FINANCIAL UNIVERSITY
UNDER THE GOVERNMENT OF THE RUSSIAN
FEDERATION
Solving Systems of Linear
Equations
Olga Evgen’evna Orel,
Ph. D., Associate Professor
Department of Mathematics-1,
olga_orel72@mail.ru
Moscow 2014
Basic Terminology
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System of Linear Equations (система |
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линейных уравнений) or Linear System: a |
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of |
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1, 2, … , |
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finite set of linear equations. |
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Solution: numbers |
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such that |
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we substitute 1 = 1, 2 = 2 |
, … , = . |
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every equation |
the system is satisfied when |
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•Solution Set (множество решений) or
General Solution (общее решение): the set of all solutions of the system.
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• An inconsistent (несовместная) system has no solutions
• A consistent (совместная) system has at least one solution
+ = 4
EXAMPLE of an inconsistent system:
+ = 3
Every system of linear equations
•has no solutions, or
•has exactly one solution, or
•has infinitely many solutions.
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An arbitrary system of m linear |
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equations in n unknowns |
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11 |
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+ 12 |
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+ + 1 |
= 1 |
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21 |
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+ 22 |
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+ + 2 |
= 2 |
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where |
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+ + = |
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1, 2, … , |
are the unknowns |
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terms |
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a's and b's are constants |
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constant |
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EXAMPLE of a general system of 3 equations in 4
unknowns (общая системаиз 3-х уравнений с 4-мя11 неизвестными1 + 12 2 + 13): 3 + 14 4 = 121 1 + 22 2 + 23 3 + 24 4 = 2
31 1 + 32 2 + 33 3 + 34 4 = 3
number of the equation |
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number of the unknown |
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5