Many problems in real-life situations can be expressed in form of simultaneous linear equations. A system of `m` linear equations in `n` variables  `x_1,x_2,* * * * x_n `  can be represented by:

`a_11x_1+a_12x_2+* * * +a_(1n)x_n=b_1`

`a_21x_1+a_22x_2+* * * +a_(2n)x_n=b_2`

`a_31x_1+a_32x_2+* * * +a_(3n)x_n=b_3`

`vdots`  ` vdots`  ` vdots`  ` vdots`

`a_(m1)x_1+a_(m2)x_2+* * * +a_(mn)x_n=b_m`

 The above system of equations can be written in matrix notation as `AX = B` where `A` is a `mxxn` matrix and `x and b` are column vectors given by:

`A=[[a_11,a_12,* * *,a_(1n)], [a_21,a_22,* * *,a_(2n)], [vdots,vdots,ddots,vdots], [a_(m1),a_(m2),* * *,a_(mn)]]` `X=[[x_1],[x_2],[vdots],[x_n]], B=[[b_1],[b_2],[vdots],[b_n]]`

 The matrix `A` is called the coefficient matrix. And the following matrix is called the augmented matrix of the system.

`A=[[a_11,a_12,* * *,a_(1n),|,b_1], [a_21,a_22,* * *,a_(2n),|,b_2], [vdots,vdots,ddots,vdots,|,vdots], [a_(m1),a_(m2),* * *,a_(mn),|,b_m]]`

 There are many methods such as Cramer's rule to solve such a system. But such methods become too cumbersome for large systems. To solve such large systems we employ numerical methods. There are two classes of numerical methods to solve such systems, viz. direct methods and iterative methods. Direct methods include Gauss Elimination method, Gauss-Jordan Elimination method etc. and iterative methods include Jacobi method and Gauss-Siedel Iteration method among others.