__Non-Homogeneous Linear Equations__:-

a_{1} x + b_{1} y + c_{1} z = d_{1}

a_{2} x + b_{2} y + c_{2} z = d_{2}

a_{3} x + b_{3} y + c_{3} z = d_{3} 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:

**Matrix Inversion Method :-**

Suppose the given system is AX = B and A is non-singular then inverse of A exists.**Cramer’s Rule :-**Suppose the given system is

a_{1}x + b_{1}y + c_{1}z = d_{1}

a_{2}x + b_{2}y + c_{2}z = d_{2}

a_{3}x + b_{3}y + c_{3}z = d_{3}and Δ, Δ_{1}, Δ_{2}, Δ_{3}are as given below, then by multiplying the equations with A_{1}, A_{2}, A_{3}and then by adding we get x(a_{1}A_{1}+ a_{2}A_{2}+ a_{3}A_{3}) + y(b_{1}A_{1}+ b_{2}A_{2}+ b_{3}A_{3}) + z(c_{1}A_{1}+ c_{2}A_{2}+ c_{3}A_{3}) = d_{1}A_{1}+ d_{2}A_{2}+ d_{3}A_{3}x.Δ + y.0 + z.0 = Δ

_{1}-> x = Δ_{1}/ ΔSimilarly multiplying the equations with B

_{1}, B_{2}, B_{3}and then by adding we get y = Δ_{2}/ Δ and by multiplying C_{1}, C_{2}, C_{3}and by adding we get z = Δ_{3}/ Δ.**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

a_{1}x + b_{1}y + c_{1}z + d_{1}= 0

a_{2}x + b_{2}y + c_{2}z + d_{2}= 0

a_{3}x + b_{3}y + c_{3}z + d_{3}= 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