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change of variables : 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. \,
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 \sqrt() into the polynomial gives
:u^2 - 9 u + 8 = 0 ,
which is just a quadratic equation with solutions:
:u = 1 \quad \text \quad u = 8.
The solutions in terms of the original variable are obtained by substituting ''x''3 back in for ''u'':
:x^3 = 1 \quad \text \quad x^3 = 8.
Then, assuming that ''x'' is real,
:x = (1)^ = 1 \quad \text \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)
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