We've done a lot of work with two
equations and two unknowns.
And where each equation and two variables
represents a line.
So you're just finding the intersection of a line.
But what happens when you start having three variables
in a linear equation?
So for example, let's say I had x plus 4y plus z is equal
to, I don't know.
Let's say it's equal to 8.
So how do we think about this?
How do we think about it visually, first of all?
Let's see if I can graph this.
So I like to draw my x- and y-axes in the traditional
manner, then have the z-axis popping in and out.
You'll see what I mean.
So that's my y-axis.
That is my x-axis.
And now my z-axis-- I can't make it.
Since this is a two-dimensional format, it's
hard to do it, but I'll draw it with a little perspective--
the z-axis, this is the back.
You can imagine this is behind your screen.
And now I'm going in front of our screen, or above our
screen, depending on how you're viewing it.
So that's the z-axis.
That is the x-axis, and that is the y-axis.
So you could almost view this as a traditional x and y.
Now z tells us how-- if this was your piece of paper-- how
much above the piece of paper we are and how much behind the
piece of paper we are.
So we're now dealing in three dimensions.
And now something to think about-- and I'll answer it for
you in about two seconds-- is, what does this represent in
three dimensions?
Does it represent a line in three dimensions?
Or something else?
Well it turns out that it actually
represents something else.
It represents a surface.
It represents-- and since the coefficients are linear and
the-- not the coefficients-- since the variables are
linear, it represents a plane.
It would represent this big curvy surface, if we had
squares in here and all of that.
But we're dealing with three-dimensional linear
equations, so this is just going to be a plane.
And let's see if we can visualize it.
The easiest way to draw this is to figure out the various
intercepts, the various axis intercepts.
So the x-intercept, the y-intercept and now the
z-intercept.
Let's do the z-intercept first. The z-intercept is when
x is equal to 0, y is equal to 0.
What is z?
So if this is 0, this is 0, the z is equal to 8.
So that's this point right here.
z is equal to 8.
Now what is the y-intercept?
That's when x is 0 and z is 0.
So if this is 0 and this is 0, what's y?
4y is equal to 8, so y would be equal to 2.
That would be right here.
That's 0, 2, 0.
0, 2, and then, no z, 0 right there.
And then, finally, let's figure out the x-intercept.
So y is 0 and z is 0, so if this is 0 and this is 0, x is
going to be equal to 8.
So that is this right here.
Now, we could draw a line connecting these.
And this wouldn't be necessarily the graph, but
it'll help you visualize the graph of the plane,
represented by this three variable linear equation.
So let me do it in a really light color, just so it
doesn't-- So up there.
And in most classes your teacher will often view this
as the graph.
But you could kind of view this-- this will be-- Let me
see if I can draw a little bit of, just to
hit the point home.
This is a surface.
So if that's our surface, so it would be the plane that
connects these three points.
And these lines right here, you can kind of view them as
where this plane is intersecting with this-- not
quadrant-- octant.
Because we have four cubic areas in the back, you could
kind of say behind the page.
And we have four quadrants above the page.
You can view it that way.
And so this is kind of the intersection of this plane
with this top, right, up octant.
Hopefully that doesn't confuse you.
But it's important to keep in mind that the
plane just keeps going.
And a plane in 3 dimensions is defined by three points.
So that's why it's useful to draw those three.
But this plane just keeps going in each direction.
And this is the best that we humans can do at graphing a
three variable linear equation on a piece of paper.
I mean, it's a lot easier if you could do it on a computer,
then you could rotate it, and you could see it a lot better.
So what does it mean if you have-- let's say you had two
equations like this.
Let's say you had this equation, and then you had
let's say another equation.
Let's say you had, I don't know, x plus y plus 3z is
equal to, I don't know, is equal to 3.
So that graph, let's see, the x-intercept, would be here.
It's going to be really messy now.
The x-intercept is there at 3.
The y-intercept is at 3.
And then the z-intercept, 3z is equal to 3.
So the z-intercept's going to be 1.
So if I were to attempt to draw that plane-- at least in
this octant that I'm participating in, and this
where it gets messy-- that plane would look like that.
And it goes off in every direction.
And now, if I had these two equations, and I have three
unknowns, what is the intersection
of these two equations?
Or what variables satisfy these two equations?
So I hope you can visualize it.
The intersection between this green plane-- which represents
by this-- and our original plane up here, is
going to be a line.
And that line is going to look something like this.
They're going to intersect right there.
There's going to be a line that is in common between this
green plane and between this blue equation,
or that blue plane.
And that might be really hard to visualize, so don't worry
about it too much if you find that confusing.
But this is the best that I can do.
So as we can see, if we have two equations and three
unknowns, the solution is a line.
And a line has an infinite number of points on it, so it
still doesn't help us much.
But it turns out, if we were to throw in a third plane in
there, or a third equation.
And I don't know, maybe it's minus x minus y minus z is
equal to 0.
That's going to be something else.
It's going to be harder to draw.
But now, this plane is going to intersect this line.
Assuming it's not parallel to one of the other two planes,
or it's not equal to one of the other two planes, this
plane is going to intersect the line that is the
intersection of this and this plane, exactly one point.
So that's just a long winded way of saying that three
planes that aren't the same plane, and that aren't
parallel to each other, will intersect
in exactly one point.
Or another way to put it, if you want to solve for you a
unique x, y and z, your constraints have to be three
variables, and three unknowns-- sorry-- three
equations with three unknowns.
So let's do one of them.
Let's see.
Image.
Actually, we're already pushing eight minutes.
This video will be about visualizing.
And then the next video, I'm going to actually solve one.
I'll see you in the next video.