System of Non-Homogeneous Linear Equations

Non-Homogeneous Linear Equations:-

a₁ x + b₁ y + c₁ z = d₁
a₂ x + b₂ y + c₂ z = d₂
a₃ x + b₃ y + c₃ z = d₃ is called a system of non-zero homogeneous linear equations.

then it can be written as AX = B where A is called co-efficient matrix.

Consistent and Inconsistent System :-

If a system of linear equations has a solution then the system is said to be consistent. Otherwise it is said to be inconsistent system.

Different Methods to Solve Non-Homogeneous System :-

The different methods to solve non-homogeneous system are as follows:

1. Matrix Inversion Method :-
   Suppose the given system is AX = B and A is non-singular then inverse of A exists.
   
2. Cramer’s Rule :- Suppose the given system is
   a₁ x + b₁ y + c₁ z = d₁
   a₂ x + b₂ y + c₂ z = d₂
   a₃ x + b₃ y + c₃ z = d₃ and Δ, Δ₁, Δ₂, Δ₃ are as given below, then by multiplying the equations with A₁, A₂, A₃ and then by adding we get x(a₁A₁ + a₂A₂ + a₃A₃) + y(b₁A₁ + b₂A₂ + b₃A₃) + z(c₁A₁ + c₂A₂ + c₃A₃) = d₁A₁ + d₂A₂ + d₃A₃

   x.Δ + y.0 + z.0 = Δ₁ -> x = Δ₁ / Δ

   Similarly multiplying the equations with B₁, B₂, B₃ and then by adding we get y = Δ₂ / Δ and by multiplying C₁, C₂, C₃ and by adding we get z = Δ₃ / Δ.

   
3. Gauss Jordan Method : For this method elementary row transformations are done. The following are called elementary row transformations:
   1-> Interchange the two rows.
   2-> Multiplication of a row with a non-zero constant K.
   3-> Addition of products of elements of a row and a constant K to the corresponding elements of some other row.
   

   Augmented Matrix :-

   For the non-homogeneous linear system AX = B, the following matrix is called as augmented matrix.

   

   Working Rule :-

   Suppose the given system is
   a₁ x + b₁ y + c₁ z + d₁ = 0
   a₂ x + b₂ y + c₂ z + d₂ = 0
   a₃ x + b₃ y + c₃ z + d₃ = 0
   By applying the elementary row transformation we have to transfer the Augmented matrix into the standard solution form as given below after which we can the solution as x = α | y = β | z = γ