NON-HOMOGENEOUS LINEAR ALGEBRAIC EQUATIONS
Non-Homogeneous Systems of Linear
Algebraic Equations
A non-homogeneous system of linear
equations with an 
coefficient matrix consists of 
linear equations and 
unknown variables, represented as:
Matrix Representation
This system can be compactly expressed in
matrix form:
where:
(Coefficient Matrix, ):
(Vector of Unknowns, 
):
(Constant Vector, 
):
Standard Form
where 
is an 
matrix, is the vector of unknowns, and 
is a non-zero constant vector.
Existence and Types of Solutions
Let 
be the augmented matrix, 
be the rank of the coefficient matrix, and 
be the number of variables:
- Inconsistent
(No Solution):
- Consistent
with a Unique Solution:
- Consistent
with Infinitely Many Solutions:
Problems:
1. solve
the system of linear equations:
sol: Given
linear system of non-homogeneous equation 
Step 1: Matrix Form &
Augmented Matrix
Augmented matrix 
: 
Step 2: finding the Rank
of the matrix
Swap Row 1 and Row 3
(
) to make the leading
coefficient 1:
Eliminate 
from Row 2 and Row 3:
,
Multiply Row 2 and Row 3
by 
(
, 
):
Eliminate 
from Row 3:
Therefore, the =3
Then the System is consistent and system has unique solution.
Step 3: Back-Substitution
Solve for 
using Row 3: 
Solve for using Row 2: 
Solve for using Row 1: 
2. solve the system of linear equations
sol: Given
linear system of non-homogeneous equation 
Step 1: Set Up the
Augmented Matrix
Step 2: finding the
rank of the matrix:
Eliminate from Row 2 and Row 3:
,
Eliminate from Row 3:
Therefore,
,
then the system is
inconsistent and system has no solution
-------------------------------------------------------------------------------------------------------------
3. solve
the system of linear equations:
Sol: Given linear
system of non-homogeneous equation
Step 1: Set Up the
Augmented Matrix
Step 2: Finding the rank
of the matrices (Row Operations)
Eliminate from Row 2 and Row 3:
, 
Eliminate from Row 3: 
Since,
Then the system is
consistent and system has infinitely many solutions.
Step 3: Parametric
Solution:
Set as a free parameter (
, where 
).
Solve for using Row 2: 
Solve for using Row 1: 
The system has infinitely
many solutions parameterized by :
4. Investigate
for what values of λ and μ the equation
x+y+z=6, x+2y+3z=10, x+2y+λz=μ
then
1. infinite number of solutions
2.unique
solution,
3.
no solution.
Sol:The system of equations is:
Step 1: Set up the
Augmented Matrix
Step 2: Perform Row
Operations
Perform and :
Now perform :
1. Infinite Number of
Solutions
For the system to have
infinitely many solutions, the bottom row must consist entirely of zeros, so 
:[ Infinite solutions:
]
2. Unique Solution
For a unique solution,
the third entry in the main diagonal of matrix must be non-zero, making 
:[ Unique solution:
3. No Solution
For the system to be
inconsistent (no solution), the coefficient part of the bottom row must be zero
while the augmented term is non-zero, giving 
and 
:[No solution: 
Problems:
1.solve 
2.solve 
3.
4.solve 
5.solve 
6. solve 5x+3y+7z=4,3x+26y+2z=9,7x+2y+10z=5 [x=
,y=
,z=k]
7.solve 
8. Investigate for what values of λ and
μ the equation 2x+3y+5z=9, 7x+3y-2z=8, 2x+3y+λz=μ then 1.
infinite number of solutions 2. unique solution, 3. no solution.
[ hint: λ=5 and μ=9]
9. Investigate
for what values of a b the equation x + 2y + 3z = 4, x + 3y + 4 = 5, x + 3y + az
= b has 1.no solution, 2. a unique solution and 3. an infinite number of
solutions [hint: a=4, b=5]
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