Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far we have seen how substitution and elimination
can be used to find a solution
for a system of three linear equations in three variables.
But how do we know that this solution is unique?
Perhaps these three planes intersect in such a way
that there are an infinite number of solutions
and this solution was only one of them.
In this lecture, we will show how to recognize
whether a system of equations has one solution, no solutions, or an infinite number of solutions
by the type of results we get when solving the system.
We will start by solving a system composed of three planes
which intersect at a single point.
We will then change one of the planes so that all three planes intersect along a line
and see how the results differ.
Finally, we will reorient the plane again to create a system with no solutions
and see what happens when we try to solve the system.
We will start with a system of three equations
whose graph is composed of three planes intersecting at a single point.
We saw in the previous lecture that systems like this can be solved
using the same basic techniques we use to solve systems of two equations in two variables.
We start by choosing two of the three equations and eliminate a variable
giving us an equation in two variables.
For example, in this system let's eliminate the variable z using the first two equations.
Multiplying the first equation by two
and adding the equations
gives us "15x - 15y = 0"
or dividing both sides of the equation by fifteen, we get "x - y = 0".
We then repeat this process with a different pair of equations
giving us a second equation in the same two variables.
This time we'll use the first and third equations and once again eliminate z.
Multiplying the first equation by three
and adding the equations
gives us "19x - 8y = 0".
We now have a system of two equations in two variables
which can be solved to find the values of x and y.
Once again, we will use the elimination method
and multiply the top equation by negative eight in order to eliminate the y terms.
Now when we add the equations
we get "11x = 0"
or dividing both sides by eleven
"x = 0".
Now that we know the value of x
we can set x to zero in either of the two equations we just added and solve for y.
Setting x to zero in the first equation
gives us "8y = 0"
or dividing both sides by eight
"y = 0".
Knowing the values of x and y
we can now set the values of x and y to zero in any of the three original equations
and find the value of z.
Setting x and y to zero in the first equation
causes the first two terms to vanish
giving us "negative z equals three"
or "z equals negative three".
So the x, y, and z values which are the solution to our system of equations
are zero, zero, and negative three.
Since each variable was forced to a single value, we know that the solution is unique.
This solution can be written as the ordered triple (0,0,-3).
These are the coordinates of the intersection
of the three planes corresponding to this system of equations.
And this point graphically represents the single unique solution of the system.
But what will happen if the system we are solving has an infinite number of solutions?
To find out, let's modify the third equation so that its plane is reoriented
causing the intersection of the three planes to form a line.
The infinite number of points on this line
correspond to the infinite number of solutions of this system.
So let's see what happens when we solve this system of equations.
Since the first and second equations are the same as the last example
using these equations to eliminate z produces the same equation in x and y as before.
Now using the first and third equations to produce another equation in x and y
we multiply the second equation by negative one before adding to eliminate the z term.
Adding the equations
we get "4x - 4y = 0"
or dividing by four "x - y = 0".
We can then try to solve the system of two equations in two variables
to find the values of x and y.
But when we multiply the second equation by negative one
and add the equations
both the x and y terms are eliminated, resulting in the equation "0 = 0".
As a general rule
if we start with a consistent system of three linear equations in three variables
and by eliminating variables, we obtain an equality between two equal numbers
then this indicates that the system has an infinite number of solutions.
You may recall from previous chapters that obtaining an equality between two equal numbers
in systems of two linear equations in two variables
also indicates an infinite number of solutions.
So what if the system we are trying to solve is inconsistent, having no solutions?
To create a system with no solutions, let's modify the third equation
so its plane is oriented in a way that creates no points common to all three planes.
Now let's see what happens when we try to solve this system of equations.
Since the first and second equations are the same as in the last two examples
using these equations to eliminate z produces the same equation in x and y as before.
Now using the first and third equations to produce another equation in x and y
we multiply the first equation by negative one before adding
so that we can eliminate z.
Now adding the equations
we get "-4x + 4y = -4"
or dividing both sides by four "-x + y = -1".
We can then try to solve this system of two equations in two variables
to find the values of x and y.
But when we add these equations both x and y are eliminated
resulting in the equation "0 = -1".
If we start with a system of three linear equations in three variables
and by eliminating variables, we obtain an equality between two unequal numbers
such as "0 = -1"
then this indicates that the system has no solutions.
As you may recall, obtaining an equality between two unequal numbers
also indicates the same thing in a system of two linear equations in two variables.
And we saw earlier that obtaining an equality between two equal numbers
indicates an infinite number of solutions.
However, in systems of three linear equations in three variables
it is possible to obtain both results.
In other words, we can obtain an equality between two unequal numbers
and an equality between two equal numbers in the same system.
This can happen when two of the equations describe parallel planes
and the third equation describes a plane which is identical to one of the other planes.
During the process of eliminating variables
the pair of equations which describe the parallel planes
will produce an equality between two unequal numbers, indicating no solutions
and the pair of equations describing identical planes
will produce an equality between two equal numbers, indicating infinitely many solutions.
However, an indication of no solutions
always trumps an indication of infinitely many solutions
since there can be no solutions for any system which contains parallel planes.
Therefore, any time we get an equality between two unequal numbers
we know the system has no solutions.
In the next lecture, we will see how to describe solution sets
which contain infinitely many solutions.