"Equations" Equations and Inequalities PPT Lesson (Lesson 1: Properties of Equations and Solution Sets of Equations)

Description

"Equations" Equations and Inequalities PPT Lesson (Lesson 1: Properties of Equations and Solution Sets of Equations)

Part One: Learning Objectives

1. Understand and be able to apply the properties of equations. (emphasis)

2. Understand the concept of identity and be able to perform identity transformations. (difficulty)

3． Can find solutions to equations. (emphasis)

core competencies

1. Use the properties of equations to cultivate the quality of logical reasoning.

2. Improve the core competencies of data analysis and mathematical operations by finding the solution set of equations.

Equation PPT, part 2 content: independent preview to explore new knowledge

A preliminary exploration of new knowledge

1. Properties of equations

Properties: (1): If the same number (or algebraic formula) is added (or subtracted) to both sides of the equation at the same time, the equation still holds.

Expressed in letters: if a=b, then for any c, a±c=_____.

Property (2): If both sides of the equation are multiplied (or divided) by the same number (or algebraic expression) at the same time (the divisor or algebraic expression is not 0), the equation still holds.

Expressed in letters: if a=b, then for any c, a×c=_____, a÷c=_____(c≠0).

2. Identity

(1) Generally speaking, for an equation containing letters, if the equation holds true when the letters take any real number, it is called an identity, and it is also called an identity on both sides of the equation. Identity is one of the basis for algebraic transformation.

(2) A frequently used identity: for any x, a, b, (x+a)(x+b)=x2+________+___.

(3) Use the "cross multiplication method" to decompose factors: ① Directly use the formula x2+(a+b)x+ab=(x+a)(x+b) to decompose;

②Use the formula acx2+(ad+bc)x+bd=(ax+b)(cx+d) to decompose.

3． The solution (or root) of an equation is the value of the unknown that makes the left and right sides of the equation equal. The process of finding solutions to equations is called solving equations. The set of all solutions to an equation is called the ______ of the equation.

First try

1. For the following deformations using the properties of equations, the correct one is ()

A． If a=b, then a+c=b-c

B． If a2=3a, then a=3

C. If a=b, then ac=bc

D. If ac=bc, then a=b

2. The following calculation formulas: (1) 3a + 2b = 5ab; (2) 5y2-2y2 = 3; (3) 7a + a = 7a2; (4) 4x2y-2xy2 = 2xy. The correct ones are ()

A． 0 B. 1 C. 2 D． 3

3． It is known that A=x3+6x-9, B=-x3-2x2+4x-6, then 2A-3B is equal to ()

A． -x3＋6x2 B． 5x3＋6x2

C. x3－6x D. -5x3＋6x2

4. The result of factoring x2-4 is ()

A． (x－2)2 B. (x-2)(x+2)

C. (x+2)2 D. (x-4)(x+4)

Equation PPT, the third part of the content: cooperative exploration to improve literacy

Application of properties of equations

[Example 1] It is known that x=y, then the following formulas: ①x-3=y-3; ②4x=6y; ③-2x=-2y; ④xy=1; ⑤x-23=y-23; ⑥xa=ya. Among them, the correct ones are ()

A． ①②③B. ④⑤⑥

C. ①③⑤ D. ②④⑥

regular method

When using the properties of equations in equation transformations, you must pay attention to the fact that the same number multiplied or divided by both sides of the equation must be a non-zero number. In addition, you must also pay attention to the conditions implicit in the equation itself.

Simplification of Identities

[Example 2] Simplify:

(1)(3a-2)-3(a-5);

(2)-3x2y+2x2y+3xy2-2xy2;

(3)2m+(m+n)-2(m+n);

(4)(4a2b-5ab2)+[-2(3a2b-4ab2)]．

[Solution](1)(3a-2)-3(a-5)=3a-2-3a+15=13.

(2)-3x2y+2x2y+3xy2-2xy2=-x2y+xy2.

(3)2m+(m+n)-2(m+n)=2m+m+n-2m-2n=m-n.

(4)(4a2b-5ab2)+[-2(3a2b-4ab2)]=4a2b-5ab2+(-6a2b+8ab2)=4a2b-5ab2-6a2b+8ab2=-2a2b+3ab2.

regular method

When removing brackets, you must first figure out whether the bracket is preceded by a "+" sign or a "-" sign. Secondly, you must pay attention to the word "both" in the rule. Whether you change the sign or not, you must treat them equally, especially the brackets. When it is preceded by a "-" sign, it is easy to make the mistake of changing only the sign of the first item in the brackets while leaving the rest of the items unchanged.

Class summary

1. Make use of the properties of equations to simplify and pay attention to whether the deformation is the same, simplify and complete the transformation, and pay attention to the transformation of symbols.

2. The steps for decomposing factors by the cross multiplication method: transfer terms → product → transform → solve.

3． The solution set of the equation should be written in the form of a set.

Equation PPT, the fourth part of the content: reaching the standard in class and solidifying the double base

1. If 3a=2b, among the following transformations, which one is incorrect ()

A． 3a＋1＝2b＋1B. 3a-1=2b-1

C. 9a=4b D． -a2＝-b3

2. The calculation result of (m+n)-2(m-n) is ()

A． 3n＋2m B. 3n＋m

C. 3n-m D. 3n＋2m

3． The correct solution to the following equation is ()

A． The solution set of x-3=1 is {-2}

B.The solution set of 12x-2x=6 is {-4}

C. The solution set of 3x-4=52(x-3) is {3}

D. The solution set of -13x=2 is -32

4. The solution set of equation 2x-1=0 is ________.

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Update Time: 2024-11-22

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