How to find equivalence when solving application problems with equations? When solving application problems, we often find out the equal relationship between quantity and quantity in application problems, which is usually called "equal relationship", and then solve the equations. Here is an example. (1) Equivalence relation and equation of simple application problem with only three quantities. A simple application problem with only three quantities. Given two quantities, find the third quantity. The equivalence relation of this kind of application problems is obvious and easy to find out. According to the equivalence relation between three quantities, three equations can often be listed. Among these three equations, one can be chosen as the equation to solve the problem. It is customary to put the unknown number on the left of the equal sign and indicate it with the letter X. For example, 1: Soybean and mung bean * * * weigh 90kg, of which 65kg is soybean and how much is mung bean? Analysis: According to the three quantities in this question, the following three equations can be listed: ① * * * weight 90kg- soybean 65kg = mung bean weight; ② mung bean weight +65kg soybean = * * * 90kg③* * * 90kg mung bean weight = 65kg soybean. If the unknown quantity is represented by X and placed to the left of the equal sign, the equation can be listed: x+65=90 or 90-x=65. Because the topic says "soybeans and mung beans weigh 90kg", it is better to list the equation as "x+65=90". Example 2: Xiao Xia's height 158 cm is higher than Xiao Yong's 13 cm. What is the height of Xiao Yong? Analysis: According to the three quantities in this question, the following three equations can be listed: ① The height of Xia is 158cm- 13cm = the height of Xiao Yong; ② Xiao Xia height158cm-Xiao Yong height13cm; ③ Xiao Yong height+13cm = Xiao Xia height 158cm. If the unknown quantity is represented by X, according to the topic "Xiao Xia's height is 158 cm, taller than Xiao Yong 13 cm", the following equation can be listed: 158-x= 13 or x+13 =/kloc-0. Analysis: According to the commonly used quantitative relationship between speed, time and distance, the following three equations can be written: ① 45 km/h× hour = distance 270 km; ② The distance is 270 km/h = 45 km/h; ③ Distance 270km/ hour = 45km/ hour. If it takes x hours to complete the journey, the equation can be listed according to the meaning of the question: 45x=270 or 270÷x=45 Example 4: The area of a rectangle is 2800 square centimeters, the length is 70 centimeters, and the width is how many centimeters? Analysis: the equivalent relationship between area and volume calculation is the calculation formula of area and volume. The problem is the area of a rectangle. According to the formula for calculating the rectangular area, the following three equations can be written: ① Length× width = rectangular area; ② Rectangular area ÷ length = width; ③ Rectangular area ÷ width = length. If the width of the rectangle is x cm, the equation can be listed according to the meaning of the question: 70x=2800. In short, when looking for equivalence relation and column equation, it is mainly based on the meaning of four operations and the quantitative relationship between application problems and column equation. But equation solution and arithmetic solution are different in solving problems. Arithmetic solution, in order to find out the unknown number, it is necessary to analyze the known number intensively, find out the relationship between the unknown number and the known number, form an expression with the known number and the operation symbol, and find out the unknown number through calculation. With regard to solving application problems with column equations, we can use letters to represent unknowns, such as x and y, so that the unknown x and the known number are in the same position and directly participate in column operations according to the equivalence relationship of the three quantities in the problem. Some problems that need "inverse solution" in arithmetic are often easier to be solved by equations. (2) Equivalence relations and equations of application problems with more than three numbers. When you encounter more than three application problems, you should carefully examine the meaning of the problems and find out what the problems are talking about, so as to analyze the relationship between known quantities and unknown quantities and list the equations. Example 1: It takes 365 days for the earth to go around the sun, which is four times longer than that of Mercury 13. How many days does it take for Mercury to go around the sun? Analysis: Because the unknown number (x) and the known number can be placed in the same position to directly participate in the column operation by solving the application problem with the column equation, the conditions described in the problem can be appropriately changed. The problem can be said as follows: four times the time (x) required for Mercury to orbit the sun plus 13 days equals 365 days. In this way, the following equation can be listed: 4x+ 13 = 365. This problem can also be said as follows: 365 days MINUS 4 times the time (x) required for Mercury to orbit the sun is equal to 13 days. In this way, the following equation can be listed: 365-4x= 13. This problem can also be said as follows: 365 days MINUS 3 days is equal to 4 times the time (x) required for Mercury to orbit the sun once. We write the unknown number (x) to the left of the equal sign, and we can get the equation: 4x=365- 13. The three different equations mentioned above are all equations for solving this application problem, so you can use any one when solving this problem. Example 2: The school bought five basketballs and seven volleyballs, which cost 355 yuan. Suppose the price of each basketball is 36 yuan, what is the price of each volleyball? Analysis: this problem, if solved by arithmetic, is the topic of "inverse solution"; If the problem is solved by equation method, it is easier to find out the equivalence relationship according to the known conditions in the topic. It is understood that the price of each basketball is 36 yuan. If the price of each volleyball is X yuan, the equation can be listed as follows: 7x+36x5 = 355 boxes. 3. Students in Grade 5 and Grade 6 of Liuchangdi Primary School planted 150 trees this year, and the number of trees planted in Grade 6 was twice that of Grade 5. How many trees have been planted in each grade? Analysis: This problem is a common and typical application problem, and it is usually called "and double problem". If you use arithmetic to solve it, it is regular. That is, the sum of two numbers ÷ (multiple+1)= 1 multiple. By using the equation method, the equivalence relation can be written directly in the order of describing the known conditions in the topic. For the convenience of calculation, we often set the number of "1 copy (1 time)" as X. In this question, if the number of trees planted in grade five is X, then the number of trees planted in grade six is 2x. The listed equation is: x+2x= 150. Example 4: The highway between Town A and Town B is 2 16km long. Two cars, A and B, start from two towns at the same time and meet three hours later. The speed of car A is 38 kilometers per hour, and how many kilometers per hour is car B? Analysis: Two cars, A and B, left two towns at the same time and met three hours later, indicating that the three-hour journey of A car+the three-hour journey of B car = the length of the road between the two towns. Let car B travel at the speed of x kilometers per hour, and the equation can be listed: 38×3+3x=2 16. This problem can also be listed according to the following equivalent relationship, that is, the length of the road between two towns-3 hours' journey of B car = 3 hours' journey of A car. The equation can be listed as follows: 2 16-3x=38×3 Two cars A and B leave at the same time and drive in opposite directions. Then, the distance traveled by two cars per hour is the sum of the speeds of two cars A and B, so we can write an equivalence relation, that is, the sum of the speeds of A and B × time = the length of the road between two towns. The equation can be listed: (38+x) × 3 = 2 16.