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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Substitution principle (mathematics) ： ウィキペディア英語版 Change of variables In mathematics, the operation of substitution consists in replacing all the occurrences of a free variable appearing in an expression or a formula by a number or another expression. In other words, an expression involving free variables may be considered as defining a function, and substituting values to the variables in the expression is equivalent to applying the function defined by the expression to these values. A change of variables is commonly a particular type of substitution, where the substituted values are expressions that depend on other variables. This is a standard technique used to reduce a difficult problem to a simpler one. A change of coordinates is a common type of change of variables. However, if the expression in which the variables are changed involves derivatives or integrals, the change of variable does not reduce to a substitution. A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth order polynomial: :$x^6\; -\; 9\; x^3\; +\; 8\; =\; 0.\; \backslash ,$ Sixth order polynomial equations are generally impossible to solve in terms of radicals (see Abel–Ruffini theorem). This particular equation, however, may be written :$(x^3)^2-9(x^3)+8=0$ (this is a simple case of a polynomial decomposition). Thus the equation may be simplified by defining a new variable ''x''3 = ''u''. Substituting ''x'' by $\backslash sqrt()$ into the polynomial gives :$u^2\; -\; 9\; u\; +\; 8\; =\; 0\; ,$ which is just a quadratic equation with solutions: :$u\; =\; 1\; \backslash quad\; \backslash text\; \backslash quad\; u\; =\; 8.$ The solutions in terms of the original variable are obtained by substituting ''x''3 back in for ''u'': :$x^3\; =\; 1\; \backslash quad\; \backslash text\; \backslash quad\; x^3\; =\; 8.$ Then, assuming that ''x'' is real, :$x\; =\; (1)^\; =\; 1\; \backslash quad\; \backslash text\; \backslash quad\; x\; =\; (8)^\; =\; 2.$ ==Simple example== Consider the system of equations :$xy+x+y=71$ :$x^2y+xy^2=880$ where $x$ and $y$ are positive integers with $x>y$. (Source: 1991 AIME) Solving this normally is not terrible, but it may get a little tedious. However, we can rewrite the second equation as $xy(x+y)=880$. Making the substitution $s=x+y,\; t=xy$ reduces the system to $s+t=71,\; st=880.$ Solving this gives $(s,t)=(16,55)$ or $(s,t)=(55,16).$ Back-substituting the first ordered pair gives us $x+y=16,\; xy=55$, which easily gives the solution $(x,y)=(11,5).$ Back-substituting the second ordered pair gives us $x+y=55,\; xy=16$, which gives no solutions. Hence the solution that solves the system is $(x,y)=(11,5)$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Change of variables」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.