First, the teaching content and requirements of this lecture

Understand the concepts of binary linear equations, binary linear equations and their solutions.

Will check whether a pair of values is the solution of a binary linear equation.

Flexible use of method of substitution to solve binary linear equations.

Understand the thinking method of solving binary linear equations with method of substitution.

Second, the focus, difficulty and key of this lecture:

1. Emphasis: Solutions to linear equations-substitution and addition and subtraction.

2. Difficulties: Choose a reasonable and simple method to solve binary linear equations.

3. Key points: Understand the thinking method of "elimination method" and try to eliminate an unknown in the equation, and transform "binary" into "unitary".

Flexible use of "substitution method" and "addition and subtraction method"

Third, this lecture focuses on important mathematical ideas:

1. We can understand the idea of transforming multivariate equations into univariate equations by learning to solve equations once.

2. Infiltrate method of substitution Thought in the training of solving complex equations.

Four, the main mathematical ability:

1. Through the training of solving binary linear equation by substitution elimination method and solving binary linear equation by addition and subtraction elimination method, the reasonable and simple method is selected to solve the equation, and the operation ability is cultivated.

2. By observing and analyzing the characteristics of unknown coefficients in the equation, it is clear that the main idea of binary linear equations is "elimination", thus promoting the transformation from unknown to known, cultivating observation ability and developing logical thinking ability.

Fifth, the idea of conversion:

The mathematical idea of "solving a problem is to reduce the practice to a solved problem" is called "transformation", which embodies that "under certain conditions, different things can be transformed into each other." Dialectical materialism is the guiding light to solve mathematical problems.

In this chapter, the outstanding application of the thought of "conversion" is as follows:

1. Turn unfamiliar into familiar. "Turn binary into binary" and "Turn ternary into binary". That is, the unfamiliar binary linear equations are transformed into familiar one-dimensional linear equations to solve. This kind of turning unfamiliar problems into familiar ones is a mathematical thought with universal guiding significance in solving mathematical problems. We should deeply understand and consciously apply it to the study of mathematics.

2. keep it simple. When solving equations, it is often difficult or inconvenient to directly eliminate binary linear equations with complex forms. Generally, it should be transformed into a simple equation and then solved by elimination method.

3. Turn practical problems into mathematical problems. When using the knowledge of linear equations to solve related application problems, the analysis method and solving steps are similar to enumerating linear equations to solve application problems. By carefully analyzing the relationship between the unknown quantity and the known quantity in the topic, find out their equal relationship and list the equations accordingly. The application problem is transformed into the problem of solving equations. Turning practical problems into mathematical problems is the basic way to solve practical problems by using mathematical knowledge.

Six, error-prone analysis:

1. To judge whether an equation is a binary linear equation, it is generally necessary to change the equation into a general form and then judge it according to the definition.

2. Solutions of binary linear equations: A binary linear equation group has countless solutions, each of which is a pair of numerical values. Solution of binary linear equation: If the unknowns in the equation are X and Y, you can take some values of X at will, and you can calculate the corresponding value of Y, so that you can get the number pairs that meet your own needs.

3. Binary linear equations: When two binary linear equations are merged, a binary linear equation group is formed. As a system of binary linear equations, two equations do not necessarily contain two unknowns, but one of them is a linear equation with one variable and the other is a linear equation with two variables.

4. Solution of binary linear equations: The values of two unknowns that make the left and right sides of two equations of binary linear equations equal are called the solutions of binary linear equations. The way to test whether a pair of values is the solution of binary linear equations is to substitute two unknowns into two equations of the equations respectively. If both equations can be satisfied, then it is the solution of the equations.

5. Matters needing attention in solving equations by substitution method:

(1) The equation after deformation cannot be substituted into the equation before deformation.

(2) Simplify the process of solving the equation by substitution method, that is, select the equation with smaller coefficient for deformation.

(3) Judge whether the result is correct.

6. Understanding the solution of binary linear equations;

The solution of (1) equations refers to the common * * * solution of each equation in the equations.

(2) The meaning of "public * * * solution" actually includes the following two aspects:

(1) Because any binary linear equation has countless solutions, the solution of the equation must be the solution of one of the equations.

(2) Moreover, this solution must satisfy any equation at the same time, so the solution of the binary linear equation group must satisfy either of the two equations in this equation group at the same time.

For example 1, it is known that the equation 3xm+3-2y 1-2n= 15 is a binary linear equation, and the values of m and n are found.

Analysis: the binary linear equation must be an integral equation that satisfies the following two conditions at the same time: ① There are two unknowns in the equation; ② The number of terms with unknowns in the equation is 1.

Solution: From the meaning of the question, m+3= 1, 1-2n= 1.

∴ m=-2，n=0

Example 2. Which of the following equations is a binary linear equation?

( 1) (2) (3) (4) (5)

Analysis: From the definition of binary linear equations, we can know: ① Every equation in the equations must be a linear equation; ② There are two unknowns in the equation * * *; ③ Two equations in the equation set must be integral equations, and the equation set (1) contains three unknowns; (2) where Xy=2 is a binary quadratic equation;

+y=6 in (5) is not an integral equation.

Solution: (3) and (4) are binary linear equations.

Example 3, the solution of the equation is ()

(A) (B) (C) (D) None of the above answers are correct.

Analysis: A pair of values of the unknowns X and Y must satisfy each equation of the known equations at the same time, which is the solution of the equations.

Solution: Substitute X =-2 and Y = 2 into Equation ①,

Left =3×(-2)+4×2=2= right,

And then substitute into equation ②,

Left =2×(-2)-2=-6, right =5.

∵ Left ≠ Right.

∴ (A) satisfies equation ①, but does not satisfy equation ②, so it is not the solution of the original equations.

Similarly, (b) satisfies both equation 1 and equation 2, so it is the solution of the original equations; And (c) satisfies equation ② but does not satisfy equation ①, so it is not the solution of the equation.

Answer option B.

Example 4. Given the solution of equation 3x-ay-2a=3, find the value of a. ..

Analysis: Because it is the solution of equation 3x-ay-2a=3, it can be understood that the values of x and y are applicable to equation 3x-ay-2a=3, that is to say, the equation holds when x takes -2 in equation 3x-ay-2a=3. In this way, x=-2 and y= can be substituted into the equation and transformed into a linear equation about A, and the value of A can be obtained.

Solution: ∫x =-2, y= is the solution of equation 3x-ay-2a=3,

∴ 3(-2)-a( )-2a=3

∴ -6- -2a=3，∴ - a=9，∴ a=-

Example 5, Solving Equation

Analysis: When solving binary linear equations by substitution method, we should try to select the equation with the absolute value of unknown coefficient of 1 for deformation. In this case, the coefficient of formula Y is-1, so we use the algebraic expression containing X to represent Y and substitute it into ① to eliminate Y..

Solution: y=5x-3 ③ From ②.

Substituting ③ into ① gives 2x+3(5x-3)=-9,

17x=0，x=0

Substitute x=0 into ③ to get y=-3.

∴

Example 6, Solving Equation

Analysis: Because the coefficient of X in both equations is 2, Equation ② can be directly substituted into Equation ① without writing x=.

Solution: Substitute ② into ① to get 3y+ 1-4y=3.

∴ y=-2

Substitute y=-2 into ② to get 2x = 3x (-2)+ 1.

∴ x=-2 ∴

Note: This problem can also be solved by substituting 2x=4y+3 from ① into ②. From the solution of this problem, we can see that adopting flexible methods according to the specific characteristics of the problem will simplify the problem.

Example 7, Solving Equation

Analysis: these two equations need to be arranged in a standard form, which helps to determine which unknown to eliminate.

Solution: sort out the original equation and get

From ④, y=3x-4 (5)

Substituting ⑤ into ⑤ gives 3x-2(3x-4)=2.

x=2

Substituting x=2 into ⑤ gives y=3×2-4=2.

∴

Exercise:

Fill in the blanks:

(1) It is known that the equation 2x2n-1-3y3m-n+1= 0 is a binary linear equation m = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

(2) Equation ① y = 3x2-x; ②3x+y = 1； ③2x+4z = 5z； ④xy = 1； ⑤+y = 0； ⑥x+y+z = 1； All landowners+x = 4, with _ _ _ _ _ _ _ _.

(3) The binary linear equation x- y=5 has _ _ _ _ _ _ solutions.

(4) Solving binary linear equations with method of substitution.

The simplest way is to use _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

Answer:

(1), 1 (2) 235 (3) Infinitely many (4)①, x, x=6-5y, ②.

test

Multiple choice

1. The solution of the equation 5x-3y=6 is ().

A, there is only one B, only two C's, countless D's, and there is no solution.

2. Equation x+2y-3=0, where both X and Y are non-negative integers, then the values of X and Y are () respectively.

A, 3,0 b, 1, 2 C, 1, 1 D, 1, 1 or 3,0.

3. Equation 3a+b=9, and the number of solutions in the positive integer range is ().

A, 1 B, 2 C, 3 D, countless.

4. If it is a set of solutions of equation 2x=3y+2k, then the value of k is ().

a、4 B、2 C、D、

5. For the equation 3x-5=7y, y is represented by an algebraic expression containing x, which is ().

a、3x=7y+5 B、y= C、7y=3x-5 D、

6. Which of the following arrays is the solution of equation 7x+2y= 19 ()?

A, B, C, D,

7. Among the following equations, the one that is not a binary linear system is ().

A, B, C, D,

8. In the four pairs of numerical values, the solution of the equations is ().

A, B, C, D,

9. If (where b≠0) is the solution of equation 5x+y=0, then ().

A, A and B must have the same symbol, and B, A and B must have different symbols.

C, A and B can have the same number, or they can have different numbers D, B ≠ 0, and A = 0.

10. Which of the following equations is a binary linear equation ()?

① ② ③ ④

a、①②③ B、②③ C、③④ D、①②

Answer and analysis

Answer: 1. C2 . D3 . B4 . b5 . B6 . a7 . D8 . d9 . b 10 . c

Analysis of senior high school entrance examination

Binary linear equations

Test point scanning:

1. Understand the concepts of binary linear equations, binary linear equations and their solutions,

2. It will be checked whether a pair of values is the solution of a binary linear equation.

The famous teacher said:

1. Binary linear equation: contains two unknowns, and the degree of the unknowns is 1. Such an equation is called an unary quadratic equation. Its standard form is: ax+by=c(a, b≠0).

2. Solution of binary linear equation: A pair of unknowns that make the left and right sides of binary linear equation equal is called the solution of binary linear equation. A binary linear equation has countless solutions. Generally speaking, given the value of an unknown quantity in an equation, the corresponding value of another unknown quantity can be found, so this logarithm is the solution of a binary linear equation.

3. Binary linear equations: When two binary linear equations are merged, a binary linear equation group is formed. Its general form is:

As two equations in binary linear equations, they don't necessarily contain two unknowns, but one of them is a univariate linear equation group and the other is a binary linear equation group.

4. Solutions of binary linear equations

The common solution of two binary linear equations is called the solution of binary linear equations. The way to test whether a pair of numerical values are the solutions of binary linear equations is to substitute two unknowns into two equations respectively. If both equations can be satisfied, it is the solution of the equation.

Note: This section is related to the concepts of binary linear equation and binary linear equation. Generally, there is no separate proposition in the senior high school entrance examination.

Solving Binary Linear Equations with method of substitution

Test point scanning:

1. Master the steps of solving binary linear equations with method of substitution.

2. Understand the basic idea of solving binary linear equations with method of substitution.

The famous teacher said:

1. method of substitution's basic idea of solving binary linear equations is "elimination", that is, "binary" is transformed into "unitary". This method embodies the basic idea of "simplifying the complex" in mathematics.

2. The steps of solving binary linear equations by substitution method are as follows:

(1) Choose an equation with a relatively simple coefficient from the equations, and use an algebraic expression containing another unknown to represent an unknown in this equation;

(2) Substituting the deformed equation into another equation to obtain a linear equation;

(3) Solve this one-dimensional linear equation and find an unknown value;

(4) Substitute the obtained value of an unknown into the formula (1) to obtain the value of another unknown, thus obtaining the solution of the equations;

3. Matters needing attention in solving equations by substitution method:

(1) The equation after deformation cannot be substituted into the equation before deformation.

(2) If one of the four coefficients of X and Y in the two equations is 1 or-1, the equation is selected for deformation, and the unknown quantity with the coefficient of 1 or-1 is represented by another unknown quantity, which can simplify the solution process of the equation.

(3) Check whether the obtained results are correct.

Typical examples of senior high school entrance examination

1. (Changsha, Hunan) Solve binary linear equations;

Test site: the solution of binary linear equations

Comment: In the second equation of this question, the coefficient of X is 1, and the coefficient of Y is-1. It's very simple. Transform the second equation, use X for Y, or use Y for X. The specific process of solving the problem is as follows.

Solution:

X=y-5 (3) from (2)

Substituting (3) into (1) gives 2y- 10+3y=40.

The answer is y= 10.

Substitute y= 10 into (3) to get x=5.

∴

Special training for real questions:

1. (Fuzhou, Fujian) If a:b=3: 1 and a+b=8, then A-B = _ _ _ _ _ _

2. The solution of (Jiangxi) equations _ _ _ _.

Answer:

1.4

Solution: Let A = 3x and B = x..

Then a+b = 3x+x = 8.

∫x = 2

∴ a=6 b=2

a-b=4

2. The process of solving the problem is as follows: starting from ①, X = 5-Y ③, and substituting ③ into ②, 5-Y-2Y =- 1.

Solution: y = 2. If y = 2 is substituted into ③, X = 5-2 = 3ⅷ Ⅷ.

Extracurricular development

Interesting talk about indefinite equation

First, from "one hundred dollars to buy one hundred chickens"

At the end of the 5th century, Zhang Qiujian, a mathematician in China, wrote a book "Calculation Classics", which has a famous title in the history of world mathematics: "Hundred Money for a Hundred Chickens";

A rooster, 5 yuan, a hen, 3 yuan, three chickens, 1 yuan, want to buy 100 chickens with 100 yuan. How many chickens do you want to buy?

If you buy X cocks, Y hens and Z chickens, then you can get the equation.

(2) × 3-( 1), that is, 14x+8y=200.

That is 7x+4y= 100..........③.

① and ② two equations form a ternary equation group, which is transformed into a binary equation ③ after elimination. In equation ③, if x takes a value, a value of y corresponding to it can be found. take for example

If x=-2, then y = 28.5.

If x=0, then y = 25.

Take x= 1.6, then y = 22.2.

If x=4, then y =18;

……

Because ③ is a binary linear equation with numerous solutions, the original equation group also has numerous solutions. take for example

∫z = 100-x-y from (1),

∴ If x=-2, then y=28.5, thus z = 73.5.

If x=0, y=25, and thus z = 75.

Take x= 1.6, then y=22.2, thus z = 76.2.

If x=4, then y= 18, thus z = 78.

……

Generally speaking, there are more unknowns than equations, so its solution is uncertain, so this kind of equation (group) is called indefinite equation (group). But under certain conditions, the indefinite equation (group) can sometimes find its definite solution, such as the problem of "one hundred chickens for one hundred dollars", because the number of chickens is a non-negative integer, so there are four groups of solutions.

Second, the integer solution of binary linear equation

We only discuss the integer solution of binary linear indefinite equation ax+by=c (where A and B are nonzero integers and C is an integer).

Example 1 Find the integer solution of the indefinite equation 7x+4y= 100.

Solution: 4y= 100-7x,

y= =25- x，

Because both x and y are integers, x must be an integer multiple of 4, that is,

x……- 16- 12-8-4 0 4 8 12 16……

y……53 46 39 32 25 18 1 1 4-3……

Looking back at the problem of "100 chickens for 100 yuan", because the number of chickens is a non-negative integer, four groups of solutions can be obtained from the solution of 1:

Example 2 Find the integer solution of the indefinite equation 6x-9y= 16.

Solution: Because the greatest common divisor of 6 and 9 is 3, divide both sides of the equation by 3, and you get it.

2x-3y=

Since both X and Y are integers, the left side of the equation (2x-3y) is also an integer, and the right side of the equation is a fraction, which is contradictory! Therefore, there is no integer solution to this problem.

From the above two examples, if the binary linear indefinite equation ax+by=c (where A and B are non-zero integers and C is an integer),

(1)a and B are coprime, and the equation has an integer solution;

(2)a and B are not coprime, but their greatest common divisor is divisible by C, and the equation has an integer solution;

(3)a and B are not coprime, and their greatest common divisor cannot be divisible by C, so the equation has no integer solution.

(Proof is abbreviated)

Third, for example

Grandma Xiao Ming sent a basket full of eggs, which can only hold about 55 at most. Xiao Ming only counted three, leaving 1. I forgot how many times, so I had to count it again. He counted five and there were only two left, but he forgot how many times. He was going to count again when Xiaoming's mother said, "Don't count." Xiao Ming looked at his mother blankly. Her eyes rolled a few times and she said, "* * * has 52 eggs." Xiao Ming was surprised and asked his mother how she knew. Her mother told Xiao Ming that the answer she got was this:

Set a basket, which can hold at most 100 eggs; Count every three times, ***x times, leaving1; Every five times, * * * y times, there are two left, namely

So there are 3x+ 1=5y+2,

Finishing y=

Because both x and y are positive integers, (3x- 1) must also be an integer multiple of 5, that is, there is.

x 2 7 12 17 22 ……

y 1 4 7 10 13 ……

m 7 22 37 52 67 ……

Because there are no more than 60 eggs in the basket, M is 52, which means there are 52 eggs in the basket.

Xiao Ming smiled with satisfaction and said, "There are also problems with indefinite equations in daily life!" " "