www.mathcentre.ac.uk/.../Equations%20of%20straight%20lines.mp4

This unit is about the equation
of a straight line.

The equation of a straight line
can take different forms

depending upon the information
that we know about the line.

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

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

a point where X is not why is 2?

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

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

them on a graph.

The first point, not 2.

Will be here.

An X coordinate of zero
and a Y coordinate 2.

The second point 1 three X
coordinate of one Y coordinate

of three. And so on 2 four.

That will be here.

And three 5.

That will be there.

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

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

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

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

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

us the equation of the line the

Y value. Is always the X
value +2.

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

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

choose on the line will have
this same relationship between

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

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

with the equation by writing it
alongside like that.

Let's look at some more straight

line graphs. Let's suppose
we start with the

equation Y equals X
or drop a table

of values and plot

some points. Again,
let's start with some

X values. Suppose the
X values run from

012 up to three.

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

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

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

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

Let's plot these points on the

graph. Access note why is not.

Is the point of the origin.

X is one. Why is one?

Will be here.

And similarly 2233.

Will be there. And there.

So we have a straight line.

Passing through the origin.

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

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

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

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

we move from one point to the

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

three? So the change in Y is 3 -

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

so the change in X is also 3 -

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

So the gradient of this line is

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

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

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

And also write the equation
of the line alongside as

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

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

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

Let's see what the Y
coordinates will be this

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

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

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

When X is one, why will be 2 * 1

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

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

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

the origin again.

When X is one way is 2. That's

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

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

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

This line passes through
the origin.

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

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

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

this point and this point.

The change in Y.

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

4 takeaway 2 which is 2?

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

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

Just two.

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

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

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

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

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

And we can plot these on
the same graph.

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

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

3 gives me this point?

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

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

a straight line passing through
the origin is a steeper line

this time. And it's equation is

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

is 3X and all these lines pass

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

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

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

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

Subtract 3 or three.

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

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

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

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

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

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

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

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

one and the gradient is one.

Now we deduce from this a
general result that whenever we

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

is a straight line.

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

And it's gradient is
M. The number multiplying

the X is the

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

worth remembering that whenever
you see why is a constant M

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

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

Let's have a
look at some

other equations of

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

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

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

some Y values. When
X is

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

when X is 0 while B1.

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

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

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

It's there.

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

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

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

Not label it. Y equals 2X
plus one.

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

Let's see what happens this

time. Let's suppose we
start with a negative

X value.

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

one will get two times minus one

is minus 2. And 4 - 2
is 2.

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

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

Let's put those points.

So if X is minus one, why is 2?

If X is zero, why is 4?

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

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

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

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

the same gradient. The number
multiplying X.

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

Again, let's have some X values.

And some Y values supposing X

is 0. Well, if X is zero and why
is 2X minus one?

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

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

And effects is 2.

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

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

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

point down here.

If X is one.

Why is one gives
me a point here?

And if X is 2 wise, three gives
me that point there.

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

this third line.

Is parallel to the previous two
lines and it's parallel because

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

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

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

different is that they're all

cutting. The Y axis
at a different point.

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

Note that the number 4
appears in the equation.

This line. Cuts the Y axis when
Y is one, and again one appears

in the equation. And again, this
line cuts the Y axis at minus

one and minus one appears in the
equation, and this gives us a

general rule. If we have an
equation of the form Y equals MX

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

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

whereabouts on the Y axis that

the graph cuts. And we call this

value. Either the for their or
the one there, or the minus one

there. We call that the vertical
intercept so the value of C is

the vertical intercept.

So now whenever you see
an equation of the form

Y equals a number times
X plus another number.

So why equals MX plus C?

That represents a straight line

graph. Where M is the gradient

of the line. And sees the
vertical intercept, which is the

place where the graph crosses
the vertical axis.

Now sometimes when we get the
equation of a straight line, it

doesn't always appear in the
form Y equals MX plus C. Let

me give you an example.

Let's consider this equation 3.
Y minus two X equals 6. Now at

first sight that doesn't look as
though it's in the form Y equals

MX plus C which is our
recognisable form of the

equation of a straight line. But
what we can do is we can do some

algebraic manipulation on this
to try to write it in this form

and one of the advantages of
doing that is that if we can get

it into this form.

We can read off what the
gradients and the vertical

intercept are, so let's work on

this. I'll start by adding 2X
to both sides.

To remove this minus 2X from
here. So if we add 2X to both

sides will get 2X plus six on

the right. And now if I divide
both sides by three, I'll get Y

on its own, which is what I'm

looking for. Dividing 2X by
three gives me why is 2/3 of

X? And if I divide 6 by
three, I'll get 2.

Now this is a much more familiar
form. This is of the form Y

equals MX plus. See where we can
read off the gradient M is 2/3

and the vertical intercept see

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

not at first sight look as
though it's a straight line

equation, but by doing some work
on it you can get it into a

recognizable form. About
another

example. Suppose we're given
some information about a

straight line graph, and we want
to try and find out what the

equation is. So, for example,
suppose we're told that a

straight line has gradient, a
fifth and were told also that

it's vertical intercept. See is

one. Let's see if we can
write down the equation.

Well, we know that a
straight line has equation

Y equals MX plus C.

So we can substitute are known
values in M is going to be 1/5.

See is going to be one. So our
equation is Y equals 1/5 of X

plus one Y equals MX plus C.

Now we might not always choose
to leave it in that form, so let

me just show you how else we
might write it. There's a

fraction here of the 5th, and if
we multiply everything through

by 5, we can remove this
fraction. So let's multiply both

sides by 5, will get 5 Y the
files or cancel. When we

multiply by 5 here just to leave

X. And five ones of five.

So this form is equivalent to
this form, but just written in a

different way. We could
rearrange it again just by

bringing everything over to the
left hand side, so we might

write 5 Y minus X. Minus five is
not, so that is another form of

the same equation and we'll see
some equations written in this

form which later on.

OK, let's have
a look at

another example. Suppose now
we're interested in trying to

find the equation of a line
which has a gradient of 1/3.

And this time, instead of being
given the vertical intercept,

we're going to be given some
information about a point

through which the line passes.
So suppose that we know that the

line passes through the points
with coordinates 12.

Let's see if we can figure
out what the equation of

the line is.

Start with our general form Y
equals MX plus C and we put in

the information that we already

know. We know that the gradient
M is 1/3, so we can put that in

here straight away.

We don't know the vertical
intercept. We're going to have

to do a bit of work to find

that. But what we do know is
that the line passes through

this point. What that means is
that when X is one, why has the

value 2? And we can use that
information in this equation.

So we're going to put, why is

2IN. X is one in home, 3
third times, one is just a

third. Let's see from this we
can workout what Sears.

So two is the same as 6 thirds
and if we take a third off, both

sides will have 5 thirds is see
so you can see we can use the

information about a point on the

line. To find the vertical
intercept, see so. Now we know

everything about this line. We
know it's vertical intercept and

we know its gradient. So the
equation of the line is why is

1/3 X +5 thirds?

I want to do that again, but I
want to do it for more general

case where we haven't got
specific values for the gradient

and we haven't got specific
values for the point. So this

time, let's suppose we've got a
straight line. This gradient is

M. But it passes through a
point with arbitrary coordinates

X one. I want.

Let's see if we can
find a formula for the

equation of the line.

Always go back to what we know.
We know already that any

straight line has this equation.
Y is MX plus C.

What do we know what we're told
the gradient is M so that we can

leave alone. But we don't know
the vertical intercept. See

let's see if we can find it.

Use what we do now. We do know
that the line passes through

this point. So that we know that
when X has the value X one.

Why has the value? Why one?

So I'm going to put those values

in here. So why has the value?
Why one when X has the value

X one? See, now we can rearrange
this to find C. So take the MX

one off both sides.

That will give me the value for
C and this value for see that

we have found, which you
realize now is made up of. Only

the things we knew. We knew the
M we knew the X one and Y one.

So in fact we know this value.
Now we put this value back into

the general equation so will
have Y equals MX plus C and see

now is all this.

And that is the equation of a
line with gradient M passing

through X one Y one. We don't
normally leave it in this form.

We write it in a slightly
different way. It's usually

written like this. We subtract
why one off both sides to give

us Y, minus Y one, and that will

disappear. And we factorize the
MX and the NX one by taking out

the common factor of M will be
left with X and minus X one.

And that is an important result,
because this formula gives us

the equation of a line.

With gradient M and which passes
through a point where the X

coordinate is X one and the Y
coordinate is why one?

Let's look at
a specific example.

Suppose we're interested in a
straight line where the gradient

is minus 2, and it passes
through the point with

coordinates minus 3 two.

We know the general form of a
straight line, it's why minus

why one is MX minus X one?
That's our general results and

all we need to do is put this
information into this formula.

Why one is the Y coordinate
coordinate of the known point

which is 2?

M is the gradient which is minus

2. X minus X one is the X
coordinates of the known point,

which is minus 3.

So tidying this up, we've got Y
minus two is going to be minus

2X plus three, and if we remove
the brackets, Y minus two is

minus 2 X minus 6.

And finally, if we add two to
both sides, we shall get why is

minus 2 X minus four, and that's
the equation of the straight

line with gradient minus 2
passing through this point. And

there's always a check you can
apply because we can look at the

final equation we've got and we
can observe from here that the

gradient is indeed minus 2.

And we can also pop in an X
value of minus three into here.

Minus two times minus three is
plus six and six takeaway four

is 2 and that's the
corresponding why value? So this

built in checks that you can

apply. Let's have a look at
another slightly different

example, and in this example
I'm not going to give you the

gradient of the line.
Instead, we're going to have

two points on the line. So
let's suppose are two points

are minus 1, two and two 4,
so we don't know the gradient

and we don't know the
vertical intercept. We just

know two points on the line,
and we've got to try to

determine what the equation
of the line is.

Now let's see how we can do

this. One thing we can do is we
can calculate the gradient of

the line because we know how to
calculate the gradient of the

line joining two points.

So let's do that. First of all,
the gradient of the line will be

the difference in the Y
coordinates, which is 4 - 2.

Divided by the difference in the
X coordinates, which is 2 minus

minus one. 4 - 2 is 2 and 2
minus minus one is 2 + 1 which

is 3. So the gradient of this
line is 2/3.

Now we know the gradients and we
know at least one point through

which the line passes. Because
we know two. So we can use the

previous formula Y minus Y one
is MX minus X one. So that's the

formula we use.

Why minus why? One doesn't
matter which of the two points

we take? Let's take the .24.

So the Y coordinate is 4.

M we found is 2/3.

X minus the X coordinate, which
is 2. So that's our equation of

the line and if we wanted to do
we can tidy this up a little

bit. Y minus four is 2/3 of X
minus 4 thirds, and if we add 4

to both sides, we can write this
as Y is 2X over 3.

And we've got minus 4 thirds
here already, and we're bringing

over four will be adding four.
So finally will have two X over

3. And four is the same as 12
thirds, 12 thirds. Subtract 4

thirds is 8 thirds. So that's
the equation of the line.

I want to do that same
argument when were given two

arbitrary points instead of
two specific points.

So suppose we have the point a
coordinates X one and Y 1.

And be with coordinates X2Y2
and. Let's see if we can figure

out what the equation of the
line is joining these two

points, and I think in this case
a graph is going to help us.

Just have a quick sketch.

So I've got
a point A.

Coordinates X One Y1.

And another point B
coordinates X2Y2 and were

interested in the equation of
this line which is joining

them.

Suppose we pick an arbitrary
point on the line anywhere along

the line at all, and let's call
that point P.

P is an arbitrary point, and
let's suppose it's coordinates

are X&Y. For any arbitrary X&Y
on the line.

And what we do know is that the
gradient of AP is the same as

the gradient of a bee.

Let me write that
down the gradient.

Of AP. Is
equal to the gradient.

Of a B.

Let's see what that means. Well,
the gradient of AP.

Is just the difference in the Y

coordinates. Is Y minus
Y one over the

difference in the X
coordinates, which is X

minus X one?

So that's the gradient of this
line segment between amp and

that's got to equal the gradient
of the line segment between A&B.

And once the gradient of the
line segment between A&B, well,

it's again. It's the difference
of the Y coordinates, which now

is Y 2 minus Y 1.

Divided by the difference in the
X coordinates which is X2 minus

X one. So that is a formula
which will tell us.

The equation of a line passing
through two arbitrary points.

Now we don't usually leave it in
that form. It's normally written

in a slightly different form,
and it's normally written in a

form so that all the wise appear
on one side and all the ex is

appear on another side, and we
can do that by dividing both

sides by Y 2 minus Y 1.

And multiplying both sides by X
minus X One which moves that up

to here. And that's the form
which is normally quoted as the

equation of a line passing
through two arbitrary points.

Let's use that in an example.

Suppose we have
two points A.

Coordinates one and minus two
and B which has coordinates

minus three and not.

Let me write down the formula
again. Why minus why one over Y

2 minus Y one is X Minus X one
over X2 minus X one?

With pop everything we know into
the formula and see what we get.

So we want Y minus Y1Y one is
the first of the Y values which

is minus 2.

Why 2 minus? Why one is the
difference of the Y values that

zero? Minus minus 2.

Equals X minus X one is the
first of the X values, which is

one and X2 minus X. One is the
difference of the X values.

That's minus three, subtract 1.

And just to tidy this up on the
top line here will get Y +2 on

the bottom line. Here will get

+2. X minus one there on
the right at the top and

minus 3 - 1 is minus 4.

Again, we can tie this up a
little bit more to will go into

minus 4 - 2 times.

And if we multiply everything
through by minus, two will get

minus two Y minus four equals X
minus one, and we can write this

in lots of different ways. For
example, we could write this as

minus two Y minus X.

And we could add 1 to both
sides to give minus 3 zero.

That's one way we could leave
the final answer.

Another way we could leave it as
we could rearrange it to get Y

equals something. So if I do
that, I'll have minus two Y

equals X and if we.

Add 4 to both sides will get
plus three there, and if we

divide everything by minus two
will get minus 1/2 X minus

three over 2. So all of these
forms are equivalent.

Now, that's not quite the whole
story. The most general form of

equation of a straight line
looks like this.

And earlier on in this unit,
we've seen some equations

written in this form. Let's look
at some specific cases. Suppose

that a this number here turns
out to be 0.

What will that mean if a is 0?

But if a is zero, we can
rearrange this and write BY.

Equals minus C.

Why is minus C Overby?

And what does this mean?
Remember the A and the beat and

the CIA just numbers their
constants. So when a is zero, we

find that this number on the
right here minus C over B is

just a constant. So what this is
saying is that Y is a constant.

Now align where why is constant.

Must be. A horizontal line,
because why doesn't change

the value of Y is always
minus C Overby.

So if you have an equation of
this form where a is zero that

represents horizontal lines.

What about if be with zero?

We're putting B is 0 in here,
will get the AX Plus Co.

And if we rearrange, this will
get AX equals minus C and

dividing through by AX is minus
C over A.

Again, a encia constants so this
time what this is saying is that

X is a constant.

Now lines were X is a constant.

Must look like this. They are
vertical lines because the X

value doesn't change.

So this general case
includes both vertical lines

and horizontal lines.

So remember, the most general
form will appear like that.

Provided that be isn't zero,
you can always write the

equation in the more familiar
form Y equals MX plus C, but

in the case in which B is 0,
you get this specific case

where you've got vertical
lines.