This presentation summarizes key concepts about systems of linear equations. It defines linear equations as those with powers of unknowns equal to 1. Linear equations can have one or multiple unknown variables. Systems of linear equations are classified as consistent or inconsistent based on whether the augmented matrix and coefficient matrix have the same rank. If the ranks are equal, the system is consistent and has either a unique solution or is redundant. If the ranks are unequal, the system is inconsistent. Real-life applications of solving systems of linear equations include event planning, finding unknown quantities, and comparing costs or nutritional information.

1. Daffodil International University
Presentation on:
System of Linear Equation

2. Name: IMAMUL KADIR
ID: 162-15-7775
Department of CSE

3. Fundamentals
An Equation in which the power of each unknown is one(1) is called a linear
equation. In a course of linear algebra unknown are usually called variables.
Example:
2x - 5 = 0 (linear equation in 1 unknown/variable)
4p – 5q + 6r = -29 (linear equation in 3 unknown/variable)
12a + 23b - 4c – 10 = 2d – 9e (linear equation in 5 unknown/variable)

4. An example with value

5. Classification
By the nature of solutions, system of linear equations are classified as follows:
Systems of Linear Equations
Inconsistent Consistent
Unique Redundant

6. Finding Solutions
Theorem: A system of linear equation is consistent iff its augmented matrix and
coefficient matrix posses the same rank.
Example: If
3a – 4b + 5c = 0
2a + 2b – 3c = 1
7a – c =2
are the linear equation then the augmented matrix of this equation is :
3 -4 5 0 1 -6 8 -1
2 2 -3 1 ………… 0 14 -19 3
7 0 -1 2 0 0 0 0
rk (augmented matrix) = 2 = rk (coefficient matrix)
the SLE is consistent.

7. If the augmented matrix and coefficient matrix posses different rank than it will be
inconsistent.
The procedures used above can be summarized as follows:
Augmented matrix
Echelon form
RREF
Equivalent system

8. The summary of result of that equation can be abridged as follows:
SLE in n variables
rk(aug mat) ≠ rk(coeff mat)
Inconsistent
rk(aug mat) = rk(coeff mat)
Consistent
rk = n
=> Unique
rk < n
=> Redundant

9. Applications in real life
1. Planning an Event – An after-school party.
2. Finding Unknown Quantities – Like objects, body etc.
3. Other Applications - Figuring out which landscaper provides the best deal, how
much different cell phone providers charge per minute, or comparing
nutritional information in recipes.

10. THANK YOU!