Put two linear equations together, then these two equations form a binary linear equation group.

An equation group consisting of several equations is called an equation group. If an equation group contains two unknowns and the number of terms of the unknowns is 1, then such an equation group is called a binary linear equation group.

Definition of bivariate linear equation: The bivariate linear integral equation with exponent 1 is called bivariate linear equation. Definition of binary linear equations: Two linear equations with two unknowns are called binary linear equations.

Solution of binary linear equation: the values of two unknowns that make the values of both sides of binary linear equation equal are called the solutions of binary linear equation.

Solutions of binary linear equations: Two common solutions of binary linear equations are called solutions of binary linear equations.

General solution and elimination: solve the unknowns in the equations one by one from more to less.

There are two ways to eliminate elements:

elimination by substitution

Example: Solve the equation x+y=5①

6x+ 13y=89②

Solution: x=5-y③ from ①.

Bring ③ into ② to get 6(5-y)+ 13y=89.

y=59/7

Bring y=59/7 into ③,

x=5-59/7

X=-24/7。

∴x=-24/7

Y=59/7 is the solution of the equation.

We call this method of eliminating the unknown quantity by "replacement method" to find the solution of the equation "replacement method" for short.

Addition, subtraction and elimination method

Example: Solving Equation x+y=9①

x-y=5②

Solution: ①+② 2x= 14

That is, x=7

Bring x=7 into ①.

Get 7+y=9.

The solution is y=-2.

∴x=7

Y=-2 is the solution of the equation.

Such a method for solving binary linear equations is called addition and subtraction, or addition and subtraction for short. There are three solutions to binary linear equations:

1. The equations x+y = 516x+13y = 892x =-24/7y = 59/7 and other solutions are the solutions of the equations.

2. There are countless groups of solutions, such as the equation group X+Y = 6 12x+2Y = 12② Because these two equations are actually an equation (also called "the equation has two equal real roots"), there are countless groups of solutions for this equation group.

3. There is no solution, such as the equation set X+Y = 4 12x+2Y = 10②, because the simplified equation ② is x+y=5, which contradicts the equation ①, so this kind of equation set has no solution.

Note: When solving problems by addition and subtraction or substitution and elimination, we should pay attention to which method is simple, so as to avoid trouble or mistakes in calculation.

Several solutions not found in textbooks

(1) Addition, subtraction and substitution mixed use method.

Example 1,13x+14y = 41(1).

14x+ 13y=40 (2)

Solution: (2)-(1) x-y =-1x = y-1(3)

Substituting (3) into (1) gives13 (y-1)+14y = 41.

13y- 13+ 14y = 4 1

27y=54

y=2

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

So: x= 1,

y=2

Features: two equations are added and subtracted, single x or single y, so the next substitution elimination method is applicable.

(2) Alternative methods

Example 2, (x+5)+(y-4)=8.

(x+5)-(y-4)=4

Let x+5 = m and y-4 = n.

The original equation can be written as m+n=8.

m-n=4

The solution is m=6,

n=2

So x+5=6,

y-4=2

So x= 1,

y=6

Features: The two equations contain the same algebraic expressions, such as x+5 and y-4 in the title, and the simplification of the equations after substitution is also the main reason.

(3) Alternative exchange of RMB

Example 3, x:y= 1:4

5x+6y=29

Let x = t and y = 4t.

Equation 2 can be written as 5t+6*4t=29.

29t=29

T= 1 so x = 1 and y = 4.

Solutions of binary linear equations

Generally speaking, 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 process of solving equations is called solving equations.

Generally speaking, binary linear equations have only one solution.

note:

Binary linear equations are not necessarily composed of two binary linear equations! It can also be composed of one or more binary linear equations alone.

★ Emphasis★ Solution of one-dimensional linear equation, one-dimensional quadratic equation and two-dimensional linear equations; Application problems related to the equation (especially travel and engineering problems) ☆ Summary ☆.

First, the basic concept of 1 Equation, its solution (root), its solution, its solution (group) 2. Classification:

Second, the basis of solving the equation-the properties of the equation 1. A = b ←→ A+C = b+C2。 A = b ←→ AC = BC (c ≠ 0)。

Third, the solution

1. Solution of linear equation with one variable: remove denominator → remove brackets → shift terms → merge similar terms → coefficient becomes 1→ solution.

2. Solution of linear equations: ① Basic idea: "elimination method" ② Method: ① Substitution method ② Addition and subtraction method.

Four. One-dimensional quadratic equation 1. Definition and general form: 2. Solution: ① Direct Kaiping method (pay attention to features) ② Matching method (pay attention to steps-knocking down the root formula) ③ Formula method: ④ Factorization method (features: left = 0) 3. The discriminant formula of roots: 4. The relationship between root and coefficient top: inverse theorem: if. 5. Common equation:

5. Equations that can be transformed into quadratic equations

1. definition of fractional equation (1) (2) basic idea: (3) basic solution: (1) denominator method (2) substitution method (for example) (4) root test and method.

2. definition of irrational equation (1) (2) basic idea: (3) basic solution: (1) multiplication (pay attention to skills! ! (2) substitution method (example), (4) root test and method.

3. A simple binary quadratic equation consisting of a binary linear equation and a binary quadratic equation can be solved by substitution method.

Six, column equation (group) to solve application problems

Solving application problems with sequence equations (groups) is an important aspect of integrating middle school mathematics with practice.

The specific steps are as follows:

(1) review the questions. Understand the meaning of the question. Find out what is a known quantity, what is an unknown quantity, and what is the equivalent relationship between problems and problems. ⑵ Set an element (unknown). ① Direct unknowns ② Indirect unknowns (often both). Generally speaking, the more unknowns, the easier it is to list the equations, but the more difficult it is to solve them.

⑶ Use algebraic expressions containing unknowns to express related quantities.

(4) Find the equation (some are given by the topic, some are related to this topic) and make the equation. Generally speaking, the number of unknowns is the same as the number of equations.

5] Solving equations and testing.

[6] answer.

To sum up, the essence of solving application problems by column equations (groups) is to first transform practical problems into mathematical problems (setting elements and column equations), and then the solutions of practical problems (column equations and writing answers) are caused by the solutions of mathematical problems. In this process, the column equation plays a role of connecting the past with the future. Therefore, the column equation is the key to solve the application problem.

Second, the common equality relationship

1. Travel problem (uniform motion) Basic relationship: S = vt (1) Meeting problem (simultaneous departure):+=;

(2) Catch-up problem (start at the same time): If Party A starts t hours later and Party B starts at point B and then catches up with Party A, then

(3) sailing in the water:

2. batching problem: solute = solution × concentration solution = solute+solvent

3. Growth rate:

4. Engineering problems: Basic relationship: workload = working efficiency × working time (workload is often considered as "1").

5. Geometric problems: Pythagorean theorem, area and volume formulas of geometric bodies, similar shapes and related proportional properties. Third, pay attention to the relationship between language and analytical formula.

For example, "more", "less", "increase", "increase to (to)", "at the same time" and "expand to) ... Another example is a three-digit number, where A has hundreds, B has tens and C has tens, so this three-digit number is: 6544.

Fourth, pay attention to writing equal relations from the language narrative.

For example, if X is greater than Y by 3, then x-y=3 or x=y+3 or X-3 = Y, and if the difference between X and Y is 3, then x-y=3.

Pay attention to unit conversion

Such as the conversion of "hours" and "minutes"; Consistency of s, v and t units, etc.

Exercise of binary linear equations

First, multiple-choice questions:

1. Among the following equations, () is a binary linear equation.

a . 3x-2y = 4z b . 6xy+9 = 0°c .+4y = 6° d . 4x =

2. In the following equations, belongs to the binary linear equations is ().

A.

3. Binary linear equation 5a- 1 1b = 2 1 ()

A. There is only one solution B. There are countless solutions C. There is no solution D. There are only two solutions.

4. The common * * solution of equations Y = 1-X and 3x+2y=5 is ().

A.

5. If │ x-2 │+(3y+2) 2 = 0, the value of is ().

A.- 1 B- 2c-3d。

6. If the solution of the equation is equal to the values of x and y, then k is equal to ().

7. In the following categories, the number of binary linear equations is ().

①xy+2x-y = 7； ②4x+ 1 = x-y； ③+y = 5； ④x = y； ⑤x2-y2=2

⑥6x-2y⑦x+y+z = 1⑧y(y- 1)= 2 y2-y2+x

A. 1

8. There are 246 students in a certain grade * * *, in which the number of boys Y is 2 times less than the number of girls X, then the following equation meets the question ().

A.

Second, fill in the blanks

9. It is known that the equation 2x+3y-4 = 0, y is expressed by an algebraic expression containing x: y = _ _ _ _ _ x is expressed by an algebraic expression containing y: x = _ _ _ _ _ _ _

10. in the binary linear equation -x+3y = 2, when x=4, y = _ _ _ _ _ _ _ when y =- 1, x = _ _ _ _.

1 1. If x3m-3-2yn- 1 = 5 is a binary linear equation, then m=____, and n = _ _ _ _.

12. If the solution of the equation is known as X-KY = 1, then k = _ _ _ _ _ _

13. If x- 1 │+(2y+ 1) 2 = 0, 2x-ky = 4, then k = _ _ _ _.

14. The positive integer solution of binary linear equation x+y=5 has _ _ _ _ _.

15. A binary linear equation whose solution is _ _ _ _ _ _ _.

16. If the solution is known, then m = _ _ _ _ _ _ and n = _ _ _ _ _ _

Third, answer questions.

17. When y =-3, the binary linear equations 3x+5y =-3 and 3y-2ax = a+2 (equations about x and y) have the same solution. Find the value of a.

18. If (a-2) x+(b+ 1) y = 13 is a binary linear equation about x and y, what conditions do A and B satisfy?

19. The values of solutions X and Y of binary linear equations are equal, so find k. 。

20. given that x and y are rational numbers, and (│ x │-1) 2+(2y+1) 2 = 0, what is the value of X-Y?

2 1. Given the equation x+3y=5, please write a binary linear equation so that the solution of the equation group formed by it and the known equation is.

22. List the equations according to the meaning of the question:

(1) Mingming went to the post office to buy *** 13 stamps. Since 0.8 yuan and 2 yuan, * * * spent money on 20 yuan and asked Mingming how many stamps he bought each.

(2) Put several chickens in several cages. If there are four chickens in each cage, one chicken has no cage. If you put five chickens in each cage, there will be no chickens in one cage. How many chickens and cages are there?

23. Does the solution of the equation satisfy 2x-y = 8? Is a pair of x and y values satisfying 2x-y = 8 the solution of the equation?

24. (Open Question) Is there an integer m that makes the equation 2x+9 = 2-(m-2) x have a solution in the integer range? How many values of m can you find? Can you find the solution corresponding to x?

Answer:

First, multiple choice questions

1.d analysis: grasp three necessary conditions for judging binary linear equation: ① there are two unknowns; ② The number of terms with unknowns is1; ③ Both sides of the equation are algebraic expressions.

2.a analysis: three necessary conditions for binary linear equations: ① there are two unknowns, ② the number of terms with unknowns is1; ③ Every equation is an integral equation.

3.b- analysis: There is no limit, a binary linear equation has countless solutions.

4.c analysis: use the exclusion method to substitute for verification one by one.

5.c analysis: using non-negative properties.

6.B

7.c analysis: According to the definition of binary linear equation, the whole equation with two unknowns and the number of unknowns does not exceed 1 is called binary linear equation. Note that it is a binary linear equation after finishing.

8.B

Second, fill in the blanks

9. 10.- 10

1 1.2 analysis: let 3m-3 = 1, n- 1 = 1, ∴m=, n = 2.

12.- 1 analysis: substitute into the equation X-ky = 1 and get -2-3k = 1, ∴ k =- 1.

Analysis of 13.4: x-1= 0,2y+1= 0,

∴x= 1, y =-, substitute it into the equation 2x-ky = 4, 2+ k=4, ∴ k = 1.

14. Solution:

Analysis: ∵x+y=5, ∴ y = 5-x, and ∵x and y are positive integers.

∴x is a positive integer less than 5. When x= 1, y = 4；; When x=2, y = 3；;

When x=3 and y = 2; When x=4, y = 1.

The positive integer solution of x+y = 5 is

15.X+Y = 12 analysis: use the quantitative relationship between x and y to form equations, such as 2x+y= 17, 2x-Y = 3, etc.

The answer to this question is not unique.

16. 14 analysis: solving in.

Third, answer questions.

17. Solution: ∫y =-3, 3x+5y =-3, ∴ 3x+5x3 =-3, ∴x=4.

Equations 3x+5y =-3 and 3x-2ax = a+2 have the same solution.

∴3×(-3)-2a×4=a+2,∴a=-。

18. solution: ∵ (a-2) x+(b+1) y =13 is a binary linear equation about x and y,

∴a-2≠0,b+ 1≠0,∴a≠2,b≠- 1

Analysis: In this question, if there are two unknowns, the coefficient of the unknowns should not be 0.

(If the coefficient is 0, the item is 0. )

19. solution: according to the meaning of the question, x=y, ∴4x+3y=7 can be changed to 4x+3x=7.

∴x= 1，Y = 1。 Substitute x= 1, Y = 1 into KX+(K- 1) Y = 3 to get K+K- 1 = 3.

∴k=2: Analysis According to the special relationship between two unknowns, one unknown can be replaced by an algebraic expression containing another unknown, and the "binary" can be changed into "one", so that the values of the two unknowns can be obtained.

20. solution: from (│ x │-1) 2+(2y+1) 2 = 0, we can get │ x │- 1 = 0 and 2y+ 1=0, ∴ x =

When x= 1, Y =-, x-y =1+=;

When X =- 1, Y =-, X-Y =- 1+=-.

Analysis: The square of any rational number is non-negative, and the sum of two non-negative numbers in the problem is 0.

Then these two non-negative numbers (│ x │- 1) 2 and (2y+ 1)2 are all equal to 0, thus obtaining │ x │-1= 0,2y+1= 0.

2 1. solution: empirical calculation is the solution of equation x+3y=5, and then write another equation, such as X-y = 3.

22.( 1) solution: suppose 0.8 yuan bought x stamps and 2 yuan bought y stamps.

(2) Solution: There are X chickens and Y cages.

23. Solution: Satisfied, not necessarily.

Analysis: The solution of ∵ is not only the solution of equation x+y=25, but also satisfies 2x-Y = 8.

The solution of equation ∴ must satisfy any one of them, but equation 2x-y = 8 has countless groups of solutions.

For example, x= 10 and y= 12 do not satisfy the equation.

24. solution: existence, four groups. ∫ The original equation can be transformed into -MX = 7,

∴ when m= 1, x =-7; When m =- 1, x = 7；; When m = 7, x =-1; X= 1 when m =-7.