Systems

Of
Week 11

Equations
GeoE 405- Advanced Engineering Mathematics for Geological Engineering
 Systems of Equations
(a) Direct Methods
Week 11

1. Gaussian-Elimination Method
2. Gauss-Jordan Method

(b)Indirect Methods

1. Jacobi’s Method
2. Gauss-Seidel Method

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Carl Friedrich Gauss lived during the
Systems of Equations

late18th century and early19th century, but

he is still considered one of the most
prolific mathematicians in history. His
contributions to the science of mathematics
and physics span fields such as algebra,
number theory, analysis, differential
geometry, astronomy, and optics, among
others. His discoveries regarding matrix
German mathematician theory changed the way mathematicians
Carl Friedrich Gauss have worked for the last two centuries.
(1777–1855)

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 What is Systems of Equations?
Systems of Equations

In mathematics, a system of equations, also known as a

set of simultaneous equations or an equation system, is a finite
set of equations for which we sought common solutions. In
systems of equations, variables are related in a specific way in
each equation

In algebra, a system of equations comprises two or more

equations and seeks common solutions to the equations.
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑
GeoE 405- Advanced Engineering Mathematics for Geological Engineering
 Writing the Augmented Matrix
of a System Equations
Systems of Equations

A matrix can serve as a device for representing and

solving a system of equations. To express a system in matrix
form, we extract the coefficients of the variables and the
constants, and these become the entries of the matrix. We use a
vertical line to separate the coefficient entries from the
constants, essentially replacing the equal signs. When a system
is written in this form, we call it an augmented matrix.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 How to: Given a system of equations,
write an augmented matrix
Systems of Equations

1. Write the coefficients of the x-terms as the

numbers down the first column.
2. Write the coefficients of the y-terms as the
numbers down the second column.
3. If there are z-terms, write the coefficients as the
numbers down the third column.
4. Draw a vertical line and write the constants to the
right of the line.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Writing the Augmented Matrix
of a System Equations
Systems of Equations

For example, consider the following 2 x 2 system of equations

3𝑥 + 4𝑦 = 7
4𝑥 − 2𝑦 = 5
We can also write a matrix containing
We can write this system as just the coefficients. This is called the
an augmented matrix: coefficient matrix.

3 4 7 3 4
4 −2 5 4 −2

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 A three-by-three system of equations such as:
Systems of Equations

3𝑥 − 𝑦 − 𝑧 = 0
𝑥+𝑦 =5
2𝑥 − 3𝑧 = 2

And is represented by the

Has a coefficient matrix augmented matrix

3 −1 −1 3 −1 −1 0
1 1 0 1 1 0 5
2 0 −3 2 0 −3 2

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 GAUSSIAN
ELIMINATION
Week 11

METHOD
GeoE 405- Advanced Engineering Mathematics for Geological Engineering
 𝑮𝒂𝒖𝒔𝒔𝒊𝒂𝒏 − 𝑬𝒍𝒊𝒎𝒊𝒏𝒂𝒕𝒊𝒐𝒏 𝑴𝒆𝒕𝒉𝒐𝒅
Systems of Equations

Learning Objectives:

• Write the augmented matrix of a system of equations.

• Write the system of equations from an augmented matrix.
• Perform row operations on a matrix.
• Solve a system of linear equations using matrices.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Performing Row Operations on a Matrix
Systems of Equations
Now that we can write systems of equations in augmented matrix form,
we will examine the various row operations that can be performed on a matrix,
such as addition, multiplication by a constant, and interchanging rows.
Performing row operations on a matrix is the method we use for solving
a system of equations. In order to solve the system of equations, we want to
convert the matrix to row-echelon form, in which there are ones down the main
diagonal from the upper left corner to the lower right corner, and zeros in every
position below the main diagonal as shown.

1 𝑎 𝑏
Row-echelon form 0 1 𝑑
0 0 1

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Performing Row Operations on a Matrix
Systems of Equations

We use row operations corresponding to equation operations to

obtain a new matrix that is row-equivalent in a simpler form. Here are
the guidelines to obtaining row-echelon form.

1. In any nonzero row, the first nonzero number is a 1. It is called

a leading 1.
2. Any all-zero rows are placed at the bottom on the matrix.
3. Any leading 1 is below and to the right of a previous leading 1.
4. Any column containing a leading 1 has zeros in all other
positions in the column.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Performing Row Operations on a Matrix
Systems of Equations

To solve a system of equations we can perform the

following row operations to convert the coefficient matrix to
row-echelon form and do back-substitution to find the
solution.

1. Interchange rows. (Notation: Ri Rj)

2. Multiply a row by a constant. (Notation: cRi)
3. Add the product of a row multiplied by a constant to
another row. (Notation: Ri + cRj)

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Performing Row Operations on a Matrix
Systems of Equations

Each of the row operations corresponds to the operations

we have already learned to solve systems of equations in three
variables. With these operations, there are some key moves
that will quickly achieve the goal of writing a matrix in row-
echelon form. To obtain a matrix in row-echelon form for
finding solutions, we use Gaussian elimination, a method that
uses row operations to obtain a 1 as the first entry so that
row 1 can be used to convert the remaining rows.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Gaussian Elimination
Systems of Equations

The Gaussian elimination method refers to a strategy used to

obtain the row-echelon form of a matrix. The goal is to write
matrix A with the number 1 as the entry down the main diagonal and
have all zeros below.

𝑎11 𝑎12 𝑎13 1 𝑏12 𝑏13

After Gaussian elimination
𝐴 = 𝑎21 𝑎22 𝑎23 𝐴= 0 1 𝑏23
𝑎31 𝑎32 𝑎33 0 0 1

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 How to: Given an augmented matrix, perform row
Gaussian-Elimination Method
operations to achieve row-echelon form

1. The first equation should have a leading coefficient of 1. Interchange

rows or multiply by a constant, if necessary.
2. Use row operations to obtain zeros down the first column below the
first entry of 1.
3. Use row operations to obtain a 1 in row 2, column 2.
4. Use row operations to obtain zeros down column 2, below the entry of
1.
5. Use row operations to obtain a 1 in row 3, column 3.
6. Continue this process for all rows until there is a \(1 in every entry
down the main diagonal and there are only zeros below.
7. If any rows contain all zeros, place them at the bottom.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Example 1:
Gaussian-Elimination Method
Solving a 2x2 System by Gaussian Elimination
Eq. 1 2𝑥 + 3𝑦 = 6 1 𝑎 𝑏
Goal:
1 0 1 𝑑
Eq. 2 𝑥−𝑦 =
2
Solution: Interchange Rows:

Eq. 2 multiply by 2 2 −2 1 2 −2 1
𝑅1 − 𝑅2
2𝑥 − 2𝑦 = 1 2 3 6 0 −5 −5
1 −1
R1 multiply by and R2 multiply by
2 5
2 3 6 1
1 1 𝑥−1=
2 −2 1 1 −1 𝑥−𝑦 = 2
2 2
3
0 1 1 𝑦=1 𝑥=
2
GeoE 405- Advanced Engineering Mathematics for Geological Engineering
 Example 2:
Gaussian-Elimination Method
Solving a 2x2 System by Gaussian Elimination
Eq. 1 4𝑥 + 3𝑦 = 11 1 𝑎 𝑏
Goal:
0 1 𝑑
Eq. 2 𝑥 − 3𝑦 = −1

Solution: 1 −3 −1
1 −3 −1 4𝑅1 − 𝑅2
4 3 11 4 3 11 0 −15 −15
1 −3 −1
Interchange Rows: R21 = 4(1) – 4 = 0
1 −3 −1 R22 = 4(-3) – 3 = -15
4 3 11 R23 = 4(-1) – 11 = -15

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 1 −3 −1
Gaussian-Elimination Method

0 −15 −15
−1
R2 multiply by
15

1 −3 −1
0 1 1

𝑥 − 3𝑦 = −1 𝑥 − 3(1) = −1
𝑦=1 𝑥 = −1 + 3
𝑥=2

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Example 3:
Gaussian-Elimination Method
Solving a 3x3 System by Gaussian Elimination
𝑥−𝑦+𝑧 =8 1 𝑎 𝑏
Goal:
2𝑥 + 3𝑦 − 𝑧 = −2 0 1 𝑑
3𝑥 − 2𝑦 − 9𝑧 = 9 0 0 1
Solution:
1 −1 1 8 1 −1 1 8
2 3 −1 −2𝑅1 + 𝑅2 0 5 −3
−2 −18
3 −2 −9 9 3 −2 −9 9

R21 = -2(1) +2 = 0
R22 = -2(-1) +3 = 5
R23 = -2(1) +(-1) = -3
R24 = -2(8) +(-2) = -18

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

Gaussian-Elimination Method
1 −1 1 8 1 −1 1 8
0 5 −3 −18 0 5 −3 −18
3 −2 −9 9 −3𝑅1 + 𝑅3 0 1 −12 −15

R31 = -3(1) +3 = 0
The easiest way to obtain a
1 −1 1 8
R32 = -3(-1) +(-2) = 1 0 1 −12 −15
1 in row 2 of column 1 is to
R33 = -3(1) +(-9) = -12 interchange R2 and R3. 0 5 −3 −18
R34 = -3(8) +9 = -15 Interchange R2 and R3.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

Gaussian-Elimination Method

1 −1 1 8 1 −1 1 8
0 1 −12 −15 0 1 −12 −15
0 5 −3 −18 −5𝑅2 + 𝑅3 0 0 57 57

R31 = -5(0) +0 = 0
R32 = -5(1) +5 = 0
R33 = -5(-12) +(-3) = 57
R34 = -5(-15) +(-18) = 57

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 1 −1 1 8 1 −1 1 8
Gaussian-Elimination Method
1 0 1 −12 −15
0 1 −12 −15 𝑅
57 3 0 0 1 1
0 0 57 57

1 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒,
R33 = 57 = 1
57
1
𝑥−𝑦+𝑧 =8
R34 = 57 = 1 𝑦 − 12𝑧 = −15
57
𝑧=1

𝑦 − 12(1) = −15 𝑥 − (−3) + 1 = 8

𝑦 = −15 + 12 𝑥 =8−3−1
𝑦 = −3 𝑥=4

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 GAUSS
JORDAN
Week 11

METHOD
GeoE 405- Advanced Engineering Mathematics for Geological Engineering
 𝑮𝒂𝒖𝒔𝒔 𝑱𝒐𝒓𝒅𝒂𝒏 𝑴𝒆𝒕𝒉𝒐𝒅
Systems of Equations

Learning Objectives:

• Represent a system of linear equations as an

augmented matrix.
• Solve the system using elementary row
operations.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 𝑮𝒂𝒖𝒔𝒔 𝑱𝒐𝒓𝒅𝒂𝒏 𝑴𝒆𝒕𝒉𝒐𝒅
Gauss - Jordan Method

In this method, we learn to solve systems of linear equation

using a process called the Gauss-Jordan Method. The process
begins by first expressing the system as a matrix, and then
reducing it to an equivalent system by simple row operations. The
process is continued until the solution is obvious from the matrix.
The matrix that represents the system is called the augmented
matrix, and the arithmetic manipulation that is used to move from
a system to a reduced equivalent system is called a row
operation.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

Gauss - Jordan Method

Once a system is expressed as an augmented matrix, the Gauss-

Jordan method reduces the system into a series of equivalent systems by
using the row operations. This row reduction continues until the system
is expressed in what is called the reduced row echelon form. The
reduced row echelon form of the coefficient matrix has 1's along the
main diagonal and zeros elsewhere. The solution is readily obtained
from this form.

1 0 0
Reduced row echelon form 0 1 0
0 0 1

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

Gauss - Jordan Method

We will next solve a system of two equations with two

unknowns, using the elimination method, and then
show that the method is analogous to the Gauss-Jordan
method.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Example 4:
Solve the following system by Elimination Method
Gaussian - Jordan Method

𝑥 + 3𝑦 = 7
3𝑥 + 4𝑦 = 11 Solve for y:
−1
Solution: −5𝑦 = −10
5
Eq. 1 multiply by -3 then add to Eq. 2 𝑦=2
−3𝑥 − 9𝑦 = −21 Solve for x from eq. 1:
3𝑥 + 4𝑦 = 11
𝑥 + 3𝑦 = 7
−5𝑦 = −10
𝑥 + 3(2) = 7
𝑥=1

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

Gauss - Jordan Method

𝑹𝒐𝒘 𝑶𝒑𝒆𝒓𝒂𝒕𝒊𝒐𝒏𝒔

1. Any two rows in the augmented matrix may be interchanged.

2. Any row may be multiplied by a non-zero constant.
3. A constant multiple of a row may be added to another row.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

Gauss - Jordan Method 𝑮𝒂𝒖𝒔𝒔 𝑱𝒐𝒓𝒅𝒂𝒏 𝑴𝒆𝒕𝒉𝒐𝒅
1. Write the augmented matrix.
2. Interchange rows if necessary to obtain a non-zero number in the first row, first
column.
3. Use a row operation to get a 1 as the entry in the first row and first column.
4. Use row operations to make all other entries as zeros in column one.
5. Interchange rows if necessary to obtain a nonzero number in the second row,
second column. Use a row operation to make this entry 1. Use row operations to
make all other entries as zeros in column two.
6. Repeat step 5 for row 3, column 3. Continue moving along the main diagonal
until you reach the last row, or until the number is zero.
1 0 0
The final matrix is called the reduced row-echelon form. 0 1 0
0 0 1
GeoE 405- Advanced Engineering Mathematics for Geological Engineering
 Example 5:
Solve the following system by Gaussian - Jordan Method
Gaussian - Jordan Method

1 0 𝑎
𝑥 + 3𝑦 = 7 Goal:
0 1 𝑏
3𝑥 + 4𝑦 = 11
Solution:
1 3 7 −3𝑅1 + 𝑅2 1 3 7
3 4 11 0 −5 −10

R21 = -3(1) +3 = 0
R22 = -3(3) +4 = -5
R23 = -3(7) +11 = -10

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 1 3 7 −1 1 3 7
𝑅2
Gaussian - Jordan Method
0 −5 −10 5 0 1 2

−1 −1 −1
𝑅21 = 0 =0 𝑅22 = −5 = 1 𝑅23 = −10 = 2
5 5 5

1 3 7 𝑅1 + −3𝑅2 1 0 1
0 1 2 0 1 2

𝑅11 = 1 + −3 0 =1 𝑅12 = 3 + −3 1 =0 𝑅13 = 7 + −3 2 =1

1 0 1 𝑥=1
0 1 2 𝑦=2

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 Example 6:
Solving a 3x3 System by Gaussian - Jordan Method
Gaussian - Jordan Method

2𝑥 + 𝑦 + 2𝑧 = 10 1 0 0
Goal:
𝑥 + 2𝑦 + 𝑧 = 8 0 1 0
3𝑥 + 𝑦 − 𝑧 = 2 0 0 1
Solution:
2 1 2 10
1 2 1 8 Interchange rows: 𝑅1 and 𝑅2
3 1 −1 2
1 2 1 8
We want a 1 in row one, column one. This can
be obtained by dividing the first row by 2, or 2 1 2 10
interchanging the second row with the first. 3 1 −1 2
Interchanging the rows is a better choice
because that way we avoid fractions.

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 1 2 1 8 1 2 1 8
Gaussian - Jordan Method
2 1 2 10 −2𝑅1 + 𝑅2 0 −3 0 −6
3 1 −1 2 3 1 −1 2

R21 = -2(1) +2 = 0 R23 = -2(1) +2 = 0

R22 = -2(2) +1 = -3 R24 = -2(8) +10 = -6

1 2 1 8 1 2 1 8
0 −3 0 −6 0 −3 0 −6
−3𝑅1 + 𝑅3 0 −5 −4 −22
3 1 −1 2

R31 = -3(1) +3 = 0 R33 = -3(1) +(-1) = -4

R32 = -3(2) +1 = -5 R34 = -3(8) +2 = -22

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 −1
1 2 1 8 𝑅2 1 2 1 8
3
2
Gaussian - Jordan Method
0 −3 0 −6 0 1 0
0 −5 −4 −22 0 −5 −4 −22
−1 −1
R21 = 0 =0 R23 = 0 =0
3 3
−1 −1
R22 = −3 = 1 R22 = −6 = 2
3 3

−2𝑅2 + 𝑅1
1 2 1 8 1 0 1 4
0 1 0 2 0 1 0 2
0 −5 −4 −22 5𝑅2 + 𝑅3 0 0 −4 −12

R11 = -2(0) +1 = 1 R31 = 5(0) +0 = 0

R12 = -2(1) +2 = 0 R32 = 5(1) + −5 = 0
R13 = -2(0) +1 = 1 R33 = 5(0) + −4 = -4
R14 = -2(2) +8 = 4 R34 = 5(2) + −22 = -12
GeoE 405- Advanced Engineering Mathematics for Geological Engineering
 1 0 1 4 1 0 1 4
0 1 0 2 −1 0 1 0 2
Gaussian - Jordan Method
𝑅3
0 0 −4 −12 4 0 0 1 3
−1 −1
R31 = 0 =0 R33 = −4 = 1
4 4
−1 −1
R32 = 0 =0 R32 = −12 = 3
4 4

1 0 1 4 −𝑅3 + 𝑅1 1 0 0 1
0 1 0 2 0 1 0 2
0 0 1 3 0 0 1 3

R11 = -0 +1 = 1 𝑥=1
R12 = -0+0 = 0 𝑦=2
R13 = -1+1 = 0
R14 = -3 +4 = 1 𝑧=3

GeoE 405- Advanced Engineering Mathematics for Geological Engineering

 I HOPE, YOU ARE NOW READY TO
SOLVE OUR ACTIVITY TODAY ☺

THE
Week 11

END THANK YOU!

GeoE 405- Advanced Engineering Mathematics for Geological Engineering