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如果两个方程组的解集相同,则称这两个方程组同解。解方程组时,通常是将原方程组逐步变形成为一个易解的方程组来解,这里的“变形”,一定要是同解变形。什么样的变形为同解变形?本文仅以二元方程组为例给出几个主要方程组的同解性定理。首先约定:以记号f(x,y)=0表为二元方程,以其中一个变量(如x)表另一个变量(如y)记为y=f(x),其余类同。定理Ⅰ:方程组{y=f(x) g(x,y)=0(*)与方程组 {y=f(x)(**)同解。 g[x,f(x)]=0 证明:设(α,β)为方程组(*)的任一解, 则有{β=f(α) g(α,β)=0, 即{β=f(α) g[α,f(α)]=0 故(α,β)亦是方程组(**)的解。
If the solution sets of the two equations are the same, the two equations are said to have the same solution. When solving a system of equations, the original system of equations is usually gradually deformed into an easily solvable system of equations. The “deformation” here must be the same solution. What kind of deformation is the same solution deformation? This paper only uses the binary equations as an example to give the homothetic theorems of several main equations. First of all, it is agreed that the table with the notation f(x,y)=0 is a binary equation, in which one variable (such as x) represents another variable (such as y) as y=f(x) and the others are similar. Theorem I: The system of equations {y=f(x) g(x,y)=0(*) is identical to the system of equations {y=f(x)(**). g[x,f(x)]=0 prove: Let (α,β) be any solution of the equation (*), then there is {β=f(α) g(α,β)=0, that is { β=f(α) g[α,f(α)]=0 So (α,β) is also the solution of the equation set (**).