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Characteristics of a Consistent System of Equations

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Submitted By aliciaortizp
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Characteristics of a Consistent System of Equations (determined): * For a given system, there exists one solution set for the different variables in the system or infinitely many sets of solution, so as long as we can find a solution for the system of equations, we refer to that system as being consistent. * For a two variable system of equations to be consistent the lines formed by the equations have to meet at some point or they have to be parallel. * For a three variable system of equations to be consistent, the equations formed by the equations must meet two conditions: 1. All three planes have to parallel 2. Any two of the planes have to be parallel and the third must meet one of the planes at some point and the other at another point.
For example; solve the system of equations below:

Using matrix method:

In the last row of the above augmented matrix, we have ended up with all zeros on both sides of the equations. This means that two of the planes formed by the equations in the system of equations are parallel, and thus the system of equations is said to have an infinite set of solutions. We solve for any of the set by assigning one variable in the remaining two equations and then solving for the other two.
For example, if we take y =3
Then:

Then using the first row equation, we solve for x

Characteristics of an Inconsistent System of Equations (underdetermined and overdetermined): * For a given set of variables, there is no set of solutions for the system of equations. * Arise when the lines or planes formed from the systems of equations don't meet at any point and are not parallel (all of them or only two and the third meets one of the planes at some point.)
For example:

Using matrix method we can solve the above as follows:

Reducing the above to Row Echelon form can be done as follows:

Adding row 2 to row 1:

The equation formed from the second row of the matrix is given as

Which means that:

But we know that the above is mathematically impossible. When we come across the above, we say that the system of equations has NO SOLUTION. Thus we refer to such systems as being inconsistent because they don't make any mathematical sense.

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