Content introduction:

Equation is an important content of junior high school algebra, and many practical problems can be solved by listing equations and solving equations. Therefore, we should seriously study the relevant knowledge of the equation.

This chapter first introduces the concept and two properties of equality, and reviews the concepts of solution and equation solution. Then learn to use the properties of equation and the law of shifting term to solve the linear equation of one yuan, and summarize the general steps to solve the linear equation of one yuan; Finally, the column equation is used to solve practical problems. One-dimensional linear equation is the basis of learning other equations and equations.

I. Equality and equality

This part of knowledge focuses on the properties of equality and the transformation of equality by using these two properties; The related concepts of the equation will test whether a number is the solution of the equation.

(1) Key points of knowledge:

1. Equation: An equation that uses an equal sign to express an equal relationship is called an equation. For example:+=, x+y=y+x, V=a3, 3x+5=9 are all called equations. Like A+B a+b, m2n, there is no equal sign, so they are not equations, but algebraic expressions.

2. The nature of the equation:

Properties of equation 1: Add (or subtract) the same number or the same algebraic expression on both sides of the equation, and the result is still an equation.

Property 2 of the equation: The result obtained by multiplying (or dividing) two sides of the equation by the same number (the divisor cannot be 0) is still an equation.

For example, x-5=4, and adding 5 on both sides gives x-5+5=4+5, that is, x=9 is still an equation; Multiplying two sides of this equation, x= 9 x, that is, x= is still an equation, so we use two properties of the equation to solve the equation.

3. Related concepts of the equation:

Equation (1): An equation with an unknown number is called an equation. Such as 5x-4=8, where x is unknown; Another example is 3x-2y=5, where x and y are unknowns.

(2) Unknown number: The unknown number before learning equation is called unknown number. For example, in 5x-4=8, x is unknown, and 5, -4 and 8 are known numbers.

(3) Solution of the equation: the value of the unknown quantity that makes the left and right sides of the equation equal is called the solution of the equation. The solution of an equation containing only one unknown number is also called a root. For example, equation 2x+5=7, when x= 1, the left side of the equation = 2x 1+5 = 7 = right side, so x= 1 is the solution of equation 2x+5=7, or x= 1 is the root of the equation.

(4) Solving the equation: the process of solving the equation.

4. Will test whether a number is the solution of the equation: substitute this number into the left and right sides of the equation to see if the left is equal to the right.

For example, check whether x=5 and x=4 are the solutions of the equation 6x-5=2x+ 1 1.

When x=5, left =6×5-5=30-5=25, right = 2× 5+11+1= 22, ∴.

When x=4, the left =4×6-5=24-5= 19, and the right = 2× 4+11= 8+1=19.

5. The equations will be listed according to the known conditions.

For example, the equation is listed according to the following conditions

(1) A number is four times smaller than it.

(2) Algebraic expression and x+ 1 are opposite.

Solution: (1) Let a certain number be x, then the equation is x=4x-8, or x+8=4x or 4x-x=8.

(2)+x+ 1=0 or =- x- 1.

6. The same solution equation:

(1) Homosolution equation: If two equations have the same solution, they are called homosolution equations. For example, the solution of 2x+3=5 is x= 1,

The solution of 3x+ 15=x+ 17 is also x= 1, so these two equations are the same solution.

(2) The same solution principle of the equation

Same solution principle 1: Add (or subtract) the same number or the same algebraic expression on both sides of the equation, and the obtained equation is the same solution as the original equation.

Principle 2: Multiply (or divide) both sides of the equation by the same number that is not equal to 0, and the obtained equation is the same solution equation of the original equation.

The process of solving the equation is the same process of solving. Solving the equation by using the property of equality mentioned in the textbook is essentially to understand the equation according to its original solution.

(2) Example:

Example 1. Determine whether the following equation holds, and explain the reasons:

( 1)3+5 = 4+4(2)2a+3b(3)x+2y = 5

(4) 3+(-2)=8-|7| (5) x+6=3x-5

A: (1) is not an equation. Because it is an equation without unknowns;

(2) It is not an equation. Because it is not an equation, it is an algebraic expression;

(3)x+2y=5 is an equation, which contains the unknowns x and y.

(4) Not an equation. Because this is an equation with no unknowns.

(5) x+6=3x-5 is an equation, which contains the unknown X.

Note: the concept of an equation has two points: ① it is an equation, ② it contains unknowns, and both are indispensable.

Example 2. Check whether x= is the solution of the following equation:

( 1)5x+2 = 2(2)3x+5 = 6(3)6x+= 4

Solution: (1) When x=, left =5× +2= +2=5 ≠ right,

∴x= not the solution of the original equation.

(2) When x=, left = 3x+5 = 2+5 = 7 ≠ right,

∴x= not the solution of the original equation.

(3) When x=, left =6×+=4+ =4 = right,

∴x= is the solution of the original equation.

Checking whether a number is the solution of an equation can be used to verify whether our process of solving the equation is correct.

Example 3. List the equations according to the following conditions

(1) 8 times a number minus 5 equals 4 times 3;

(2) A certain number is larger than it by 7;

(3) The square of the sum of a number and 3 is greater than its square by 4;

(4) Three times the difference between a certain number and 5 is equal to 33;

(5) The sum of a certain number and -7 is the antonym of the sum of a certain number;

(6) The square of a number is 8 times larger than its own square.

Solution: Let a certain number be x, then the equation listed according to the conditions is:

(1) 8x-5=4x+3 (2) x- x=7 or x= x+7.

(3) (x+3)2-x2=4 (4) 3(x-5)=33

(5) (x-7)+(x+ )=0 (6) x2=2x+8

Example 4. Name the basis of the following deformation:

( 1) 2x-5=3，2x=8

(2) 3x=27，x=9

(3) -3x=，x=-

(4) - x=4，x=- 12

(5) =2，x+3= 10

(6) =x+6，x-2=3x+ 18

Solution: (1) According to the basic properties of equation 1, 2x-5+5=3+5 and 2x=8 are obtained.

(2) According to the basic properties of Equation 2, 3x× = 27x and x=9.

(3) According to the basic property 2 of the equation, -3x×(- )= ×(-), and x=-

(4) According to the basic property 2 of the equation, -x×(-3)=4×(-3), x=- 12 is obtained.

(5) According to the basic property of the equation 2,5× () = 2× 5, x+3= 10 is obtained.

(6) According to the basic property of Equation 2, 3× () = 3× (x+6), x-2=3x+ 18 is obtained.

Attention: ① When applying the principle of the same solution of the equation, we should pay attention to the same change on both sides of the equation at the same time, and don't just pay attention to one side and forget the other. (2) When one side of the equation is a polynomial, we should pay attention to the use of the distribution law to avoid such mistakes: (6) The term =x+6 is multiplied by 3 to get x-2=3x+6.

Example 5. Given that x=-4 is the solution of equation 2x+3|a|=x- 1, find the value of a. ..

Analysis: It is known that x=-4 is the solution of the equation, so put x=-4 into the equation, and the left and right sides are equal, so there is 2×(-4)+3|a|=-4- 1, which is an equation about |a|, so we can find out the value of |a| and further determine the value of a.

Solution: ∫x =-4 is the solution of equation 2x+3|a|=x- 1

∴ 2×(-4)+3|a|=-4- 1，

∴ -8+3|a|=-5，

From the basic properties of the equation 1: -8+8+3|a|=-5+8,

That is, 3|a|=3,

According to the basic properties of equation 2:| a | = 1, ∴ a = 1.