System of linear non homogeneous equations

Non Homogeneous Equations

a1x+b1y=c1

a2x+b2y=c2

If there is n equation & n unknown you will not find the unique solution always.

eg. x+y=1

x+y=10 there is no solution for these equation.

Working rule for finding the solution of linear non homogeneous equation (AX=B)

1. Suppose the coefficient matrix 'A' is of type M*N
2. Write the augmented matrix [AB] and reduce to echlon form by applying only row transformations.
3. This echelon form will enable us to know the ranks of the augmented matrix [AB] and the coefficient matrix A. Then the following different cases arise

Case1:  Rank A < Rank [AB] ⇒AX=B is inconsistant   ⇒ No solution

Case2: Rank A = Rank [AB]=r(say)

a)  r=n  ⇒ unique solution

b) r<n   ⇒ n-r linearly independnt solution

⇒n-r variable will be assigned random values

⇒infinite no of  solutions

Example:

x+y+z=9

2x+5y+7z=52

2x+y-z=0

Solution:

[AB] =

R2-->R2-2R1

R3-->R3-2R1

R2⇔R3

R3-->R3+3R2

r(A)=r(AB)=3

n=3

n=r

therefor ,unique solution

x+y+z=9

-y-3z=-18

-4z=-20

z=5 , y=3, x=1