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.