Enzyme plays an important role in the living system as well as life process, and is involved in almost every chemical reaction; thus the enzyme kinetics is fundamental to the research of the chemical reactions. Therefore, from the early part of twentieth century, many researchers were dedicated to the study of the analysis of kinetics of enzymatic system, and a lot of excellent work has been done, because the analysis of the enzyme kinetics can help us find how the activity of enzyme is controlled, how the drug inhibits the enzyme, its role in metabolism, and so on. In this paper, we present a method by rigorous mathematics to analyze the general enzyme kinetics model, which is concerned with a single substrate catalyzed by enzyme and transformed into a single product, and it is based on computer algebra technique and the qualitative theory of dynamical system. Our method does not need any assumption and all the solutions here are analytical. Finally, some examples are given to test this method and make it readable by simulated progress curves, so that almost everyone may understand it. Furthermore, the techniques in this study can be used to the research of other disciplines, such as biophysics, and so on.