A descriptive presentation on System of Linear Equations (Business Mathematics) for Undergrad Students as well as School Students. Good Luck!!

1. Business Mathemetics
system of linear equations

2. System of Linear
Equations
Consistent
System
Unique Solution
Infinitely Many
Solutions
Inconsistent
System
No Solution
Consistent & Inconsistent System

3. Lets assume two linear equations i.e. a1x + b1y + c1 = 0 & a2x + b2y + c2 = 0
These equations can have –
• No Solution 
𝐚 𝟏
𝐚 𝟐
=
𝐛 𝟏
𝐛 𝟐
≠
𝐜 𝟏
𝐜 𝟐
• Unique Solution 
𝐚 𝟏
𝐚 𝟐
≠
𝐛 𝟏
𝐛 𝟐
• Infinitely Many Solutions 
𝐚 𝟏
𝐚 𝟐
=
𝐛 𝟏
𝐛 𝟐
=
𝐜 𝟏
𝐜 𝟐
3
Consistent System - A system of linear equations is said to be consistent if
it processes at least one solution.
Inconsistent System - A system of linear equations which is not consistent
is said to be inconsistent.

4. Homogeneous & Non-Homogeneous System
Lets consider the system of n linear equations in n unknowns –
The system of equations is said to be Homogeneous if B = O.
The system of equations is said to be Non-Homogeneous if B ≠ O.

5. Methods of Solving a
System of Linear
Equations
Matrix Inverse Method Cramer’s Rule

6. Matrix Inverse Method
6
Let’s consider the system of n linear equations in n unknowns –
----- (1)
This system can be represented by single matrix equation -
AX = B ----- (2)
where, A is a coefficient matrix of the system.

7. 7
If A is Non-Singular i.e., |A| ≠0, then A-1 exists
Hence from equation (2), we obtain
X = A-1B -----(3)
which gives the required solution of the system.
Since A-1 is unique, the solution of the system of equations of (3) is also unique.
NOTE: How to calculate A-1
• First of all, calculate determinant of A (i.e. |A|)
• Then calculate adj(A)
So it is calculated by
A-1 =
1
𝐴
𝑎𝑑𝑗 𝐴

8. 8
Criterion For Consistency or Inconsistency -
• If |A| ≠ 0, then system is consistent and has a Unique solution, given by
X= A-1B
• If |A| = 0 & (adj A)B = O, then system is consistent & has infinitely
many solutions.
• If |A| = 0 & (adj A)B ≠ O, then system is inconsistent.

9. 9

10. 10
Applications
of Matrix
Inverse
Method
Equilibrium
Investment
Inter-
Holding &
Profit
Sharing
Cost
Determina
-tion
Production
Sales &
Profit

11. 1. Equilibrium
11

12. 12

13. 2. Investment
13

14. 14

15. 3. Inter-Holding & Profit Sharing
15

16. 16

17. 17

18. 4. Cost Determination
18

19. 19

20. 5. Production
20

21. 21

22. 22

23. 6. Sales & Net Profit
23

24. 24

25. 25
Cramer’s Rule (Determinant Method)
Determinant can also be applied for solving a system of n linear equations in n
variables. The method for solving the system of equations by making use of
determinant was given by mathematician “Gabriel Cramer” and is referred to
as Cramer’s Rule.

26. 26
Steps in solving problem through Cramer’s Rule –
Let three equations be a1x + b1y + c1z = k1, a2x + b2y + c2z = k2 &
a3x + b3y + c3z = k3
Taking as - AX = B, where
A =
a1 b1 c1
a2 b2 c2
a3 b3 c3
, X =
x
y
z
& B =
k1
k2
k3

27. 27
Step 1 - Find |A| = D
Step 2 - Replace 1st column of A matrix with B and then find value of |A| and
mark it as D1.
Step 3 - Repeat the same process with 2nd & 3rd column and mark it as D2
and D3 respectively.
Step 4 - Now we will be able to find values of X,Y & Z and is equal to
X =
𝐷1
𝐷
Y =
𝐷2
𝐷
Z =
𝐷3
𝐷

28. 28

29. 29

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31. 31

32. 32

33. 33

34. Efforts By -
GAUTAM VARSHNEY
Sri Guru Gobind Singh College of Commerce
University of Delhi
34
Thanks!
ANY QUESTIONS?