The so-called overall idea is to consider the problem from the whole, and regard a set of numbers or an algebraic formula as a whole when solving problems. By studying the overall form, overall structure, and the internal connection between the whole and the local part of the problem, we can obtain the problem-solving methods. Using this method of thinking can often simplify the complex and make the difficult and easy. Now we will give examples to illustrate the application of the overall idea in solving systems of equations for reference when learning.
Type 1 Use the overall idea to solve the equation system
1, substitute the whole into
Comment: The trick of this question is to substitute (x+y) as the whole into directly find the value of y, and then obtain the value of x.
2, overall addition and subtraction
Comment: In view of the coefficient characteristics in this question, the two equations are cleverly added (subtracted) in the whole, and then quickly solved.
3, overall conversion
Comment: If you expand this equation system without brackets , there will be a large amount of calculation. Through observation, we can find that the brackets in the two equations in the original system of equations are the same, which can be regarded as a whole element solution, and then the values of x and y are found.
4, overall eliminates the yuan
Comment: Treat a certain algebraic formula as a "whole", substituting it as a whole or adding or subtracting it as a whole, so as to achieve the purpose of eliminating the yuan. Generally speaking, when both systems of equations contain the same algebraic formula, this algebraic formula is often regarded as a whole. To simplify operations.
5, Overall superposition
Comment: When unknown numbers in the system of equations have rotational symmetry, the equations in the system of equations can be added and subtracted directly, which is the so-called overall superposition method .
Type 2 Use the overall idea to evaluate
Type 3. Use the overall idea to solve the actual
8. (2018 Spring· Zizhong County mid-term) Problems: There are three commodities: A, B and C. ① Purchase 3 pieces A, 5 pieces B, and 7 pieces C. It costs 490 yuan in total; ② Purchase 4 pieces A, 7 pieces B, and 10 pieces C. It costs 690 yuan in total; ③ Purchase 2 pieces A, 3 pieces B, and 1 piece C. How much does it cost to purchase one item of three products, A, B and C?
Xiao Ming said: "You can list the three-element equations of according to three conditions, and find out how much it costs to purchase one piece A, B, and C, and then add it to get the answer." After some thought, Xiaoli said: "This question can remove the conditions ③, and only use ①②, and you can still get the answer." Regarding Xiaoli's speech, the students had a warm discussion.
(1) Please follow Xiao Ming’s idea to solve the problem.
(2) Is Xiaoli’s statement correct? If it is correct, please complete the answering process; if it is incorrect, please explain the reason.
(3) Please solve the following problems based on the experience accumulated in the above-mentioned problem solving: the school purchases four types of teaching aids A, B, C, and D. The first time you buy 1 A teaching aid, 3 B teaching aids, 4 C teaching aids, and 5 D teaching aids, a total of 2,018 yuan will be spent; the second time you buy 1 A teaching aid, 5 B teaching aids, 7 C teaching aids, and 9 D teaching aids, a total of 3,036 yuan will be spent. How much does it cost to purchase 5 pieces of A teaching aids, 3 pieces of B teaching aids, 2 pieces of C teaching aids, and 1 piece of D teaching aids?
Comment: The solution to this question uses the overall structure of the thinking method, which looks ingenious and incompetent, reflecting the agility, flexibility and leapiness of the overall thinking method.
