Linear system of equations 
A system of m linear equations in n unknowns has a solution if and only if the
rank r of the augmented matrix equals that of the coefficient matrix.
If the two matrices have the same rank r and r = n, the solution is unique.
If the two matrices have the same rank r and r < n, then at least one set of r
of the unknowns can be solved in terms of the remaining (n r) unknowns.
Coefficient matrix is the matrix which contains the valriables coefficiens.
Constant matrix is the array of the free column values.
Augmented matrix is the combined matrix which contains the Coefficient matrix and Constant matrix side by side.

Cramer's rule 
Another way to solve linear system of equation is by Cramer's rule which
involves only determinants. Consider the system of equations:
The solution is given by the equation: 

D_{i} is the determinant esteblished by changing the ith column with the free values column b_{i}

Example: Solve the system of linear equations of three variables x, y and z:
x y + 2z = 2 
3x 2y + 4z = 5 
2y 3z = 2 

Consider the system of linear equations containing three variables x, y and z.
this equations cab be written in matrix notation as:
One way to solve the equations is by row transformation on the augmented matrix.
From the third line it is obvious that z = 0 
From second line: y 2z = 1 → y = 1 
From first row: x y + 2z = 2 → x = 1 

Solve the above equations by Cramer' s rule. 

Example: Solve the system of the linear equations:
x + 2y 3z = 4 
x + 3y + z = 11 
2x + 5y 4z = 13 
2x + 6y + 2z = 22 

After performing rows operations we get the upper triangular matrix:
It is seen that r = n (rank = number of variables) therfore the equations have a unique solution:
z = 1 y = 3 x = 1 or in vector representation (1, 3, 1)

Example: Solve the system of the linear equations:
x + 2y 2z + 3w = 2 
2x + 4y 3z + 4w = 5 
5x + 10y 8z + 11w = 12 

The result is that r < n (r = 2 n = 4)
r Rank of the matrix total non zero rows 
n Number of variables 
Because r n = 4 2 = 2 two variables are dependent on the other two variables and we have to choose certain values for two variables
for example: w = a and y = b (a, b are any number) then:
The solution space vector is (4 + a 2b, b, 1 + 2a, a)
for example if we choose: a = 1 b = 1 then the solution is: (7, 1, 3, 1).

Example: Solve the system of the linear equations:
2x + y 2z + 3w = 1 
3x + 2y z + 2w = 4 
3x + 3y + 3z 3w = 5 

From the last row it is observed that: 0 = 8 this is clearly impossible,and
this set of equations don't have a solution.
