For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
Polynomials And Linear Equation of Two VariablesAnkur Patel
A complete description of polynomials and also various methods to solve the Linear equation of two variables by substitution, cross multiplication and elimination methods.
For polynomials it also contains the description of monomials, binomials etc.
Students learn to define and identify linear equations. They also learn the definition of Standard Form of a linear equation.
Students also learn to graph linear equations using x and y intercepts.
This the presentation prepared by SIDI DILER the student of CIVIL ENGINEERING at Government Engineering College BHUJ under the fulfillment of the Progressive Assessment component of the Course of Vector Calculus and Linear Algebra with code 2110015.
A shuttle valve is a type of valve which allows fluid to flow through it from one of two sources. Generally a shuttle valve is used in pneumatic systems, although sometimes it will be found in hydraulic systems.
Reciprocating engine cylinders are often classified by whether they are single- or double-acting, depending on how the working fluid acts on the piston.
Gas Metal Arc Welding or MIG welding .
Gas metal arc welding (GMAW), sometimes referred to by its subtypes metal inert gas (MIG) welding or metal active gas (MAG) welding, is a welding process in which an electric arc forms between a consumable wire electrode and the workpiece metal(s), which heats the workpiece metal(s), causing them to melt and join
pulsed spray
globular spray
'The Road Not Taken' is one of the most famous poems written by the American poet, Robert Frost. The poem describes a person standing at a fork in the road in a wood, unsure which one to take. Frost uses the road as a metaphor for the journey of life
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
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Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
2. 1.1 Introduction to Systems of Linear Equations
a linear equation in n variables:
332211 bxaxaxaxa nn =++++
a1
,a2
,a3
,…,an
, b: real number
a1
: leading coefficient
x1
: leading variable
Notes:
(1) Linear equations have no products or roots of variables and
no variables involved in trigonometric, exponential, or
logarithmic functions.
(2) Variables appear only to the first power.
3.
Notes:
Every system of linear equations has either
(1) exactly one solution,
(2) infinitely many solutions, or
(3) no solution.
4.
Equivalent:
Two systems of linear equations are called equivalent
if they have precisely the same solution set.
Notes:
Each of the following operations on a system of linear
equations produces an equivalent system.
(1) Interchange two equations.
(2) Multiply an equation by a nonzero constant.
(3) Add a multiple of an equation to another equation.
5. (4) For a square matrix, the entries a11, a22, …, ann are called
the main diagonal entries.
1.2 Gaussian Elimination and Gauss-Jordan Elimination
m×n matrix:
mnmmm
n
n
n
aaaa
aaaa
aaaa
aaaa
321
3333231
2232221
1131211
rowsm
columnsn
(3) If , then the matrix is called square of order n.nm =
Notes:
(1) Every entry aij in a matrix is a number.
(2) A matrix with m rows and n columns is said to be of size
m×n .
6.
Ex 1: Matrix Size
]2[
00
00
−
2
1
031
− 47
22
πe
11×
22×
41×
23×
Note:
One very common use of matrices is to represent a system
of linear equations.
7.
a system of m equations in n variables:
mnmnmmm
nn
nn
nn
bxaxaxaxa
bxaxaxaxa
bxaxaxaxa
bxaxaxaxa
=++++
=++++
=++++
=++++
332211
33333232131
22323222121
11313212111
=
mnmmm
n
n
n
aaaa
aaaa
aaaa
aaaa
A
321
3333231
2232221
1131211
=
mb
b
b
b
2
1
=
nx
x
x
x
2
1
bAx =Matrix form:
9.
Elementary row operation:
jiij RRr ↔:(1) Interchange two rows.
ii
k
i RRkr →)(:)(
(2) Multiply a row by a nonzero constant.
jji
k
ij RRRkr →+)(:)(
(3) Add a multiple of a row to another row.
Row equivalent:
Two matrices are said to be row equivalent if one can be obtained
from the other by a finite sequence of elementary row operation.
10.
Row-echelon form: (1, 2, 3)
(1) All row consisting entirely of zeros occur at the bottom
of the matrix.
(2) For each row that does not consist entirely of zeros,
the first nonzero entry is 1 (called a leading 1).
(3) For two successive (nonzero) rows, the leading 1 in the higher
row is farther to the left than the leading 1 in the lower row.
Reduced row-echelon form: (1, 2, 3, 4)
(4) Every column that has a leading 1 has zeros in every position
above and below its leading 1.
12.
Gaussian elimination:
The procedure for reducing a matrix to a row-echelon form.
Gauss-Jordan elimination:
The procedure for reducing a matrix to a reduced row-echelon
form.
Notes:
(1) Every matrix has an unique reduced row echelon
form.(2) A row-echelon form of a given matrix is not unique.
(Different sequences of row operations can produce
different row-echelon forms.)
13.
−−
−
−
456542
1280200
28124682
12r
The first nonzero
column
Produce leading 1
Zeros elements below leading 1
leading 1
Produce leading 1
The first nonzero column
Ex: (Procedure of Gaussian elimination and Gauss-Jordan elimination)
−−
−
−
456542
28124682
1280200
−−
−
−
456542
1280200
1462341)(
1
2
1
r
−−
−
−
24170500
1280200
1462341)2(
13
−
r
Submatrix
18.
Homogeneous systems of linear equations:
A system of linear equations is said to be homogeneous
if all the constant terms are zero.
0
0
0
0
332211
3333232131
2323222121
1313212111
=++++
=++++
=++++
=++++
nmnmmm
nn
nn
nn
xaxaxaxa
xaxaxaxa
xaxaxaxa
xaxaxaxa
19.
Trivial solution:
Nontrivial solution:
other solutions
0321 ===== nxxxx
Notes:
(1) Every homogeneous system of linear equations is consistent.
(2) If the homogenous system has fewer equations than
variables,
then it must have an infinite number of solutions.(3) For a homogeneous system, exactly one of the following is true.
(a) The system has only the trivial solution.
(b) The system has infinitely many nontrivial solutions in
addition to the trivial solution.
20.
Ex 9: Solve the following homogeneous system
032
03
321
321
=++
=+−
xxx
xxx
− 0110
0201
)1(
21
)(
2
)2(
12 ,, 3
1
rrr −
Let tx =3
Rttxtxtx ∈==−= ,,,2 321
solution)(trivial0,0When 321 ==== xxxt
Sol:
−
0312
0311
matrixaugmented
form)echelon
-row(reduced
3
21
variablefree
,variableleading
x
xx
:
:
22. Example 3: Describe all solutions for
−=−+
−=+−−
=−+
486
1423
7453
321
321
321
xxx
xxx
xxx
Solutions of Nonhomogeneous Systems
i.e. Describe all solutions of whereAx=b
−
−−
−
=
816
423
453
A
and b =
7
−1
−4
Geometrically, what does the solution set represent?