Knowledge point induction
First, the related concepts of the equation
1. Equation: An equation with an unknown number is called an equation.
2. One-dimensional linear equation: contains only one unknown (element) X, and the exponents of the unknown X are all 1 (degree). This equation is called one-dimensional linear equation. For example,1700+50x =1800,2 (x+1.5x) =1
3. Solution of the equation: The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation.
Note: The solution of the (1) equation and the solution of the equation are different concepts. The solution of the equation is essentially the result of the solution, which is a numerical value (or several numerical values), and the meaning of solving the equation refers to the process of finding the solution of the equation or judging that the equation has no solution. (2) The test method of the equation solution is to substitute the unknown value into the left and right sides of the equation to calculate its value, and then compare the values on both sides to draw a conclusion.
Second, the nature of the equation.
Properties of equation (1): Add (or subtract) the same number (or formula) on both sides of the equation, and the results are still equal. It is expressed in the form of a formula: if a=b, then a c = b c
(2): When both sides of the equation are multiplied by the same number or divided by the same number that is not 0, the results are still equal, which is expressed in the form of a formula: if a=b, then ac = bc If a=b(c≠0), then ac=bc.
Third, the law of shift term: moving the symbol on one side of the equation to the other side is called shift term.
Fourth, the rule of removing brackets.
1. The factors outside the brackets are positive numbers, and the symbols of the items after removing the brackets are the same as those of the corresponding items in the original brackets.
2. The factor outside the bracket is negative, and the sign of each item is changed by the sign of the corresponding item in the original bracket after the bracket is removed.
Five, the general steps to solve the equation
1, denominator (least common multiple of denominator on both sides of the equation)
2. Parenthesis deletion (according to the rules of parenthesis deletion and distribution)
3. Move the term (move the term containing the unknown to one side of the equation, and all other terms will be moved to the other side of the equation. Moving the term will change the sign).
4. Merge (transform the equation into ax=b(a≠0))
5. The coefficient is 1 (divide the unknown coefficient a on both sides of the equation to get the solution of equation x=ba).
Sixth, the general steps to solve practical problems with equation thought.
1. Examination: Examination of questions, analysis of what is known and what is sought in questions, and clarification of the relationship between quantity and quantity.
2. Setting: setting unknown (can be divided into direct efforts and indirect efforts)
3. Column: List the equations according to the meaning of the question.
4. Solution: Solve the listed equations.
5. Check: Check whether the solution meets the meaning of the question.
6. Answer: Write the answer (some units should indicate the answer)
Seven, the relationship between common application types and quantity.
1, sum, difference, multiplication and division:
(1) multiplicity relation: it is reflected by the key words "how many times, how many times, how many times, what percentage, growth rate …".
(2) How much relationship: it is reflected by the key words "more, less, harmony, difference, lack, surplus ……".
2. Equal area deformation: "Equal area deformation" is based on the premise that the shape changes but the volume remains unchanged. Common equivalence relations are:
(1) The shape area has changed, but the perimeter has not changed;
② Raw material volume = finished product volume.
3, labor deployment: this kind of problem to find out the number of changes, common problems are:
(1) can be transferred in and out;
(2) Only the transfer-in did not turn out, the transfer-in part changed, and the rest remained unchanged;
(3) Only the transfer-out has not been transferred in, some of the transfers have changed, and the rest remain unchanged.
4. Quantity problem
(1) Need to know the representation method of numbers: the hundredth digit of a three-digit number is A, the tenth digit is B, and the first digit is C (where A, B and C are integers, 1≤a≤9, 0≤b≤9, and 0≤c≤9), then this three-digit number is represented as: 65438.
(2) Some representations in the number problem: the relationship between two consecutive integers, the larger one is larger than the smaller one1; Even numbers are represented by 2n, and continuous even numbers are represented by 2n+2 or 2n-2; Odd numbers are represented by 2n+ 1 or 2n- 1.
5. Engineering problems: The three quantities in engineering problems and their relationships are: total work = working efficiency × working time.
6. Travel problems:
(1) Three basic quantities in the travel problem and their relationships: distance = speed × time.
(2) The basic types are
(1) meeting problems;
(2) follow up the problem; Common ones are: running for opponents; Navigation problems; Circular runway problem.
7, the problem of commodity sales
Correlation: commodity profit = commodity price-commodity purchase price = commodity price × discount rate-commodity purchase price; Commodity profit rate = commodity profit/commodity purchase price; Commodity price = commodity price × discount rate
8. Savings problem
(1) The money deposited by the customer in the bank is called the principal, and the reward paid by the bank to the customer is called interest. Principal and interest are collectively referred to as the sum of principal and interest, the time of deposit in the bank is called the number of periods, and the ratio of interest to principal is called the interest rate. Interest tax is paid at 20% interest.
(2) Interest = principal × interest rate × number of periods
Sum of principal and interest = principal+interest
Interest tax = interest × tax rate (20%)