Hello. I'm Professor Von Schmohawk and welcome to Why U.
When we first introduced systems of linear equations
we started by studying simple examples
involving systems of two equations in two variables.
We then introduced the concepts of dependent and independent systems.
We showed that an independent system of two equations in two variables
always has a graph consisting of two distinct lines.
Those lines can intersect at a single point, giving the system a single unique solution
or the lines can be parallel, and in this case the system will have no solutions.
We also showed that if a system of two equations is dependent
it will consist of two equations that describe the same line.
In this case, since the two graphs intersect at every point
the system will have infinitely many solutions.
Additionally, since the two equations describe the same line
the equations must be either identical or multiples of each other.
This makes it easy to tell if a system of two equations is dependent.
If the equations are multiples of each other
then the system is dependent and must have infinitely many solutions.
This is true of any dependent system of two equations
regardless of the number of variables.
However, in systems of more than two equations
there are ways that the equations can be dependent
without being multiples of each other.
Take for example, this dependent system of three linear equations in three variables.
No equations in this system are multiples of each other
yet they are all still dependent.
So what property makes equations dependent if they're not multiples of each other?
Mathematically speaking, in a group of equations
a dependent equation is "a linear combination of one or more other equations in the group".
So what exactly is a "linear combination" of equations?
A linear combination of equations is produced
by first multiplying each equation by a constant
and then adding the results.
For example, this equation
is a linear combination of the two equations on the right
since if we multiply the top equation by 3
and the bottom equation by 2
and then add the results
we get the equation on the left.
Since this equation was produced
by multiplying the other equations by constants and then adding the results
it is a "linear combination" of those two equations.
Of course, different linear combinations of those two equations could be created
by choosing different multipliers.
For example, instead of multiplying the top equation by 3 and the bottom equation by 2
if we multiply the top equation by 4 and the bottom equation by 1
and then add the results
we get a different equation, which is another linear combination of the same two equations.
Linear combinations can be made by adding multiples of any number of equations.
For instance, we could add a multiple of negative one times a third equation
producing a linear combination of three equations.
So let’s take a look at a dependent system of three linear equations in two variables.
The three equations in this system are all dependent.
Each equation can be derived by multiplying the other two equations by constants
and then adding the results.
For instance, adding one times the equation "x = 2"
to one times the equation "y = 3"
produces the equation "x + y = 5".
Likewise, adding one times the equation "x + y = 5"
to negative one times the equation "y = 3".
produces the equation "x = 2".
And adding one times the equation "x + y = 5"
to negative one times the equation "x = 2"
produces the equation "y = 3".
Since each one of these three equations is a "linear combination" of the other two
all three equations are dependent.
Notice that even though this system is dependent
it has a single unique solution
corresponding to the common intersection point of the graphs of the three equations.
This shows how it is possible for a system with more than two equations to be dependent
without necessarily having infinitely many solutions.
Let's take a look at a few more examples.
As we saw, this system of two equations is dependent, with infinitely many solutions.
But let's say that this system contained a third equation
whose graph intersects the graph of the other two equations at a single point.
This system is still dependent, since it contains a group of two dependent equations
but now, instead of an infinite number of solutions, it has a single unique solution
since there is only one point which satisfies all the equations in the system.
It is also possible for a dependent system of more than two equations to have no solutions.
For example, if the graph of the third equation
was parallel and distinct from the graphs of the other equations
this system would still be dependent
since it contains a group of two dependent equations
but since no solution can satisfy all the equations in the system
it is also inconsistent, having no solutions.
Likewise, dependent systems of more than two equations with three or more variables
may or may not have infinitely many solutions.
For example, this dependent system is made up of a group of three dependent equations
and the system has infinitely many solutions
represented by the intersection line of the three planes.
But let's say a fourth independent equation was added
whose graph intersects the other planes' line of intersection at a single point.
Then although this system contains a group of three dependent equations
and is therefore a dependent system
it has a single unique solution
since only one point exists which satisfies all the equations in the system.
Likewise, if the fourth plane was distinct and parallel to one of the other planes
then this system would still be dependent but inconsistent
since no solution exists which can satisfy all the equations.
So although dependent systems can have infinitely many solutions
it is also possible for dependent systems with more than two variables
to have a single unique solution
or no solutions.
Saying that a system is dependent
simply means that some of the equations in the system are dependent.
And saying that a system is independent
simply means that none of the equations in the system are dependent.
So how do we know whether or not a system of equations is dependent?
It is usually easy to spot when two equations in the system are multiples of each other
but it is not always easy to tell when equations are linear combinations
of two or more other equations.
Fortunately, the process of Gauss-Jordan elimination automatically does this work for us.
In the next lecture, we will see how this happens.