Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last lecture, we saw that just as the graph of a linear equation in two variables
creates a line which extends infinitely in both directions
the graph of a linear equation in three variables
creates a plane which extends infinitely in all directions
Since the graph of a linear equation in three variables is a plane
a system of three linear equations in three variables
can be represented as a group of three planes.
Each plane may be oriented at any angle
and positioned anywhere in space relative to the other planes.
Since each plane represents the set of solutions to one of the three equations
any point which simultaneously falls on all three planes
corresponds to a solution which simultaneously satisfies all three equations.
That point will therefore be a solution of the system.
In this lecture, we will look at different ways
in which three planes can be oriented in 3-dimensional space
and different types of solutions which can exist for a system
of three linear equations in three variables.
We saw that systems of two linear equations in two variables
can have three types of solution sets
depending upon the ways in which the lines are oriented in the Cartesian plane.
If the lines have different slopes
then they must intersect at a single point
and the system will have a single unique solution.
If the lines are distinct and parallel, then there are no points of intersection
and the system has no solutions.
And if the lines are identical, the two graphs coincide
intersecting at an infinite number of points.
The system therefore has an infinite number of solutions.
Likewise, systems of three linear equations in three variables
can have a single unique solution
no solutions
or an infinite number of solutions
depending on the ways in which the three planes are oriented.
Let's consider all the possible ways in which three planes can intersect or not intersect.
We will number these planes 1, 2, and 3.
One possibility is that no two planes in the system are parallel
and all three planes intersect at only one point.
In this case, the system will have one unique solution.
A second possibility is that two of the planes are not parallel
and so their points of intersection form a line.
If the third plane intersects the other two along this same line
then the common points of intersection for all three planes are represented by that line.
The system will therefore have an infinite number of solutions
corresponding to every point on the line.
The third plane does not necessarily have to be distinct from both of the other planes.
If the third plane is identical to one of the other two planes
then the points common to all three planes
are still represented by the same line of intersection
and the system will have an infinite number of solutions.
An infinite number of solutions will also exist if a system consists of three identical planes.
In this case, any point which lies on one plane is common to all three planes.
The system will therefore have an infinite number of solutions
corresponding to every point on the plane.
In addition, there are several orientations of the planes
which will result in a system with no solutions.
Any time a system has two distinct parallel planes, there can be no solution.
Since distinct parallel planes have no points in common
regardless of the orientation of the third plane
there can be no points which lie on all three planes.
The third plane can be distinct and parallel to the other two
identical to one of the other two
or not parallel to either of the other planes
thus intersecting both planes.
Regardless of the orientation of the third plane
since there are no points in common to all three planes, the system has no solutions.
One additional configuration of the planes which will result in a system with no solutions
is if the planes are oriented so that their intersection points
lie along three distinct parallel lines.
Once again, since there are no points in common to all three planes, the system has no solutions.
So we have seen eight ways in which the three planes
in a system of three linear equations in three variables can be oriented.
These orientations can create a system with one solution
an infinite number of solutions
or no solutions.
These are the same three general categories of solutions which can exist
for systems of two linear equations in two variables.
Systems in two variables have a single unique solution
when the two lines in the system are not parallel and thus intersect at a single point.
Likewise, systems in three variables have a unique solution
when none of the three planes are parallel and intersect at a single point.
Systems in two variables have no solutions when the two lines are parallel and distinct.
And systems in three variables have no solutions when any two planes are parallel and distinct
or when the planes are oriented so that their intersection points
lie along three distinct parallel lines.
An infinite number of solutions exist for systems in two variables
when the two lines coincide, thus intersecting at an infinite number of points.
And systems in three variables have an infinite number of solutions
when all three planes either coincide
or intersect along a line.
Regardless of the number of variables or equations in a system
if the system has one or more solutions, it is called "consistent"
and if the system has no solutions it is "inconsistent".
Likewise, two or more equations whose graphs are identical, are called "dependent"
and equations with distinct graphs are "independent".
Now that we have seen how systems of three linear equations in three variables
can be visualized
in the next lecture, we will see how these systems can be solved.