"Equations" PPT lesson on equations and inequalities (the solution set of the system of equations in lesson 3)

Description

"Equations" PPT lesson on equations and inequalities (the solution set of the system of equations in lesson 3)

Part One: Learning Objectives

1. Understand the definition and representation method of the solution set of a system of equations. (difficulty)

2. Master the method of finding solutions to systems of equations using the elimination method. (emphasis)

3． Can use the knowledge of equation systems to solve some simple practical problems. (main difficulty)

core competencies

1. Cultivate mathematical abstract literacy by understanding the definition of a system of equations.

2. Improve the subject literacy of data analysis and mathematical operations by finding the solution set of the system of equations.

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

A preliminary exploration of new knowledge

1. solution set of a system of equations

Generally speaking, by combining multiple equations, a system of equations can be obtained. In a system of equations, the ______ obtained from the solution set of each equation is called the solution set of this system of equations.

2. The basis for finding the solution set of the system of equations is the properties of the equations, etc. The commonly used method is the ______ method.

3． The solution set of a system of linear (quadratic) equations of two variables is expressed as {(x, y)|(a, b),...}, where a and b are definite real numbers, and the solution set of a system of linear equations of three variables The expression method is {(x, y, z)|(a, b, c),...}, where a, b, c are definite real numbers.

First try

1. When using the substitution method to solve the system of equations y=1-xx-2y=4, the correct substitution is ()

A． x-2-x＝4　B． x-2-2x=4

C. x-2+2x=4 D. x-2+x=4

2. It is known that the system of linear equations of two variables 2x+y=7, x+2y=8, and the solution set is ()

A． {(x,y)|(2,3)} B． {(x,y)|(3,2)}

C. {(x,y)|(-2,3)} D． {(x,y)|(-2,-3)}

3． It is known that A={(x, y)|x+y=5}, B={(x, y)|2x-y=4}, then A∩B=()

A． {(x,y)|(1,4)} B． {(x,y)|(2,3)}

C. {(x,y)|(3,2)} D. {(x,y)|(4,1)}

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

Solution set of linear equations in two variables

【Example 1】Find the solution set of the following system of equations.

(1)x+y=4, ①2x-3y=3.②

(2)3x-7y=-1, ①3x+7y=13.②

[Solution] (1) From ①, we get y=4-x.③

Substituting ③ into ②, we get 2x-3(4-x)=3.

Solving this equation, we get x=3.

Substituting x=3 into ③, we get y=1.

Therefore, the solution set of the original system of equations is {(x, y)|(3,1)}.

(2) Method 1: ①＋②, we get 6x=12, so x=2.

Substituting x=2 into ②, we get 3×2+7y=13, so y=1.

Therefore, the solution set of the original system of equations is {(x, y)|(2,1)}.

Method 2: ①-②, get -14y=-14, so y=1.

Substituting y=1 into ①, we get, 3x-7×1=-1, so x=2.

So the solution set of the original system of equations is {(x, y)|(2,1)}.

regular method

Commonly used methods to find the solution set of a system of linear equations of two variables include addition, subtraction, elimination, and substitution and elimination. You must be able to select the appropriate method according to the characteristics of the system of equations to be solved, and pay attention to the representation form of the solution set.

Solution set of linear equations in three variables

[Example 2] Find the solution set of the following system of equations.

(1)x+y+z=12, ①x+2y+5z=22, ②x=4y.③

(2)2x+y+3z=11, ①3x+2y-2z=11, ②4x-3y-2z=4.③

[Solution] (1) Method 1: Substituting ③ into ①② respectively, we get

5y+z=12, 6y+5z=22, the solution is y=2, z=2,

Substituting y=2 into ③, we get x=8.

Therefore, the solution set of the original system of equations is {(x, y, z)|(8,2,2)}.

Method 2: ②-①, get y+4z=10, ④

②-③, get 6y+5z=22, ⑤

Combining ④⑤, we get y+4z=10, 6y+5z=22, and we get y=2, z=2,

Substituting y=2 into ③, we get x=8.

Therefore, the solution set of the original system of equations is {(x, y, z)|(8,2,2)}.

regular method

The basic idea of ​​finding the solution set of a system of linear equations of three variables is: eliminate elements through "substitution" or "addition and subtraction", convert "ternary" into "binary", and convert the system of linear equations of three variables into a linear equation of two variables. group, and then converted into a linear equation of one variable to solve.

Find the analytical formula of the function using the method of undetermined coefficients

[Example 3] It is known that the image of the quadratic function y=ax2+bx+c passes through the points (-1,2), (2,8), (5,158). Find the analytical formula of this quadratic function.

[Ideas] Consider a, b, and c as three unknown numbers, and substitute three sets of known values ​​of x and y respectively to get a three-dimensional linear equation system. Solve this equation system You can find the values ​​of a, b, c.

Class summary

1. Commonly used methods to find the solution set of a system of linear equations of two variables include addition, subtraction, elimination, and substitution and elimination. You must be able to select the appropriate method according to the characteristics of the system of equations to be solved, and pay attention to the expression form of the solution set.

2. The undetermined coefficient method finds the analytical formula of the function. The way to solve this kind of problem is to form a system of equations based on the coordinates of the points on the image, solve the system of equations to obtain the values ​​of the letter coefficients, and then determine the analytical formula of the function being sought.

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

1. The solution set of the system of linear equations of two variables x+3y=7, y-x=1 is ()

A． {(x,y)|(1,2)}B． {(x,y)|(1,0)}

C. {(x,y)|(-1,2)} D． {(x,y)|(1,-2)}

2. When finding the solution set of the equation system x+y-z=11, x+z=5, x-y+2z=1, to make the operation simple, the elimination method should be ()

A． Eliminate x first B. Eliminate y first

C. Eliminate z first D. None of the above statements are correct

3． There are three cups A, B, and C on the table. The three cups originally contained some water. First pour all the water from cup A into cup C. At this time, the amount of water in cup C is 40 ml more than twice the original amount of water in cup A. Then pour all the water from cup B into cup C. At this time, the amount of water in cup C is The original amount of water in cup B is 3 times less than 180 ml. If the water does not overflow during the process, the original amount of water in the two cups A and B is different ()

A． 80 ml B. 110ml

C. 140 ml D． 220ml

4. Design a system of quadratic equations of two variables so that the solutions of this system of quadratic equations of two variables are x=2, y=3 and x=-3, y=-2. Try to write a system of equations that meets the requirements ________.

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Update Time: 2024-07-04

This template belongs to Mathematics courseware People's Education High School Mathematics Edition B Compulsory Course 1 industry PPT template

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