Soution of Linear Equations
Download as DOCX, PDF0 likes228 views
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}.
![8
x+y+z2 y+3 zx-y+z2
Solve[x+y+z2 y+3 zx-y+z2,{x,y,z}]
{{x4/3,y0,z2/3}}
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](https://image.slidesharecdn.com/internshipreport-151101080825-lva1-app6891/85/Soution-of-Linear-Equations-8-320.jpg)
Soution of Linear Equations
- 3. 3 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
- 5. 5 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
- 7. 7 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:
- 8. 8 x+y+z2 y+3 zx-y+z2 Solve[x+y+z2 y+3 zx-y+z2,{x,y,z}] {{x4/3,y0,z2/3}} 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