CHAPTER 05.03: SYSTEM OF EQUATIONS: Consistent and inconsistent system of equations Theory



In this segment we’ll talk about what does it mean when somebody says the system of equations is consistent or inconsistent.  So if somebody gives you let’s suppose a set of equations - n equations, n unknowns - and says go ahead and solve it. Now there is possibility in this case is that either this is a consistent system or it is an inconsistent system. So if somebody gives you an equation of unknowns you’ll put them in a matrix form then this particular system of equations can be considered to be a consistent system or it would be an inconsistent system.


So how do we know whether a particular system of equations is consistent or inconsistent?  Consistent means that a solution exists. What that means is that it has a solution; it doesn’t mean that it has only one solution but that if a system of equations does have a solution that is considered to be a consistent system of equations.  Inconsistent system is that no solution exists.  So if that is the case then what do we mean by a solution exists? There are two possibilities when a solution is existing.  The idea of a unique solution or you have infinite solutions.  So there might be solution which is existing but what you may find is that there will be two cases either the system of equations has a unique solution or it has infinite solutions.  Now people will say hey is there a possibility of solutions being finite number, like we only take saying that hey either there is zero solutions, one solution or infinite solutions. Is there a possibility to have five solutions we’ll suppose?  That is not the case for simultaneous linear equations, there is a theorem to prove that and we’ll do that later. 


But at this time, what you got to understand is that if somebody gives you a system of equations and unknowns, three possibilities arise. You’re either going to get a unique solution, or you’re going to get an infinite solution or you’re going to get no solution. Now if the system is consistent and it has a solution, either a unique solution or infinite solutions then it’s called a consistent system of equations. If no solutions exist, then it is called an inconsistent system of equations. We’ll see an example of all three in the next segment. And that’s the end of this segment.