The document discusses solving systems of linear equations using Mathematica. It begins with an introduction to linear equations and systems of linear equations. It then discusses Mathematica, including its history and features. The work section demonstrates using Mathematica's Solve function to find the solution to a system of 3 linear equations in 3 variables. It inputs the 3 equations into Solve to return the solution set {x→4/3, y→0, z→2/3}.

1. 1
Solutionof Systemof LinearEquations

2. 2
Tableof Contents:
Outline……………………………………………………………………………………………
Introduction…………………………………………………………………………………..
Literature Review…………………………………………………………………………..
Work………………………………………………………………………………………………
Conclusion……………………………………………………………………………………..

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Outline:
 Introduction
 Literature Review
 Work
 Conclusion

4. 4
Solution of System of Linear Equations
Introduction:
In everyday life, wedeal with the differentphysicalsituations that can be
modelled by different systemof linear equations. Linear equations have many
more applications in everyday life. Linear equations are used in Linear equations
can have one or more variables. Linear equations occur abundantly in most
subareas of mathematics and especially in applied mathematics. While they arise
quite naturally when modeling many phenomena, they are particularly useful
since many non-linear equations may be reduced to linear equations by assuming
that quantities of interest vary to only a small extent fromsome"background"
state. Linear equations do not include exponents. Linear equations are very
important and are widely used in our daily life. In our daily life, textiles, education,
computing, softwaredeveloping etc , mathematics and linear equations are used
everywhere. Wecan solve the systemof linear equations using software
“Mathematica”. Details of the title is given below:
LinearEquation:
A linear equation has the form
a1x1 +a2x2 +…+anxn =K
where“a”” is a real number, “x” is a variable and “k” is a constant.
In a linear equation, each variable has an exponent of exactly 1.
Examples are
2x +3y= = 12
3x + 4y =5
Reference : Mary Jane Sterling, ”Linear Algebra For Dummies”, 2009 by Willey
Publishing
A linear equation in variables x1 ,…, xn is an equation of the form
a1x1 +…+ anxn = b

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Where a1,…,an and b are real or complex numbers. The numbers a1, a2,…,an are
the coefficients of the equation and b is the constant form.
Examples are
9x +4y =6
2x + 7y = 4
Reference : Fred Szabo, “Linear Algebra, An Introduction using Mathematica”
,2000 by Academic Press.
LinearSystemof equation:
A finite set of linear equations is called a systemof linear equations or linear
system.
Here is a systemof linear equations
3x + 4y +5z =15
5x + 13y = 6
4y + 6z = 12
Has a solution {𝑥 →
218
1239
, 𝑦 → −
38
413
, 𝑧 →
978
413
}
Reference : S.K.Kate, “ Engineering Mathematics-I” , 2009 by Technical
Publications.
Solution of system of Equations:
A solution of systemof equations is n unknowns is an ordered set of n numbers
that when substituted for unknowns makeall the equations of the systemtrue.
6x + y +5z =5
5x + 6z = 6
y + 5z = 2
has a solution {𝑥 → −
10
409
, 𝑦 → −
12
409
, 𝑧 →
178
409
}
Reference : Patricia Clark Kenschaft, “ Linear Mathematics: A Practical Approach”

6. 6
Mathematica:
Mathematica is a computational softwareprogramused in
many scientific, engineering, mathematical and computing fields, based on
symbolic mathematics. Itwas conceived by Stephen Wolframand is developed by
WolframResearch of Champaign, Illinois. The WolframLanguageis the
programming languageused in Mathematica.
History:
Mathematica is the world's mostpowerfulglobal computation system. First
released in 1988, it has had a profound effect on the way computers are used in
technical and other fields. In 2008, following a dramatic reinvention in 2007,
Mathematica continued the momentum of innovation by bringing major new
application areas into its integrated framework.
Itis often said that the release of Mathematica marked the beginning of modern
technical computing. Ever sincethe 1960s individualpackages had existed for
specific numerical, algebraic, graphical, and other tasks. But the visionary concept
of Mathematica was to create once and for all a single systemthat could handle
all the various aspects of technical computing--and beyond--in a coherent and
unified way. The key intellectual advancethat made this possiblewas the
invention of a new kind of symbolic computer language that could, for the first
time, manipulate the very wide rangeof objects needed to achieve the generality
required for technical computing, using only a fairly small number of basic
primitives.
Features
 Elementary and Special mathematical function libraries
 Matrix and data manipulation tools including supportfor sparsearrays
 Supportfor complex number, arbitrary precision, intervalarithmetic and
symbolic computation
 2D and 3D data, function and geo visualization and animation tools
 Solvers for systems of equations, diophantine equations, ODEs, PDEs, DAEs,
DDEs, SDEs and recurrencerelations
 Numeric and symbolic tools for discrete and continuous calculus

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 Multivariate statistics libraries including fitting, hypothesis testing, and
probability and expectation calculations on over 140 distributions.
 Supportfor censored data, temporal data, time-series and unit based data
 Calculations and simulations on randomprocesses and queues
 Tools for visualizing and analysing directed and undirected graphs Number
theory function library Linear and non-linear Control systems librarieS
Reference: Wolfram, Stephen Mathematica Turns 20 Today, Wolfram, retrieved
16 May 2012
Literature Review:
A literature review on severalexisting linear regression equations for correlating
water activity (aw) and refractometric moisturecontent in floral honeys was
performed in order to providea weighted average regression equation. For this
purpose, a metal analysis of the literature linear equations was made which takes
into account the number of data points as well as the precision involved in the
different literature studies. The weighted average linear regression equation is a
general linear relationship to calculate the honey water activity on the basis of
the honey moisturecontent Itwas obtained fromcomparing and combining the
results of ten independent studies involving 638 observations and including
honeys fromArgentina, Colombia, Spain, Germany, Czech Republic, China,
México, Cuba, Brazil, El Salvador, India and Vietnam. Itwas also found the
proposed equation describes satisfactorily the water activity of concentrated (and
supersaturated) glucose/ fructose solutions, which would confirmthat these
sugars arethe main determinants of water activity in honey.
Work:
Mathematica can solvesystems of linear equations. To solvethe equations
x + y + z = 0, x + 2y + 3z = 1, and x – y + z = 2, we can use the Mathematica
function solve. To do this, make an equation of the list of the left hand sides of
the equations and the list of right hand sides as follows:

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x+y+z2 y+3 zx-y+z2
Solve[x+y+z2 y+3 zx-y+z2,{x,y,z}]
{{x4/3,y0,z2/3}}
x y z 0
x 2y 3z 1
x y z 2
x y z 0 x 2 y 3 z 1 x y z 2