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## Solutions to linear equations

# Intro to the coordinate plane

## Video transcript

This right here is a
picture of Rene Descartes. Once again, one
of the great minds in both math and philosophy. And I think you're seeing
a little bit of a trend here, that the great
philosophers were also great mathematicians
and vice versa. And he was somewhat of a
contemporary of Galileo. He was 32 years younger,
although he died shortly after Galileo died. This guy died at a
much younger age. Galileo was well into his 70s. Descartes died at what
is only 54 years old. And he's probably most known in
popular culture for this quote right over here, a
very philosophical quote. "I think,
therefore I am." But I also wanted to throw in,
and this isn't that related to algebra, but I just thought
it was a really neat quote, probably his least famous
quote, this one right over here. And I like it just because
it's very practical, and it makes you realize
that these great minds, these pillars of
philosophy and mathematics, at the end of the day, they
really were just human beings. And he said, "You
just keep pushing. You just keep pushing. I made every mistake
that could be made. But I just kept pushing." Which I think is very,
very good life advice. Now, he did many things in
philosophy and mathematics. But the reason why
I'm including him here as we build our
foundations of algebra is that he is the
individual most responsible for a very strong connection
between algebra and geometry. So on the left over here, you
have the world of algebra, and we've discussed
it a little bit. You have equations
that deal with symbols, and these symbols
are essentially they can take on values. So you could have something
like y is equal to 2x minus 1. This gives us a relationship
between whatever x is and whatever y is. And we can even set up a table
here and pick values for x and see what the
values of y would be. And I could just pick
random values for x, and then figure out what y is. But I'll pick relatively
straightforward values just so that the math
doesn't get too complicated. So for example, if
x is negative 2, then y is going to
be 2 times negative 2 minus 1, which is negative 4
minus 1, which is negative 5. If x is negative
1, then y is going to be 2 times negative
1 minus 1, which is equal to-- this is negative
2 minus 1, which is negative 3. If x is equal to 0, then y is
going to be 2 times 0 minus 1. 2 times 0 is 0 minus
1 is just negative 1. I'll do a couple more. And I could have
picked any values here. I could have said,
well, what happens if x is the negative
square root of 2, or what happens if x is
negative 5/2, or positive 6/7? But I'm just picking
these numbers because it makes the
math a lot easier when I try to figure out
what y is going to be. But when x is 1, y is going
to be 2 times 1 minus 1. 2 times 1 is 2 minus 1 is 1. And I'll do one more. I'll do one more in a color
that I have not used yet. Let's see, this purple. If x is 2, then y is
going to be 2 times 2-- now our x is 2-- minus 1. So that is 4 minus
1 is equal to 3. So fair enough. I just kind of sampled
this relationship. I said, OK, this describes
the general relationship between a variable
y and a variable x. And then I just made it a
little bit more concrete. I said, OK, well, then
for each of these values of x, what would be the
corresponding value of y? And what Descartes
realized is is that you could visualize this. One, you could visualize
these individual points, but that could also
help you, in general, to visualize this relationship. And so what he
essentially did is he bridged the worlds
of this kind of very abstract, symbolic algebra
and that and geometry, which was concerned with
shapes and sizes and angles. So over here you have
the world of geometry. And obviously, there are people
in history, maybe many people, who history may have forgotten
who might have dabbled in this. But before Descartes,
it's generally viewed that geometry
was Euclidean geometry, and that's essentially the
geometry that you studied in a geometry class in
eighth or ninth grade or 10th grade in a traditional
high school curriculum. And that's the
geometry of studying the relationships between
triangles and their angles, and the relationships between
circles and you have radii, and then you have triangles
inscribed in circles, and all the rest. And we go into some depth in
that in the geometry playlist. But Descartes
said, well, I think I can represent this visually
the same way that you could with studying these
triangles and these circles. He said, well, if we
view a piece of paper, if we think about a
two-dimensional plane, you could view a
piece of paper as kind of a section of a
two-dimensional plane. And we call it two
dimensions because there's two directions that
you could go in. There's the up/down direction. That's one direction. So let me draw that. I'll do it in blue because we're
starting to visualize things, so I'll do it in
the geometry color. So you have the
up/down direction. And you have the
left/right direction. That's why it's called
a two-dimensional plane. If we're dealing in
three dimensions, you would have an
in/out dimension. And it's very easy to do
two dimensions on the screen because the screen
is two dimensional. And he, says, well, you know,
there are two variables here, and they have this relationship. So why don't I associate
each of these variables with one of these
dimensions over here? And by convention, let's make
the y variable, which is really the dependent variable--
the way we did it, it depends on what x is-- let's
put that on the vertical axis. And let's put our
independent variable, the one where I just
randomly picked values for it to see what y
would become, let's put that on the horizontal axis. And it actually
was Descartes who came up with the convention of
using x's and y's, and we'll see later z's, in
algebra so extensively as unknown variables
or the variables that you're manipulating. But he says, well, if we
think about it this way, if we number these
dimensions-- so let's say that in the x direction,
let's make this right over here is negative 3. Let's make this negative 2. This is negative 1. This is 0. Now, I'm just numbering
the x direction, the left/right direction. Now this is positive 1. This is positive 2. This is positive 3. And we could do the
same in the y direction. So let's see, so this could be,
let's say this is negative 5, negative 4, negative
3, negative-- actually, let me do it a little
bit neater than that. Let me clean this
up a little bit. So let me erase this
and extend this down a little bit so I can go all
the way down to negative 5 without making it
look too messy. So let's go all
the way down here. And so we can number it. This is 1. This is 2. This is 3. And then this could be
negative 1, negative 2. And these are all
just conventions. It could have been
labeled the other way. We could've decided to put
the x there and the y there and make this the
positive direction and make this the
negative direction. But this is just the
convention that people adopted starting with Descartes. Negative 2, negative 3,
negative 4, and negative 5. And he says, well, I
think I can associate each of these pairs of values
with a point in two dimensions. I can take the x-coordinate, I
can take the x value right over here, and I say, OK,
that's a negative 2. That would be right over there
along the left/right direction. I'm going to the left
because it's negative. And that's associated
with negative 5 in the vertical direction. So I say the y
value is negative 5, and so if I go 2 to
the left and 5 down, I get to this point
right over there. So he says, these two values,
negative 2 and negative 5, I can associate it with
this point in this plane right over here, in this
two-dimensional plane. So I'll say that point has
the coordinates, tells me where to find that point,
negative 2, negative 5. And these coordinates are
called Cartesian coordinates, named for Rene
Descartes because he's the guy that came up with these. He's associating, all of a
sudden, these relationships with points on a
coordinate plane. And then he said, well,
OK, let's do another one. There's this other
relationship, where I have when x is
equal to negative 1, y is equal to negative 3. So x is negative
1, y is negative 3. That's that point
right over there. And the convention
is, once again, when you list the
coordinates, you list the x-coordinate,
then the y-coordinate. And that's just what
people decided to do. Negative 1, negative
3, that would be that point right over there. And then you have the point
when x is 0, y is negative 1. When x is 0 right
over here, which means I don't go to the left
or the right, y is negative 1, which means I go 1 down. So that's that point
right over there, 0, negative 1, right over there. And I could keep doing this. When x is 1, y is 1. When x is 2, y is 3. Actually, let me do it in
that same purple color. When x is 2, y is 3, 2 comma 3. And then this one right over
here in orange was 1 comma 1. And this is neat by itself. I essentially just
sampled possible x's. But what he realized
is, not only do you sample these possible
x's, but if you just kept sampling x's, if you
tried sampling all the x's in between, you would actually
end up plotting out a line. So if you were to
do every possible x, you would end up getting a
line that looks something like that right over there. And any relation, if you
pick any x and find any y, it really represents
a point on this line. Or another way to think about
it, any point on this line represents a solution to this
equation right over here. So if you have this
point right over here, which looks like it's about
x is 1 and 1/2, y is 2. So me write that, 1.5 comma 2. That is a solution
to this equation. When x is 1.5, 2 times
1.5 is 3 minus 1 is 2. That is right over there. So all of a sudden,
he was able to bridge this gap or this relationship
between algebra and geometry. We can now visualize
all of the x and y pairs that satisfy this
equation right over here. And so he is responsible
for making this bridge, and that's why the
coordinates that we use to specify these points are
called Cartesian coordinates. And as we'll see, the first
type of equations we will study are equations of
this form over here. And in a traditional
algebra curriculum, they're called linear equations. And you might be saying,
well, OK, this is an equation. I see that this
is equal to that. But what's so linear about them? What makes them
look like a line? And to realize why
they are linear, you have to make this jump
that Rene Descartes made, because if you were to plot
this using Cartesian coordinates on a Euclidean plane,
you will get a line. And in the future,
we'll see that there's other types of equations where
you won't get a line, where you'll get a curve or something
kind of crazy or funky.