I. Equation

1. Definition: an equation with an unknown number.

2. The properties of the equation: 1) Add or subtract the same number or the same algebraic expression on both sides of the equation, and the equation remains unchanged.

2) Both sides of the equation are multiplied or divided by the same non-zero number at the same time, and the equation remains unchanged.

Note: When using nature, be sure to pay attention to the simultaneous changes on both sides of the equal sign; We must pay attention to the number 0 when using property 2.

Second, one-dimensional linear equation

1. Definition: An integral equation with unknown number 1

Note: 1) The unknown formula is an algebraic expression; 2) There is only one unknown, and the number of unknown is 1.

2. General steps of one-dimensional linear equation

Matters needing attention in specific methods of deformed names

The denominator is multiplied by the least common multiple of the denominator on both sides of the equation, and items without denominator cannot be omitted;

The numerator is a whole, and brackets should be added after removing the denominator.

To delete the brackets, first delete the brackets, then delete the brackets, and finally delete the braces. Don't omit the items in brackets;

Don't get the symbols wrong.

The shift term moves the term containing the unknown to one side of the equation, and all other terms are moved to the other side of the equation (the shift term needs to change its sign);

Don't lose things.

Merging similar terms transforms the equation into the form of AX = B (A ≠ 0), and its exponent remains unchanged.

Convert the coefficient to 1 and divide it by the unknown coefficient a on both sides of the equation to get the solution of the equation.

Don't confuse numerator and denominator.

Third, the application problem.

1. Several common problems and their characteristics

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Matters needing attention in the equivalence relationship between quantity and formula in type questions

The problems of sum multiple and difference multiple can be understood from the topic, such as the relationship between "multiple" and "more and less"

The perimeter formula and volume formula of equal product problem should distinguish radius and diameter under the condition that perimeter and volume are unchanged.

line

travel

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Problem Encountered Problem Distance = Speed Time

Time = distance speed

Speed = travel time

Fast line spacing+slow line spacing = opposite original spacing. Pay attention to the departure time and place.

Catch up with the problem fast line spacing-slow line spacing = the original distance is the same direction. Pay attention to the departure time and place.

The deployment problem is to find the direction and quantity of the deployment object flow from the quantitative relationship after deployment.

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Matters needing attention in the equivalence relationship between quantity and formula in type questions

The total amount of the proportional distribution problem = the sum of the components, and part of it is set as X.

Engineering problem workload = working efficiency working time

Work efficiency = workload working hours

Working hours = workload and work efficiency

The sum of the workload completed by two or more objects with different work efficiency is equal to the total workload. The total workload is generally set to 1.

Profit rate problem The profit rate of goods =

Commodity profit = commodity price-find out the relationship between profit and profit rate when the commodity is purchased, and sell it at a percentage of the original price.

Let A and B be the single digit and the tenth digit of a two-digit number respectively, then this two-digit number can be expressed as

10 b+ An indirect unknown is set by the problem.

3. General steps to solve application problems with equations:

1) Set an unknown number appropriately.

2) Find the equivalence relation

3) Solving equations

4) Answer according to the actual situation

Note: 1) When setting an unknown number to write an answer, the unit should write clearly.

2) When listing equations, the quantities expressed on both sides of the equation should be the same, and the units of each item should be the same.

3) When looking for the relationship of equality, we should make full use of the conditions given in the problem, and don't omit them, but we can't use the same condition repeatedly, otherwise we will get an identity and not get the solution of the application problem.

4) The solution of the equation depends on its practical significance, and then the solution of the application problem can be determined.

The review outline of binary linear equation contains two unknowns. The equation with one unknown term is called binary linear equation, and the value of a pair of unknowns that make the values on both sides of binary linear equation equal is called a solution of binary linear equation. An equation group consisting of two linear equations and containing two unknowns is called a binary linear equation group. At the same time, the solution of each equation in binary linear equations is called the solution of binary linear equations.

Basic ideas: the steps of solving practical problems by using the equations of binary linear equations (reviewing questions, finding out the known and unknown, and analyzing the quantitative relationship), making plans (considering how to set elements and list equations according to equivalence relations), implementing plans (listing equations and solving them to get answers), and reviewing (checking and reflecting on the scale of solving problems). The main methods and skills are to solve binary linear equations by substitution and addition and subtraction, and to solve simple practical problems by using binary linear equations. Chapter 5: Exponential powers of integers and the basic rules of their operations. The multiplication rule of algebraic expressions is to multiply the coefficients and the monomials with the same base respectively, and the remaining letters, together with their exponents, remain unchanged as polynomials of factorial monomials and products, that is, each term of polynomials is multiplied by the monomials, and then the obtained products are added. Polynomial times polynomial. First, each term of one polynomial is multiplied by each term of another polynomial. Then, the products are added and divided by the division rule of algebraic expression, and the coefficient and the same base power are used as the factors of quotient respectively. For letters contained only in the division formula, they are used as factors of quotient together with its index. Polynomial divided by monomial, first divide each term of this polynomial by this monomial, and then add the obtained quotients.