Generally speaking, in the test of mathematical operation, we divide the frequently tested knowledge points into multiple problems, such as common "trip problem", "engineering problem" and "inclusion problem" ... Equation method has no fixed problem-solving object. Generally speaking, as long as there is an equivalence relationship in the stem of the problem and a quantitative relationship between the unknowns, we can list the problem of solving the equation with the idea of constructing the equation. Let's look at the application of equation method in various questions.
Example 1 The total age of mother, sister and younger sister is 64 years old. When my mother is three times as old as my sister, my sister is six years old. When my sister is twice as old as my sister, my mother is 34 years old. How old is my mother now?
The analysis of this topic is a question of age. In the process of solving the age problem, we often use the divisibility method and the equation method. In the column equation, the most obvious equivalent relationship in the age problem is that the age difference is equal. In this problem, we can find two groups of equal relationships through analysis: the age difference between mother and sister, and the age difference between sister and sister. With the equal age difference between these two parts, we can list the equations to solve.
By making the following table, we can clearly find the quantitative relationship of this problem:
In this problem, we can obviously get two equations: 3x-x = 34-2y; X-6=2y-y and y=4 can be easily solved by these two equations. As can be seen in the third line, when the mother was 34 years old, her sister was 8 years old and her sister was 4 years old, and the total age was 46 years old. The difference between her and 64 years old was 18 years old, so no one was 6 years old. Now the mother is 34+6=40 years old.
Equation method is not only suitable for problems without specific methods, but also for problems with fixed solutions, such as summation for maximum value. For example, in solving and determining extreme value problems, we often construct arithmetic progression to solve the common problems in interpretation, but when the questions are complex, it is difficult to solve them with common methods, and equation method can easily solve this complex problem.
Example 2 Students in seven classes of a grade planted 304 trees. It is known that at least 20 trees are planted in each class, and we don't want to wait for trees. According to the number of people, they are classified as 1 class to class 7. It is also known that the number of species of 1 is the sum of 2 and 3 species, and the number of species of 2 species is the sum of 4 and 5 species. So how many trees can Class Three plant at most?
The analysis requires that Class Three should plant as many trees as possible, so other classes should plant as few trees as possible, so Class Six and Class Seven should plant 20 trees and 2 1 tree respectively. Suppose three classes of X trees, then two classes of x+ 1 tree, one class of 2x+ 1 tree, four classes of x+ 1 tree, and five classes of X+1tree, so the equation 2x+1tree is obtained. Substituting 4 and 5 kinds of planted trees can meet the requirements of the problem.