Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far, we have studied systems of linear equations
which have single unique solutions
infinitely many solutions
or no solutions.
And we have seen that when those systems are represented as augmented matrices
it is easy to tell which type of solutions the system has
once the matrix has been transformed to reduced row echelon form
through the process of Gauss-Jordan elimination.
In three-variable systems we have also seen how the planes
which graphically represent those equations change during that process
when the system has a single unique solution
or infinitely many solutions.
But we have not yet seen how the planes change when a system has no solutions.
Systems of linear equations with no solutions are called "inconsistent"
since no variable values exist
that are consistent with the solutions for every equation in the system.
In the previous lecture, we saw that when a matrix representing an inconsistent system
is reduced using Gauss-Jordan elimination
at some point during the process, a row will be produced with all zero coefficient entries
and a non-zero constant entry.
In this lecture, we will examine different ways
that a system of linear equations in three variables can be inconsistent
and how the planes which graphically represent this system
change during the process of Gauss-Jordan elimination.
We have seen that any system whose graph contains two distinct parallel planes will be inconsistent
since distinct parallel planes have no points in common
and therefore the equations which those planes represent can have no common solutions.
Regardless of the orientation of the other planes in the system
no point can exist which simultaneously lies on all planes.
Another configuration which produces an inconsistent system
is a system containing a group of three planes which are not parallel
but whose intersections lie along three parallel lines.
Since in this configuration, no point can simultaneously lie on all three planes
the system has no solutions and is therefore "inconsistent".
We will refer to this configuration of planes as a "triangular tube".
No matter how many equations and planes are included in a system
as long as any two of the planes are parallel
or any three planes form a triangular tube
it is impossible for a point to simultaneously lie on every plane in the system
and so the system will have no solutions.
So let's see what happens to these parallel planes and triangular tubes
when Gauss-Jordan elimination reduces an inconsistent system in three variables.
Before we start, it is useful to know that equations representing parallel planes
can often be spotted without having to go through the process of Gauss-Jordan elimination.
If this is the case, we can save a lot of time
since we know that the system has no solutions.
As an example, let's take a very simple system of two parallel planes.
Notice that both equations have the same "x" coefficient of "one"
but the constants in the equations are different.
Now let's make the equations a bit more complicated by adding a "y" term.
These equations still describe parallel planes
and just as before, the variable coefficients are the same in both equations
while the constants are different.
In fact, any two equations whose coefficients are the same but whose constants are different
will describe parallel planes.
Now let's change the second equation by multiplying it by 2.
Since multiplying both sides of an equation by a non-zero constant
doesn't change its solutions
the two equations still describe the same planes.
This might make the situation a little less obvious.
However, if we notice that the coefficients in the second equation
are all twice the value of the coefficients in the first equation
but the constants are not
we could still recognize that the equations describe parallel planes.
So when the coefficients of two equations in a system are the same multiple of each other
and the constants are not
this tells us that the system is inconsistent.
Likewise, if we also multiply the first equation by 3
the coefficients in the second equation are all 2/3 of the coefficients in the first equation
and the constants are not
so once again, this indicates an inconsistent system
but now this relationship between the two equations is even less obvious.
Fortunately, even if we don't notice
that two equations in the system describe parallel planes
Gauss-Jordan elimination will take care of this for us.
So let's see what happens when we represent the system in matrix form
and use Gauss-Jordan elimination to convert the system to reduced row echelon form.
We start by changing the leading entry of the first row to a "one"
using a scale operation to multiply row one by 1/6.
We then change the entry below that "one" to a zero
using a pivot operation adding negative 4 times row one
to row two.
This changes all the coefficient entries in row two to zero
but not the constant entry.
At this point we can stop, since we know that the system has no solutions.
Or if we like, we can continue the process of Gauss-Jordan elimination
until the matrix is in reduced row echelon form.
To do this, we must change the leading entry of row two to a "one"
using a scale operation to multiply row two by negative 1/40.
And since each leading entry must be the only non-zero entry in its column
the entry above that leading entry must be zeroed
using a pivot to add negative 6 times row two
to row one.
Since every row's leading entry is a "one"
positioned to the right of the leading entry above
and every leading entry is the only non-zero entry in its column
the matrix is now in reduced row echelon form.
So if a system contains parallel planes
during the process of Gauss-Jordan elimination
a row of all zero coefficient entries with a non-zero constant entry will be produced
indicating that the system is inconsistent.
But as we have seen, parallel planes are not the only configuration
which can cause a system to be inconsistent.
Three planes forming a "triangular tube" will also create an inconsistent system.
So how do we recognize this type of system?
Unfortunately, a system of equations whose graph contains a triangular tube
cannot typically be spotted by simply looking at the equations.
If this system represented in matrix form is reduced through Gauss-Jordan elimination
what sort of indication of inconsistency will be produced?
To find out, let's reduce this matrix.
We start by changing the first row's leading entry to a "one"
using a scale operation to multiply row one by 1/4.
We then change the entry below the "one" to a zero
using a pivot operation to add negative 12 times row one
to row two.
The bottom entry in that column is then changed to a zero
using another pivot operation to add negative 4 times row one
to row three.
We then change the leading entry of row two to a "one"
using a scale operation to multiply row two by 1/16.
And the entry below that is then zeroed
using another pivot operation adding negative 16 times row two
to row three.
Row three now contains all zero coefficient entries
and a non-zero constant entry
indicating that this system is inconsistent.
At this point we can stop, since we know that the system has no solutions
or we can continue until the matrix is in reduced row echelon form.
To do this we change the leading entry of row three to a "one"
using a scale operation to multiply row three by negative 1/32.
The leading entries of each row are now ones
but to be in reduced row echelon form
each leading entry must be the only non-zero entry in its column.
We therefore must zero the entries above the leading entry of row three
and row two.
We start by zeroing the entry above the leading entry of row three
using a pivot to add negative 5/2 times row three to row two
and zero the entry above that
using a pivot to add 3/2 times row three to row one.
Then we zero the entry above the leading entry of row two
using a pivot to add row two
to row one.
Since every row's leading entry is a "one"
positioned to the right of the leading entry above
and every leading entry is the only non-zero entry in its column
the matrix is now in reduced row echelon form.
So just as in the previous example
Gauss-Jordan elimination produced a matrix row containing all zero coefficient entries
with a non-zero constant entry.
But considering that this system
contained no parallel planes
how did we end up with the same type of result
as the system which contained parallel planes?
Surprisingly, during the process of Gauss-Jordan elimination
two of the planes in the system became parallel.
So why did the planes described by rows two and three become parallel?
When Gauss-Jordan elimination reduces a matrix
representing equations which describe a triangular tube
during the reduction process
coefficients of the same variable will be zeroed in two of the equations
causing the corresponding planes to become parallel
to the same coordinate axis.
In this example, the zeroed coefficient entries were in the "x" column
so the corresponding planes each rotated around their parallel lines of intersection with plane one
becoming parallel to the x-axis
and therefore, parallel to each other.
Then, just as in any system containing parallel planes
a pivot operation acting on rows corresponding to the parallel planes
produces a row of all zero coefficient entries
and a non-zero constant entry
indicating that the system is inconsistent.
So far we have seen two configurations of planes
that will produce an inconsistent system of three linear equations in three variables.
However, in three-variable systems with more than three equations
there are additional configurations that produce inconsistent systems
which don't contain parallel planes or triangular tubes.
But in all cases, at some point during the process of Gauss-Jordan elimination
an equivalent system with parallel planes will always be produced
resulting in an indication of inconsistency.
One quick example will show how this happens.
Consider a system containing the equations "x equals zero"
"y equals zero"
and "z equals zero".
This system has a single unique solution
corresponding to the common point at the origin shared by all three planes.
Now let's add a fourth equation "x plus y plus z equals three".
Since this fourth plane does not include the single point shared by the other three planes
there is no point common to all four planes.
This system is therefore inconsistent
even though it contains no parallel planes or triangular tubes.
Now let's see what happens when we use Gauss-Jordan elimination to reduce this system.
Since the leading entry of the first row is already a "one"
we only need to zero the other non-zero entry in that column.
We do this using a pivot to add negative one times row one
to row four.
Since the x-coefficient of row four is now zero
this causes plane four to become parallel to the x-axis.
The leading entry of the second row is also already a "one"
so we only have to zero the other non-zero entry in that column
adding negative one times row two
to row four.
Since the y-coefficient of row four is now zero
this causes plane four to also become parallel to the y-axis.
Since plane four is now parallel to the x and y axes
it is has become horizontal so planes three and four are now parallel.
Continuing with Gauss-Jordan elimination
we see that the leading entry of the third row is already a "one"
so we zero the other non-zero entry in that column
adding negative one times row three
to row four.
This gives us the result that is always produced
when a pivot acts on two matrix rows corresponding to parallel planes
a row containing all zero coefficient entries and a non-zero constant entry
indicating an inconsistent system.
In the last several lectures, we have seen how systems of linear equations can be solved
using matrices and the process of Gauss-Jordan elimination.
This technique is especially useful in systems of more than three variables
where the substitution or elimination methods can become cumbersome.
But what type of actual applications
can be solved by systems of linear equations with more than three variables?
In the next several lectures, we will take a look at some real-world applications
which can be modeled by systems of linear equations with four or more variables.