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解某些分式方程时,人们常用合比定理、分比定理、合分比定理将原方程变形,再由变形所得的方程求解。其实,这些定理都是关于比例性质的定理,并不是方程变形的定理。众所周知,通过方程变形来解方程,必须注意变形前后的方程是否同解。现在要问:用上述比例性质定理将方程变形,能否保证变形前后方程同解?如果会增根或遗根,那么怎样排除增根和拾回遗根呢?本文试对此问题作一肤浅的探讨,供参考。
When solving some fractional equations, people usually use the theorem of proportionality, the theorem of ratios, and the ratio-theorem theorem to distort the original equation, and then solve the equation obtained by the deformation. In fact, these theorems are all theorems on proportional properties and are not theorems of equation deformation. As we all know, to solve the equation by equation deformation, we must pay attention to whether the equations before and after the deformation are the same solution. Now we must ask: Using the above proportionality property theorem to distort the equation, can we guarantee the same solution before and after the deformation? If it will increase roots or roots, then how to eliminate rooting and picking up roots? This article tries to make a shallow question on this issue. The discussion is for reference.