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] = \begin{bmatrix} 1 & 1 &1 &: & 9\\ 2& 5 &7 & : & 52\\ 2& 1 &-1 & : & 0 \end{bmatrix}

 

R2-->R2-2R1

R3-->R3-2R1

\begin{bmatrix} 1 & 1 &1 &: & 9\\ 0& 3 &5 & : & 34\\ 0& -1 &-3 & : & 18 \end{bmatrix}

R2⇔R3

\begin{bmatrix} 1 & 1 &1 &: & 9\\ 0& -1 &-3 & : & 18\\ 0& 3 &5 & : & 34\\ \end{bmatrix}

 

 

R3-->R3+3R2

\begin{bmatrix} 1 & 1 &1 &: & 9\\ 0& -1 &-3 & : & 18\\ 0& 0 &-4 & : & -20\\ \end{bmatrix}

 

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

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