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Bussiness Mathematics
Shanza Ashraf (29960
Pazerish Hareem (29952)
Ma'am Sumaira Nadeem
Gauss Elimination Method
Introduction
 Gaussian Elimination Method is another method to find the solution of a system. It is
performed on an augmented matrix and we use row operations to find the solution of
a specific system.
 Gaussian elimination method attempts to transform the system into diagonal form.
The transformation should occur column by column moving from left to right.

𝑎1 𝑏1 𝑐1
𝑎2 𝑏2 𝑐2
𝑎3 𝑏3 𝑐3
𝑑1
𝑑2
𝑑3

1 0 0
0 1 0
0 0 1
𝑣1
𝑣2
𝑣3
Original system
Transformed system
 The operations in the Gauss elimination are called elementary operations.
 Elementary operations for rows are:
1. Interchange of two rows.
2. Multiplication of a row by a nonzero constant.
3. Addition of a constant multiple of one row to another row.
 Two matrices are said to be row equivalent if one matrix can be obtained from the
other using elementary row operations
Gauss elimination for solving a system of equations
 Write the augmented matrix of the system.
 Use elementary row operations to construct a row equivalent matrix in row-echelon
form.
 Write the system of equations corresponding to the matrix in row-echelon form.
 Use back-substitution to find the solutions to this system.
Question
Method: Gaussian elimination
 The house garden requires 18 units of Mineral 1, 13 units of Mineral 2, and 14 units of
Mineral 3. These are three types of fertilizer available. Each bag of fertilizer A contains
2 unit of mineral 1, 1 units of mineral 2, and 3 units of mineral 3. Each bag of fertilizer
B contains 2 units of mineral 1, 3 units of mineral 2, and 1 unit of mineral 3. Each of
fertilizer C contains 4 units of mineral 1, 2 units of mineral 2, and 3 units of mineral 3.
How many bags of each fertilizer should the garden utilize to attain the required
amount of minerals?
Item Fertilizer A Fertilizer B Fertilizer C
Total amount
minerals
Mineral 1 2 2 4 18
Mineral 2 01 3 2 13
Mineral 3 3 01 3 14
Solution to the problem
 Set up variables
Let x be the number of bags of fertilizer A
Let y be the number of bags of fertilizer B
Let z be the number of bags of fertilizer C
 Set up equations
2x+2𝑦 + 4𝑧 = 18
x+3y+2z=13
3x+y+3z=14
2x+2𝑦 + 4𝑧 = 18
x+3y+2z=13
3x+y+3z=14
 Set up augmented Matrix
 AX=B
2 2 4
1 3 2
3 1 3
𝑥
𝑦
𝑧
18
= 13
14
Eq#1
Eq#2
Eq#3
𝑅1 2 2 4
𝑅2 1 3 2
𝑅3 3 1 3
18
13
14
 Interchange R1 and R2
1 3 2
2 2 4
3 1 3
13
18
14
Now do Elementary operations and make 2 and 3 equal to 0.
1
2
∗ 𝑅1
1 3 2
0 −4 0
0 −8 −3
13
−8
−25
R2 -2R1
R3 -3R1
1 3 2
2 2 4
3 1 3
13
18
14
 Write it down in equations form.
 y = 2
 -3z = -9
z =3
 Put value of z = 3 in eq 1 (x+2z = 7)
x+2z = 7
x+2(3)=7
x+6=7
x=7-6X
x =1
1
−4
∗ 𝑅2
R1 -3R2
R3 +8R2
1 3 2
0 −4 0
0 −8 −3
13
−8
−25
1 3 2
0 1 0
0 −8 −3
13
2
−25
1 3 2
0 1 0
0 −8 −3
13
2
−25
1 0 2
0 1 0
0 0 −3
7
2
−9
Solution/Result
 X = 1
Y = 2
Z = 3
 Therefore, the house garden requires 01 bag of fertilizer A, 02 bags of fertilizer
B and 03 bags of fertilizer C to attain 18 units of Mineral 1, 13 units of Mineral
2, and 14 units of Mineral 3

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shanza project maths.pptx

  • 1. Bussiness Mathematics Shanza Ashraf (29960 Pazerish Hareem (29952) Ma'am Sumaira Nadeem
  • 3. Introduction  Gaussian Elimination Method is another method to find the solution of a system. It is performed on an augmented matrix and we use row operations to find the solution of a specific system.  Gaussian elimination method attempts to transform the system into diagonal form. The transformation should occur column by column moving from left to right.  𝑎1 𝑏1 𝑐1 𝑎2 𝑏2 𝑐2 𝑎3 𝑏3 𝑐3 𝑑1 𝑑2 𝑑3  1 0 0 0 1 0 0 0 1 𝑣1 𝑣2 𝑣3 Original system Transformed system
  • 4.  The operations in the Gauss elimination are called elementary operations.  Elementary operations for rows are: 1. Interchange of two rows. 2. Multiplication of a row by a nonzero constant. 3. Addition of a constant multiple of one row to another row.  Two matrices are said to be row equivalent if one matrix can be obtained from the other using elementary row operations Gauss elimination for solving a system of equations  Write the augmented matrix of the system.  Use elementary row operations to construct a row equivalent matrix in row-echelon form.  Write the system of equations corresponding to the matrix in row-echelon form.  Use back-substitution to find the solutions to this system.
  • 5. Question Method: Gaussian elimination  The house garden requires 18 units of Mineral 1, 13 units of Mineral 2, and 14 units of Mineral 3. These are three types of fertilizer available. Each bag of fertilizer A contains 2 unit of mineral 1, 1 units of mineral 2, and 3 units of mineral 3. Each bag of fertilizer B contains 2 units of mineral 1, 3 units of mineral 2, and 1 unit of mineral 3. Each of fertilizer C contains 4 units of mineral 1, 2 units of mineral 2, and 3 units of mineral 3. How many bags of each fertilizer should the garden utilize to attain the required amount of minerals? Item Fertilizer A Fertilizer B Fertilizer C Total amount minerals Mineral 1 2 2 4 18 Mineral 2 01 3 2 13 Mineral 3 3 01 3 14
  • 6. Solution to the problem  Set up variables Let x be the number of bags of fertilizer A Let y be the number of bags of fertilizer B Let z be the number of bags of fertilizer C  Set up equations 2x+2𝑦 + 4𝑧 = 18 x+3y+2z=13 3x+y+3z=14
  • 7. 2x+2𝑦 + 4𝑧 = 18 x+3y+2z=13 3x+y+3z=14  Set up augmented Matrix  AX=B 2 2 4 1 3 2 3 1 3 𝑥 𝑦 𝑧 18 = 13 14 Eq#1 Eq#2 Eq#3
  • 8. 𝑅1 2 2 4 𝑅2 1 3 2 𝑅3 3 1 3 18 13 14  Interchange R1 and R2 1 3 2 2 2 4 3 1 3 13 18 14 Now do Elementary operations and make 2 and 3 equal to 0. 1 2 ∗ 𝑅1 1 3 2 0 −4 0 0 −8 −3 13 −8 −25 R2 -2R1 R3 -3R1 1 3 2 2 2 4 3 1 3 13 18 14
  • 9.  Write it down in equations form.  y = 2  -3z = -9 z =3  Put value of z = 3 in eq 1 (x+2z = 7) x+2z = 7 x+2(3)=7 x+6=7 x=7-6X x =1 1 −4 ∗ 𝑅2 R1 -3R2 R3 +8R2 1 3 2 0 −4 0 0 −8 −3 13 −8 −25 1 3 2 0 1 0 0 −8 −3 13 2 −25 1 3 2 0 1 0 0 −8 −3 13 2 −25 1 0 2 0 1 0 0 0 −3 7 2 −9
  • 10. Solution/Result  X = 1 Y = 2 Z = 3  Therefore, the house garden requires 01 bag of fertilizer A, 02 bags of fertilizer B and 03 bags of fertilizer C to attain 18 units of Mineral 1, 13 units of Mineral 2, and 14 units of Mineral 3