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PROFESSOR: Hi.
Our lecture for today probably
should be entitled it
should've been functions, but
it's analytic geometry
instead, or a picture is
worth a thousand words.
What we hope to do today is
to establish the fact that
whereas in the study of calculus
when we deal with
rate of change we are interested
in analytical
terms, that more often than not,
we prefer to visualize
things more intuitively in
terms of a graph or other
suitable visual aid, and that
actually, this is not quite as
alien or as profound as it
may at first glance seem.
Consider, for example,
the businessman who
says profits rose.
Profits rose.
Now, you know, profits don't
rise unless the safe blows up
or something like this.
What profits do is they increase
or they decrease.
The reason that we say profits
rise is that when the profits
are increasing, if we are
plotting profit in terms of
time, the resulting graph
shows a rising tendency.
As the profit increases,
the curve rises.
And in other words then, we
begin to establish the feeling
that we can identify the
analytic term increasing with
the geometric term rising.
And this identification, whereby
difficult arithmetic
concepts are visualized
pictorially, is something that
begins not only very early in
the history of man, but very
early in the development of the
mathematics curriculum.
Oh, as a case in point, consider
the problem of 5
divided by 3 versus
6 divided by 3.
I remember when I was in grade
school that this particular
problem always seemed more
appealing to me than this
problem, that 6 divided by 3
seemed natural, but 5 divided
by 3 didn't.
And the reason was is that in
terms of visualizing tally
marks, it was much easier to see
how you divide six tallies
into three groups than five
tallies into three groups.
However, as soon as we
pick a length as our
model, the idea is this.
Either one can divide a line
into three parts of equal
length or one can't
divide the line.
Now, if I can geometrically
divide this line into three
equal parts, and in plain
geometry we learn to do this,
then the fact is that I can
divide this line segment into
three equal parts regardless
of how long this
line happens to be.
Oh, to be sure, if this line
happens to be 6 units long,
this point is named 2.
And if, on the other hand, the
line happens to be only 5
inches long, the resulting
point is named 5/3.
But notice that in either case,
I can in a very natural
way define or identify either
ratio as a point on the line.
And this idea of identifying
numerical concepts called
numbers with geometric concepts
called points is a
very old device and a device
that was used and still is
used in the curriculum today
under the name of the number
line, under the name of graphs,
and what we will use
as a fundamental building block
as our course develops.
Now, you know, in the same way
that we can think of a single
number as being a point on the
line, we can think of a pair
of numbers, an ordered pair
of numbers, as being a
point in the plane.
This is Descartes geometry,
which we can call coordinate
geometry, the idea being that
in the same way as we can
locate a number of along the
x-axis, shall we say, we could
have located a numbered pair
as a point in the plane.
Namely, 2 comma 3 would mean the
point whose x-coordinate
was 2 and whose y-coordinate
was 3.
By the way, the reason that we
say ordered pairs is, if you
observe, 2 comma 3 and
3 comma 2 happen to
be different pairs.
And notice again how vividly the
geometric interpretation
of this is.
Namely, in terms of locating a
point in space, it's obvious
that the point named 2 comma 3
is not the same as the point
named 3 comma 2.
Again, the important
thing is this.
When I think of 5 divided by 3,
when I think of the ordered
pair 2 comma 3, I do not have
to think of a picture.
I can think of these things
analytically.
But the picture gives me certain
insights that will
help me with my intuition, an
aid that I don't want to
relinquish.
For example, going back
to the graph again.
Thinking of the analytic term
greater than, notice how much
easier it is to think of, for
example, higher than, see, one
point being higher than another
or to the right of.
You see, geometric concepts to
name analytic statements, or
instead of increasing,
as we mentioned
before, to say rising.
And there will be many, many
more such identifications as
we go along with our course.
At any rate, let's continue
to see then what is the
relationship then between
functions that we talked about
and graphs?
The idea is something
like this.
Let's return to our friend of
the first lecture: s equals
16t squared.
We can think of a distance
machine being the function
where the input will be time and
the output will be what?
The square of the input
multiplied by 16.
Observe that from this, I do not
have to have any picture
to understand what's
happening here.
Namely, I can measure an input,
measure an output, and
observe analytically
what is happening.
However, as we saw last time,
our graph sort of shows us at
a glance what seems to be
happening, that we can
identify rising and falling with
increasing and decreasing
and things of this type.
We will explore this, of course,
in much more detail as
we continue in our course.
By the way, there's no reason
why the input has to be a
single number.
For example, why couldn't
the input be an
ordered pair of numbers?
Among other things, let's take
a simple geometric example.
Consider, for example, finding
the volume of a cylinder in
terms of the radius of its
base and the height.
We know from solid geometry
that the volume is
pi r squared h.
We could therefore think of a
volume machine where the input
is the ordered pair r comma h,
and the output is the single
number pi r squared h.
By the way, notice here the
meaning of ordered pair.
You see, if the pair 2 comma 3
goes into the machine, notice
that the recipe here
says what?
You square the first
member of the pair.
In other words, if 2 comma 3
is the input, we square 2,
which is 4, multiplied by 3,
which is 12, and 12 times pi,
of course, is 12 pi.
On the other hand, if the input
is 3 comma 2, the first
number is 3.
Our recipe squares
the first number.
That would be 9, times 2 is
18, times pi is 18 pi.
But again, observe that I at
no time needed a picture to
visualize what was
happening here.
Of course, if I wanted a
picture, I could try to plot
this also, but notice now that
my graph would probably need
three dimensions to draw.
And why would it need
three dimensions?
Well, notice that my input has
two independent measurements r
and h, and therefore, I would
need two dimensions just to
take care of r and h.
Then I would need a third
dimension to plot v.
And by the way, notice
the next stage.
If I had an input that consisted
of three independent
measurements, this would still
make sense, but now I would be
at a loss for the picture.
In other words, what I'm trying
to bring out next is
the fact that whereas pictures
are a tremendous help, maybe a
second subtitle to our lecture
should've been a picture is
worth a thousand words
provided you
can could draw it.
Because, you see, if we needed
three independent dimensions
to locate the input and then
a fourth one to locate the
output, how would we
draw the picture?
By the way, this happens in high
school algebra again, if
you want to see the analogy.
Look at, for example,
the algebraic
equation a plus b squared.
We learned in algebra that this
is a squared plus 2ab
plus b squared.
Observe that we do not need to
have a picture to understand
how this works.
Oh, to be sure, if we had a
picture, we get a tremendous
amount of insight as to
what's happening here.
Namely, let's visualize a square
whose side is a plus b.
On the one hand, you see, the
area of the square would be a
plus b squared, you see,
the side squared.
On the other hand, if we now
subdivide this figure this
way, we see that that same
square is made up of four
pieces having one piece of area
a squared, two pieces of
area ab, and one piece
of area b squared.
And so we see, on the other
hand, that the area of the
square is a squared plus
2ab plus b squared.
And so again, observe that
whereas this stands on its own
two legs, the picture helps
us quite a bit.
By the way, we could continue
this with a cube over here.
Namely, it turns out that a plus
b cubed is a cubed plus
3a squared b plus 3ab squared
plus b cubed.
And again, if we wanted to,
and we won't take the time
here, but if we wanted to, we
could now draw a cube whose
side is a plus b.
Namely, we could take the same
diagram that we had before and
now make a third dimension
to it.
And if you did that,
you would see what?
That the cube whose volume is a
plus b cubed is divided into
eight pieces, one of size a by
a by a, three of size a by a
by b, three of size a by
b by b, and one of
size b by b by b.
Again, the picture is a
tremendous visual aid.
Now, the key step is this.
If we were to now give in to
our geometric intuition and
say, lookit, why worry about
algebra when it's so much
easier to do this thing by
geometry, the counterexample
is consider a plus b to
the fourth power.
Now, by the binomial theorem,
and I might add, the same
binomial theorem that allowed
us to get these results, we
can also write that
this is what?
It's a is the fourth plus 4a
cubed b plus 6a squared b
squared plus 4ab cubed
plus b to the fourth.
Now, it's not important how
we get this result.
The important point is that
analytically, we can raise a
number to the fourth power just
as easily as we can to
the third power or
the second power.
The only difference is that in
the case of the third power,
we had a picture that
we could use.
In the fourth power case, we
didn't have a picture.
And what a tragedy it would
have been to say, hey, we
can't solve this problem
because we
can't draw the picture.
And by the way, as a rather
interesting aside, notice the
geometric influence on
how we read this.
This is called a plus b
to the fourth power.
But somehow or other, we don't
call this one a plus b to the
third power.
We call it a plus b cubed,
suggesting the geometric
configuration of the cube.
And here we don't usually say a
plus b to the second power.
We say a plus b squared.
You see, the idea is that when
the picture is available, it
gives us a tremendous insight as
to what can be done, and it
helps us learn to visualize
what's happening analytically.
In fact, what usually happens
is we use the picture to
justify what's happening
analytically when we can see
the picture and then just
carry the analytic part
through unimpeded in
the case where we
can't draw the picture.
Let me give you another
example of this.
Let's look at the
following set.
That also should review our
language of sets for us.
Let S be the set of all ordered
pairs x comma y such
that x squared plus y
squared equals 25.
Question: Does the ordered pair
3 comma 4 belong to S?
Answer: Yes.
How do we know?
Well, we have a test
for membership.
We're supposed to do what?
Square each of the entries, each
of the numbers, add them,
and if the answer is 25, then
that ordered pair belongs to
S. 3 squared plus 4 squared is
25, so this pair belongs to S.
How about 1 comma 2?
Well, 1 squared plus 2 squared
is 1 plus 4, which is 5.
5 is not equal to 25, so 1 comma
2 does not belong to S.
Well, did we need any geometry
to be able to
visualize this result?
Hopefully, one did not
need any geometry to
visualize this result.
On the other hand then, what
does it mean in analytic
geometry when we say that x
squared plus y squared equals
25 is a circle?
As badly as I draw, x squared
plus y squared equals 25 looks
less like a circle than the
circle I drew over here.
You see, what we really
mean is this.
Consider all the points in the
plane x comma y for which x
squared plus y squared
equals 25.
These are precisely the points
on this particular circle.
And the easiest way to see that,
of course, is that since
the radius of the circle is 5
and the point x comma y means
that this length is x and this
length is y, notice that from
the Pythagorean theorem, we see
it once, that x squared
plus y squared equals 25.
Now again, the solution set to
this equation is our set S
whether we're thinking of this
thing algebraically or
geometrically.
On the other hand, watch what
our picture seems to give us
that we didn't have before.
Let's return to our point 1
comma 2, which we saw didn't
belong to S. Well,
look at this.
Where would 1 comma 2 be?
1 comma 2 would be inside
the circle.
Why is that?
Because, you see, if we take
the point 1 comma 2, if we
take, say, for example, the
point 1 comma 2, notice that
the distance from the origin
to the point 1 comma 2
is less than 5.
In other words, the distance is
less than 5 so the square
of the distance is
less than 25.
In other words, not only can we
say that 1 comma 2 does not
belong to S, which we could have
said without the picture,
we can now say what?
1 comma 2 is--
and notice the geometric
language here--
is inside the circle.
In other words, the study of
inequalities can now be reduced.
Instead of talking about less
than and greater than, we can
now talk about such things
as inside and outside.
You see, inside the circle,
which is a simple geometric
concept, just means what?
A set of all points for which
x squared plus y squared is
less than 25.
Outside that circle, x squared
plus y squared is
greater than 25.
On the circle, x squared plus
y squared equals 25.
Again, a nice identification
between numbers and pictures,
analysis and geometry.
Well, this then shows us why
we want to study pictures
rather than functions.
Now, if we look at any textbook
in which we deal with
graphs, it always seems that
we start with graphs of
straight lines.
And the question is what is so
great about a straight line?
After all, pictures in general
are going to be much more
complicated than that.
What advantage is there in
starting with straight lines?
And again, we begin to realize
how straight lines are the
backbone of all types of
analytical procedures and all
types of curve plotting.
For example, let's suppose we
were studying this particular
curve, and we wanted to know
what was happened to that
curve in the neighborhood
around the point p.
Let's draw in the tangent line
to the curve at the point p.
Notice that this line that
we've drawn serves as a
wonderful approximation curve
itself if we stay close enough
to the point of tangency.
In other words, notice how much
we can deduce about this
curve if we study only the
straight line segment at the
point of tangency.
Of course, the approximation
gets worse and worse as we
move further and further out.
But in the neighborhood of the
point of what's going on,
notice again then that the
straight line is an important
building block.
By the way, again we use
straight lines in a rather
subtle way in something
called interpolation.
For example, let's suppose I go
to a log table and I look
up the log of 2.
I find that the log
of 2 is 0.301.
I look up the log of 4.
I find that's 0.602.
Now I look for the log of 3,
and I see that it's been
obliterated.
I don't know what it is.
So I say, well, let me guess.
3 is halfway between 2 and 4.
Therefore, I would suspect that
the log of 3 is halfway
between the log of 2
and the log of 4.
And so, halfway between here
would be about what?
0.452, roughly speaking.
All of a sudden, the
obliteration on
my book clears up.
And I look, and I don't
find 0.452.
Instead I find--
well, let's write
it over here.
What I find is 0.477.
Now, you know, this is a pretty
big error to attribute
to slide rule inaccuracy
or trouble in
rounding off the tables.
What really went wrong
over here?
And the answer comes up again
that unless otherwise
specified, the process known
as interpolation hinges on
replacing a curve by a straight
line approximation.
In fact, you see, if we were
to draw the curve of the
logarithm function, we would
find that the picture is
something like this.
And when we looked up the log
of 2, this height is what we
found in the table.
When we looked up the log of
4, this height is what we
would have found in the table.
If we had looked up in the table
the log of 3, this is
the height that we
would have found.
This is the height
that's 0.477.
Notice that in general, if we go
halfway from here to here,
we do not go halfway
from here to here.
It depends on the shape
of the curve.
The only time you can be sure
that you have proportional
parts is if the curve that
joined these two points was a
straight line.
And notice, by the way, that
by the shape of this curve,
the straight line falls below
the curve, and therefore, the
height that we found
was to the straight
line, not to the curve.
That was the point 0.452.
So, in other words, notice that
we got smaller than the
right answer because we
approximated as if it was a
straight line that was
joining the curve.
You see, what interpolation
hinges on is that the size of
the interval is very small and
that you can assume that for
the accuracy that you're
interested in that the
straight line approximation to
the curve is sufficiently
accurate to represent
the curve itself.
Well, enough about that.
Once we've talked about why
straight lines are important,
the next thing is how do we
measure straight lines?
See, another interesting point
to something like this.
Many times we know what
something means subjectively,
but we don't know what
it means objectively.
For example, one way of finding
a line is to know two
points on the line.
Another way is to know
one point and the
slant of the line.
The question that comes up
is how do you measure
the slant of a line?
In other words, shall you say
that the line is very slanty?
And if the answer to that is
yes, how do you distinguish
between slanty and very slanty,
steep and very steep,
very steep and very,
very steep?
We need something
more objective.
And the way we get around
this is as follows.
Given a line, we define
the slope as follows.
We pick any two points
on the line.
And from those two points, we
can measure what we call the
run of the line, in other words,
how far you've gone
this way, and the rise
of the line, how much
it's risen this way.
And what we do is we define
the slope to be the rise
divided by the run, or without
the delta notation in here, y2
minus y1 over x2 minus x1.
By the way, there are little
problems that come up.
After all, our answer should
not depend on the picture.
It should be sort of
self-contained analytically.
The question comes up is what
if I had labeled this point
x2, y2 and this point x1, y1?
What would have happened then?
And observe that as long as we
keep the pairs straight, it
makes no difference whether you
write this or whether you
write this.
Because, you see, in each
case, all you've done is
change the sign, and negative
over negative is positive.
So certainly our answer to what
a slope is is objective
enough, so it does not depend on
how the points are labeled.
A second objection that most
people have is they say
something like, well, who are
you to say that we pick these
two points?
What if I came along and picked
these two points and I
now computed the slope by taking
this as my delta y and
this as my delta x?
Obviously, it would be a tragedy
if the answer to the
problem depended on which pair
of points you picked since a
line should have
but one slope.
Again, notice that our high
school training in geometry,
similar triangles, motivates
why we pick ratios.
Namely, while this delta y and
this delta y may be different
and this delta x and this delta
x may be different, what
is true is that the ratio of
this delta y to this delta x
is the same as the ratio of this
delta y to this delta x.
And that's why we
pick the ratio.
By the way, another way of
talking about ratio is if you
look at delta y divided by delta
x and you've had some
trigonometry, it reminds
you of a trigonometric
relationship.
Namely, you look at delta y, you
look at delta x, and you
say, my, isn't that just
the tangent of
this particular angle?
Couldn't I define the slope?
And by the way, the general
symbol for slope, for better
or for worse, just happens
to be letter m.
Why couldn't I define m to be
the tangent of phi, where phi
is the angle that the straight
line makes with
the positive x-axis?
And, of course, there is a
little subtlety here that we
should pay attention to.
This would be an ambiguous
definition if the scale on the
x- and the y-axis were
not the same.
In other words, notice that by
changing the scale here, I can
distort the same analytic
information.
So if I agree, however, that the
unit on the x-axis is the
same as the unit on the y-axis,
then I can say, OK,
the slope is also tangent
of the angle phi.
I much prefer to say it's delta
y divided by delta x,
because then if I forget
the scale,
I'm still in no trouble.
On the other hand, if we use the
tangent definition, we can
utilize all we know about
trigonometry to get some other
interesting results.
Namely, the question that might
come up is can we study
the slopes of two different
lines very conveniently in
terms of our definition
of slope?
And the answer is this.
If we imagine now that our lines
are drawn to scale here,
and here are two different
lines, which I'll call l1 and
l2, and we'll call the angle
that l1 makes with the
positive x-axis phi 1, the angle
that l2 makes with the
positive x-axis phi 2.
Therefore, what? m1
is tan phi 1.
m2 is tan phi 2.
Notice that our formula for the
tangent of the difference
of two angles-- you see,
notice that this
angle here is what?
Since this angle is the sum of
these two, this angle here is
phi 2 minus phi 1 or the
negative of phi 1 minus phi 2.
I should have had this phi 2
minus phi 2, but since that
just changes the sign, that will
not have any bearing on
the point I want to make.
Let's continue this way.
Tangent of phi 1 minus phi 2 is
tan phi 1 minus tan phi 2
over 1 plus tan phi
1 tan phi 2.
On the other hand, by our
definitions of m1 and m2, this
is m1 minus m2 over
1 plus m1 m2.
Now, this tells me how to find
the angle between two lines
just in terms of knowing
the slope.
Two very special interesting
cases as extremes suggest
themselves right away.
One case is what happens if
the lines are parallel?
If the lines are parallel, you
see, phi 1 equals phi 2, in
which case phi 1 minus phi
2 is 0, in which case the
tangent of phi 1 minus phi
2 had better be 0.
But the only way a fraction can
be 0 is for the numerator
to be 0, and that means
that m1 must equal m2.
In other words, in terms of
slopes, we can study parallel
lines just by equating
their slopes.
A less obvious relationship
that's equally important is
how do you measure whether two
lines are perpendicular?
The answer is if they're
perpendicular, the angle
between them is 90 degrees.
The tangent of 90 degrees is
infinity, as we learned.
That's equivalent
to saying what?
That the denominator is 0.
See, the only way a fraction
blows up is for the
denominator to be 0.
But the only way that 1 plus
m1 m2 can be 0 is for what?
m1 m2 to be equal to minus 1.
And this gives us the other very
well-known result that in
terms of slopes, to study
whether two lines are
perpendicular, all we need to
investigate is whether one
slope is the negative reciprocal
of the other.
Well, again, the textbook
will bring out
slopes in more detail.
The next question that we'd like
to bring up in terms of a
picture is worth a thousand
words is how do you identify
the straight line with an
algebraic equation?
What do we mean by the equation
of a straight line?
Well, again, there are
two possibilities.
The first possibility
is that the line is
parallel to the y-axis.
If the line is parallel to the
y-axis, if the line goes
through the point a comma 0,
notice that the only criteria
for the point to be on
that line is that its
x-coordinate equal a.
By the way, this is often
abbreviated in the textbook as
the line x equals a.
Many a student says how do
you know this is a line?
Why isn't this the
point x equals a?
And here again is a good review
of why we stress the
language of sets.
Here again is a good reason why
we express the language of
sets so strongly.
Namely, go back to the universe
of discourse here.
When you see the set of all
ordered pairs x comma y for
which x equals a, this gives
you the hint that you're
talking about pairs of points,
and that tells you that you
have numbers in the plane, not
on the line, not on the
x-axis, a two-dimensional
interpretation over here.
You see, if this said the set of
all x such that x equals a,
it would just be a point.
But notice the hint over here.
At any rate, this then becomes
the equation of a straight
line if the straight line is
parallel to the y-axis.
Of course, the other possibility
is what if the
line isn't parallel
to the x-axis?
And here, too, we say OK,
suppose we know a point on the
line and suppose we know
the slope of the line.
What we will do is pick any
other point on the plane,
which we will label arbitrarily
x comma y, and see
what the equation x comma
y has to satisfy.
How do the coordinates have to
be related to be on this line?
Well, we already know that slope
does not depend on which
two points you pick.
Consequently, since the slope
of this line is m, the slope
must also be what?
y minus y1 over x minus x1.
And this becomes the fundamental
definition for the
equation of a line which is not
parallel to the y-axis.
And by the way, again, I think
m's and x1's and y1's tend to
give you a bit of hardship
at first until
you get used to them.
Let's illustrate this thing
with a specific example.
Suppose I say to you I am
thinking of the line whose
slope is 3 and which passes
through the point 2 comma 5.
And notice the language
of sets here.
To say that 2 comma 5 is on the
line is the same as saying
that 2 comma 5 belongs
to the set of points
determined by the line.
Drawing a rough sketch
over here--
and by the way, notice something
very important here.
I never have to draw to scale.
Because, you see, all I'm going
to use is the analytic
terms, and 2 comma 5 is still 2
and 5, no matter how I draw
the picture.
So, for example, if I say OK,
let's see what it means for
the point x comma y to belong
here, I say, well,
what does that mean?
My slope is going to have
to be what? y minus 5.
That's my rise.
My run is x minus 2, and
that must equal 3.
And if I clear this of
fractions, I get what? y is
equal to 3x minus 1.
By the way, does
this check out?
If x is 2, 2 times 3
is 6, minus 1 is 5.
2 comma 5 is on the line.
You see, here's the thing.
We talked about the line
geometrically.
Now I have an algebraic
equation.
I no longer have to refer
to the picture.
I have something analytic now.
For example, suppose a person
says to me I wonder if the
point 8 comma 23 is
on this line?
I don't have to draw
a picture to scale.
I don't have to waste
any time.
I know that the equation of my
line is y equals 3x minus 1.
By the way, if y equals 3x minus
1, as soon as x is 8,
what must y equal?
y must equal what?
23?
Is that right?
And so is the point 8 comma
23 on the line?
Yes.
How about 8 comma 12?
8 comma 12 isn't on the
line because 3 times 8
minus 1 is not 12.
But notice that we can even see
algebraically that 8 comma
12 must be below the line.
In other words, our study of
equations allows us not only
to visualize lines as equations,
but we can also
visualize inequalities
as pictures.
In other words, if we have the
equation of a line, if this is
the line y equals, say, 3x plus
1 or something like this,
then what is this region here?
These are all those values which
lie-- whose heights lie
below the height to
be on the curve.
Again, not a very clear example
in the sense of
drawing the picture neatly for
you, but our main aim is not
to draw neat pictures here.
Our main aim is to show how
analytical terms can be
studied very conveniently
in terms of pictures.
In fact, perhaps to conclude
today's lesson, what we should
talk about is an old algebraic
concept called
simultaneous equations.
Suppose you're asked to solve
this pair of equations.
You say, well, let's see, if y
equals 3x minus 1 and it's
also equal to x plus 1, that
says that x plus 1
equals 3x minus 1.
I now solve this thing
algebraically.
I get 2x equals 2,
so x equals 1.
Knowing that x equals 1, I can
see that y equals 2, and I see
that 1 comma 2 is my solution.
In other words, if I wound up
with this thing algebraically,
1 comma 2 is the only member
that belongs to both of these
two solution sets.
Now, again, notice how I can
solve this purely analytic
problem without recourse
to a picture.
On the other hand, if I want
to think of this thing
pictorially, notice that y
equals 3x minus 1 is the
equation of a particular
straight line, and y equals x
plus 1 is also the equation
of a line.
Notice that since these two
lines are not parallel, they
intersect at one particular
point.
And the geometric problem that I
solved on the previous board
turns out to be that the point
1 comma 2 is the point that
both of these lines
have in common.
In fact, if we call this line
as before l1 and if we name
this l2, in the language of
sets, 1 comma 2 is what?
The point which is the
intersection of the two lines
l1 and l2, again, a geometric
interpretation
for an analytic problem.
In fact, notice also how much
mileage I can get out of the
geometric picture.
For example, notice that this
region here has a very nice
geometric interpretation.
It's the set of what?
All points which are below this
line and above this line.
In other words, what?
To be below this line, y must be
less than x plus 1, and to
be above this line, y must be
greater than 3x minus 1.
Notice then that a pair of
simultaneous inequalities,
which may not be that easy to
handle, are very easy to
handle in terms of regions
in the plane.
Notice also that since two lines
can either be parallel
or not parallel, we also get a
nice geometric interpretation
as to why simultaneous
equations
may have one solution.
Namely, the lines are
not parallel,
and hence, they intersect.
Or no solutions, the lines
could've been parallel without
intersecting.
Or infinitely many solutions,
the two lines could have been
different equations.
In effect, I should say what?
The two equations could've been
different equations for
the same line.
Again, this may seem a little
bit sketchy and rapid, but all
we want to do is give
the overview.
The reading assignment in the
text goes into great detail on
the points that we've
mentioned so far.
But again, in summary, what our
lesson was supposed to be
today was to indicate the
importance of being able to
visualize and to identify
analytic results with
geometric pictures.
And so, until next
time, goodbye.
Funding for the publication of
this video was provided by the
Gabriella and Paul Rosenbaum
Foundation.
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