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.

non-homogeneous-linear-equation-matrices

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.
    Matrix Inversion Method
  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.

    Cramer's Rule of Solving Non-Homogeneous System
  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.

    Augmented-Matrix of the system AX=B

    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

    Gauss-Jordan-Method-working-rule

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