A less cumbersome way to represent a linear system is to write it as an augmented matrix. For example the linear suppose you want to find the solution for x,y,z in ℤ_{5} to the system

It has exactly the same information as the original system but here the columns correspond to the variables and the rows correspond to the equations in the system. Corresponding to the elementary operations, we have row operations.
We can use Gauss elimination in the usual way. Take 3 = −2 times the top equation and add to the second.

Now switch the bottom two rows.

Then take 4 times the bottom row and add to the top two.

Next multiply the second row by 3

Now take 2 times the second row and add to the top.

Therefore, the solution is x = y = z = 3. How do you know when to stop? You certainly should stop doing row operations if you have gotten a matrix in row reduced echelon form described next.
Definition 2.2.2 An augmented matrix is in row reduced echelon form if
Echelon form means that the leading entries of successive rows fall from upper left to lower right.
Example 2.2.3 Here are some matrices which are in row reduced echelon form.

Example 2.2.4 Here are matrices in echelon form which are not in row reduced echelon form but which are in echelon form.

Example 2.2.5 Here are some matrices which are not in echelon form.

The following is the algorithm for obtaining a matrix which is in row reduced echelon form.
This algorithm tells how to start with a matrix and do row operations on it in such a way as to end up with a matrix in row reduced echelon form.
Sometimes there is no solution to a system of equations. When this happens, the system is said to be inconsistent.
Here is another example based on the use of row operations.
Example 2.2.7 Give the complete solution to the system of equations, 3x−y−5z = 9, y−10z = 0, and −2x + y = −6.
The augmented matrix of this system is

After doing row operations, to obtain row reduced echelon form,

The equations corresponding to this reduced echelon form are y = 10z and x = 3 + 5z. Apparently z can equal any number. Lets call this number t. ^{1}Therefore, the solution set of this system is x = 3 + 5t,y = 10t, and z = t where t is completely arbitrary. The system has an infinite set of solutions which are given in the above simple way. This is what it is all about, finding the solutions to the system.
In summary,
Definition 2.2.8 A system of linear equations is a list of equations,

where a_{ij} are numbers, and b_{j} is a number. The above is a system of m equations in the n variables, x_{1},x_{2}

It is desired to find
As illustrated above, such a system of linear equations may have a unique solution, no solution, or infinitely many solutions and these are the only three cases which can occur for any linear system. Furthermore, you do exactly the same things to solve any linear system. You write the augmented matrix and do row operations until you get a simpler system in which it is possible to see the solution, usually obtaining a matrix in echelon or reduced echelon form. All is based on the observation that the row operations do not change the solution set. You can have more equations than variables, fewer equations than variables, etc. It doesn’t matter. You always set up the augmented matrix and go to work on it.
Definition 2.2.9 A system of linear equations is called consistent if there exists a solution. It is called inconsistent if there is no solution.
These are reasonable words to describe the situations of having or not having a solution. If you think of each equation as a condition which must be satisfied by the variables, consistent would mean there is some choice of variables which can satisfy all the conditions. Inconsistent would mean there is no choice of the variables which can satisfy each of the conditions.