
the gradient of AP.

402
00:28:06,400 --> 00:28:08,332
Is just the difference in the Y

403
00:28:08,332 --> 00:28:12,841
coordinates. Is Y minus
Y one over the

404
00:28:12,841 --> 00:28:16,625
difference in the X
coordinates, which is X

405
00:28:16,625 --> 00:28:18,044
minus X one?

406
00:28:19,130 --> 00:28:23,123
So that's the gradient of this
line segment between amp and

407
00:28:23,123 --> 00:28:27,479
that's got to equal the gradient
of the line segment between A&B.

408
00:28:28,060 --> 00:28:31,723
And once the gradient of the
line segment between A&B, well,

409
00:28:31,723 --> 00:28:35,386
it's again. It's the difference
of the Y coordinates, which now

410
00:28:35,386 --> 00:28:37,384
is Y 2 minus Y 1.

411
00:28:37,390 --> 00:28:42,898
Divided by the difference in the
X coordinates which is X2 minus

412
00:28:42,898 --> 00:28:47,488
X one. So that is a formula
which will tell us.

413
00:28:48,160 --> 00:28:50,870
The equation of a line passing
through two arbitrary points.

414
00:28:50,870 --> 00:28:54,122
Now we don't usually leave it in
that form. It's normally written

415
00:28:54,122 --> 00:28:57,103
in a slightly different form,
and it's normally written in a

416
00:28:57,103 --> 00:29:01,168
form so that all the wise appear
on one side and all the ex is

417
00:29:01,168 --> 00:29:04,420
appear on another side, and we
can do that by dividing both

418
00:29:04,420 --> 00:29:06,317
sides by Y 2 minus Y 1.

419
00:29:07,010 --> 00:29:14,056
And multiplying both sides by X
minus X One which moves that up

420
00:29:14,056 --> 00:29:21,090
to here. And that's the form
which is normally quoted as the

421
00:29:21,090 --> 00:29:26,400
equation of a line passing
through two arbitrary points.

422
00:29:28,850 --> 00:29:32,126
Let's use that in an example.

423
00:29:32,690 --> 00:29:38,996
Suppose we have
two points A.

424
00:29:39,000 --> 00:29:44,350
Coordinates one and minus two
and B which has coordinates

425
00:29:44,350 --> 00:29:46,490
minus three and not.

426
00:29:47,220 --> 00:29:52,576
Let me write down the formula
again. Why minus why one over Y

427
00:29:52,576 --> 00:29:58,344
2 minus Y one is X Minus X one
over X2 minus X one?

428
00:29:59,010 --> 00:30:02,429
With pop everything we know into
the formula and see what we get.

429
00:30:03,200 --> 00:30:10,340
So we want Y minus Y1Y one is
the first of the Y values which

430
00:30:10,340 --> 00:30:11,768
is minus 2.

431
00:30:12,000 --> 00:30:16,940
Why 2 minus? Why one is the
difference of the Y values that

432
00:30:16,940 --> 00:30:20,400
zero? Minus minus 2.

433
00:30:20,910 --> 00:30:27,126
Equals X minus X one is the
first of the X values, which is

434
00:30:27,126 --> 00:30:32,898
one and X2 minus X. One is the
difference of the X values.

435
00:30:32,898 --> 00:30:35,118
That's minus three, subtract 1.

436
00:30:35,980 --> 00:30:41,164
And just to tidy this up on the
top line here will get Y +2 on

437
00:30:41,164 --> 00:30:43,108
the bottom line. Here will get

438
00:30:43,108 --> 00:30:47,317
+2. X minus one there on
the right at the top and

439
00:30:47,317 --> 00:30:49,396
minus 3 - 1 is minus 4.

440
00:30:51,870 --> 00:30:56,112
Again, we can tie this up a
little bit more to will go into

441
00:30:56,112 --> 00:30:57,627
minus 4 - 2 times.

442
00:30:58,190 --> 00:31:02,370
And if we multiply everything
through by minus, two will get

443
00:31:02,370 --> 00:31:07,690
minus two Y minus four equals X
minus one, and we can write this

444
00:31:07,690 --> 00:31:12,250
in lots of different ways. For
example, we could write this as

445
00:31:12,250 --> 00:31:14,150
minus two Y minus X.

446
00:31:15,050 --> 00:31:18,690
And we could add 1 to both
sides to give minus 3 zero.

447
00:31:18,690 --> 00:31:21,210
That's one way we could leave
the final answer.

448
00:31:22,430 --> 00:31:26,252
Another way we could leave it as
we could rearrange it to get Y

449
00:31:26,252 --> 00:31:29,528
equals something. So if I do
that, I'll have minus two Y

450
00:31:29,528 --> 00:31:30,893
equals X and if we.

451
00:31:31,670 --> 00:31:37,091
Add 4 to both sides will get
plus three there, and if we

452
00:31:37,091 --> 00:31:41,678
divide everything by minus two
will get minus 1/2 X minus

453
00:31:41,678 --> 00:31:45,848
three over 2. So all of these
forms are equivalent.

454
00:31:47,480 --> 00:31:53,852
Now, that's not quite the whole
story. The most general form of

455
00:31:53,852 --> 00:31:58,100
equation of a straight line
looks like this.

456
00:31:58,430 --> 00:32:03,510
And earlier on in this unit,
we've seen some equations

457
00:32:03,510 --> 00:32:09,098
written in this form. Let's look
at some specific cases. Suppose

458
00:32:09,098 --> 00:32:14,178
that a this number here turns
out to be 0.

459
00:32:14,740 --> 00:32:16,468
What will that mean if a is 0?

460
00:32:17,000 --> 00:32:20,180
But if a is zero, we can
rearrange this and write BY.

461
00:32:20,920 --> 00:32:22,930
Equals minus C.

462
00:32:24,010 --> 00:32:27,520
Why is minus C Overby?

463
00:32:29,270 --> 00:32:32,942
And what does this mean?
Remember the A and the beat and

464
00:32:32,942 --> 00:32:36,614
the CIA just numbers their
constants. So when a is zero, we

465
00:32:36,614 --> 00:32:40,592
find that this number on the
right here minus C over B is

466
00:32:40,592 --> 00:32:44,876
just a constant. So what this is
saying is that Y is a constant.

467
00:32:45,720 --> 00:32:48,426
Now align where why is constant.

468
00:32:49,000 --> 00:32:53,810
Must be. A horizontal line,
because why doesn't change

469
00:32:53,810 --> 00:32:57,230
the value of Y is always
minus C Overby.

470
00:32:59,080 --> 00:33:04,218
So if you have an equation of
this form where a is zero that

471
00:33:04,218 --> 00:33:05,319
represents horizontal lines.

472
00:33:06,530 --> 00:33:09,578
What about if be with zero?

473
00:33:09,910 --> 00:33:15,370
We're putting B is 0 in here,
will get the AX Plus Co.

474
00:33:16,060 --> 00:33:20,980
And if we rearrange, this will
get AX equals minus C and

475
00:33:20,980 --> 00:33:24,670
dividing through by AX is minus
C over A.

476
00:33:25,990 --> 00:33:30,566
Again, a encia constants so this
time what this is saying is that

477
00:33:30,566 --> 00:33:31,974
X is a constant.

478
00:33:32,740 --> 00:33:35,575
Now lines were X is a constant.

479
00:33:36,240 --> 00:33:40,959
Must look like this. They are
vertical lines because the X

480
00:33:40,959 --> 00:33:42,246
value doesn't change.

481
00:33:42,250 --> 00:33:46,362
So this general case
includes both vertical lines

482
00:33:46,362 --> 00:33:47,904
and horizontal lines.

483
00:33:49,030 --> 00:33:52,660
So remember, the most general
form will appear like that.

484
00:33:54,200 --> 00:33:57,130
Provided that be isn't zero,
you can always write the

485
00:33:57,130 --> 00:34:00,646
equation in the more familiar
form Y equals MX plus C, but

486
00:34:00,646 --> 00:34:04,455
in the case in which B is 0,
you get this specific case

487
00:34:04,455 --> 00:34:05,920
where you've got vertical
lines.