Students often have some problems in the learning process of the first three chapters of seventh grade mathematics. Let's summarize it for students' reference in the learning process. I hope that students will correct me if they have it, encourage them if they don't, and truly master this knowledge! I. Error-prone knowledge induction 1. The concept knowledge of opposites: the sum of two opposites is zero. For example, if the distance between points representing two different numbers a+3 and 2a- 1 on the number axis is equal to the origin, then A = _ _ _ _ _. Analysis: Two numbers with the same distance from the origin on the number axis are in opposite directions, so the sum of a+3 and 2a- 1 is zero, and the value of a can be found. Exercise: 1) If the absolute values of a+3 and 2a- 1 are equal, then a = _ _ _ _ _. (Hint: the absolute values are equal, and two numbers may be equal or opposite) 2) a and 2a- 1 opposite. Then their absolute values are _ _ _ _ _ _ _ _ .......................................................................................................................... ............................, at the same time, solving application problems by using equations can also cultivate our ability to analyze and solve problems, so I hope students can pay attention to this part and understand it thoroughly! We know that the key to solving application problems is to establish equations, so when reading problems, we should first find out the unknown and known quantities in the problems, analyze the relationship between quantities and find out the equivalence relationship. Then list the linear equations and solve them. Let's analyze it briefly: 1. Profit rate problem: 1) Keywords: purchase price (or cost), selling price, price tag, discount, profit, etc. 2) Equivalence: a) Profit = selling price-purchase price = list price × discount-purchase price b) Profit rate = profit. Price = list price × discount 3) Typical example: Example 1: A new store is opened, and all the goods are 20% off to attract customers. It is known that if 60 yuan buys a pair of "Shiny First Step" brand leather shoes, the merchant can still make a profit of 40% after 20% discount. What is the price tag of this kind of leather shoes? What's the preferential price? [Resolution] By analyzing the known conditions in the list, the equivalence relation is found: purchase price discount, marked price (preferential price), profit rate 60 yuan 20%, X yuan 80%, X yuan 40%, equivalence relation: profit rate = profit/purchase price solution: if the set price is X yuan, then: x= 105, excellent. Preferential price 84 yuan. Exercise: A store will increase the purchase price of a certain clothing by 40%, then mark the price and sell it at a 20% discount. In this way, every garment is still profitable 15 yuan. What is the purchase price of each piece of clothing? ……