CHAPTER 05.04: SYSTEM OF EQUATIONS: Consistent and inconsistent system of equations Example
In this segment, we will look at some examples of inconsistent and consistent systems of equations. So, we know that, hey a particular system of equations is to be consistent if it has a solution. And that solution can either be unique or infinite. And with inconsistent system of equations will be inconsistent if it has no solutions at all. So letís look at this example right here. 2, 4, 1, 3. X, Y is equal to 6 and 4. This set of equations is considered to be a unique solution. How do I know that it is a unique solution? That is something we will talk about in the future segments. So this one has a unique solution and if you are going to solve these equations- 2 unknowns and 2 equations- 2equations 2 unknowns by any method you will get 1 and 1 as a solution. You can also look at, hey, if I substitute 1 and 1 here, I will get 6 and 4. So that is one of the solutions, but it is a unique solution. So this system of equations is considered consistent because it has a solution and that solution we are telling you at this time is unique.
Now if I- weíll look at this particular system of equations like 2, 4, 1, 2, X, Y is equal to 6 and 3. This one has infinite solutions. This one has infinite solutions. So like, for example, I can have 1, 1. 1, 1 is a solution. Another solution for this set of equations is 0, 1.5. And so on and so forth. In fact, you can put 1 and 1- you get 6 and 3- put 0, 1.5 you get 6 and 3. But the number of solutions, which you can find, which look like this, is infinite. So both of these- this system of equations and this system of equations are consistent. Because they have a solution. Any why does this have infinite solutions? If you write this as an equation 2x plus 4y is equal to 6. And you get x plus 2y is equal to 3. You can very well see that this equation is simply a multiple of this equation. That if I take this 2nd equation and I multiply by 2 I get the 1st equation. So if we were going to draw it on a piece of graph paper, you would find that this line is on top of this line. They are not parallel lines. They are on top of each other. Hence you have infinite solutions. So that is just a way of looking at 2 equations with 2 unknowns-then well get to 3 equation, 3 unknowns, 4 equations 4 unknowns. It will be harder to visualize that.
Letís look at another set of equations. Letís suppose I have the same thing- 2, 4, 1, 2, x, y is equal to 6 and 2. And in this case I will have no solutions. There is no solution which exists for this set of equations right here. And people might say, hey, what is different about this set of equations and this set of equations? It is that if I were to expand I would get 2x plus 4y equal to 6. x plus 2y equal to 2. If you want to draw this on a piece of graph paper, you will find 2 parallel lines, which will never intersect with each other, hence there is no solution for this set of equations. Letís suppose this one you have two parallel line but they are on top of each other, so thatís why you have infinite solutions. So these two right here, these are consistent system of equations. And right here you get inconsistent system of equation. And one of things you are finding out here is that for the same coefficient matrix here and here, that you can possibly get inconsistent or no solution, which is inconsistent or you can get infinite solutions, which is also a part of the consistent solution.
So one of the things that I am trying to say here is just be aware of the fact that-here we got two sets of equations with the same coefficient matrix, in one case we are getting infinite solutions and in another case we are getting no solutions. But if I was going to I would not be able to develop a set of equations with this as the coefficient matrix while I get a unique solution. So think about it for a little bit-about why I am saying those things. But it will be much clearer when we talk about how do we figure out if somebody gives a set of equations - not 2 equations and 2 unknowns but a thousand equations and a thousand unknown, letís suppose- how do I make the distinction between it being consistent or inconsistent. And once I have made the determination that it is consistent- how do I know if it has infinite solutions or a unique solution. And that is the end of this segment.