Algorithm for solving systems of rational equations in the substitution method. How is the system of equations solve? Methods for solving equation systems. Visual System Solution Method

1 . FULL NAME. Teachers: ____ Tkachuk Natalia Petrovna _____________________________________________________________________________________________

2. Class: _8 Date: .11.03 ________ subject_-mathematics, №71 lesson on schedule:

3. Theme lesson Solution Systems in the method of substitution 4 . The place and role of the lesson in the topic studied :. Lesson consolidating knowledge. The purpose of the lesson :

Educational: develop the knowledge of solving systems of equations by the substitution method. Know / understand: if graphics have common points, the system has solutions; If graphs do not have common points, then the system of solutions does not have; Algorithm for solving systems of equations.Be able to solve systems by substitution Promote the development of skills to apply the knowledge gained in non-standard (typical) conditionsDeveloping: Promote the development of students' skills to summarize the knowledge gained, conduct analysis, synthesis, comparisons, make the necessary conclusions. Promote the development of skills to apply the knowledge gained in non-standard and typical conditions.Educational: Contribute to the development of creative attitude towards learning activities

Characteristics of the stages of the lesson

Activity

pupils

Self-determination.

Activate cognitive activity

Solve the system

verbal

Frontal

Greeting students. Conduct. Creating a spelling situation for a lesson, success in the upcoming lesson.

Check the readiness for the lesson.

2. Actualization of knowledge.

Identify the quality and level of mastering knowledge and skills obtained in previous lessons on the topic

Find out whether a pair of numbers by solving the system. x \u003d 5 y \u003d 9

What operations can be done with equations?

(multiply both parts of the equation for the same number, to divide the number not equal to zero ....)

Work in the group

Frontal. Gupper-analysis of algorithms for solving problems;

If necessary, asks leading questions.

Answer questions.

3. Suit of the learning task, lesson purposes.

Formation

and development of skills

determine and formulate

problem, goal and topic

for studying lines

How the system of equations is solved by the method of addition, the method of substitution.

What method is appropriate to apply when solving. This system?

Work in the group.

Individual.

Frontal.

What activities did we do to find out the cost of buying?

What topic will we study?

Spoke.

4. Stage of actualization of knowledge on the topic

Promote the development of skills to distinguish and compare lines. Provide conditions for the development of skills competently, clearly and accurately express their thoughts.

№ 621

Find out the mutual location of direct

2x + 0.5u \u003d 1.2 and x- 4u \u003d 0

Is it possible to determine the direct or not on their coefficients?

2. Make an equation of direct which are parallel to each other.

Working with scientist

Work in pairs with self-test

Frontal, individual. Workshop on solving problems

If necessary, asks leading questions. Conducts parallel with previously studied material.

Provides motivating the execution of proposed tasks.

Takes students to the conclusion about the existence of formulas.

We solve the tasks, answer the questions of the teacher, if necessary, perform an exercise in the notebook.

In turn comment, analyze, determine the causes and ways to solve.

5. The work on independent

application of knowledge gained. Actualization of knowledge and skills in solving problems.

Formation and development of the reading abilities of numbers. Elap of its activities to solve the task, control the result, correction of the result obtained, regulation itself

1 var -

2 Var.

Independent work. Checking a neighbor.

« brainstorm»,

Controls performance.

Performs: individual control; Selective control.

Moves to the statement of his opinion.

Decide tasks. Carry out: self-esteem; mutual test; Exhibit a preliminary assessment.

6. Zoom of the lesson, self-esteem.

The formation and development of the ability to analyze and comprehend their achievements.

The ability to determine the level of mastering by educational material.

Evaluation of intermediate results and regulation itself to increase the motivation of educational activities

Evaluation at each stage

1. Do you know how to build graphs of linear equations?

2. Do you determine whether they intersect or not.

3. Do you know the algorithm solving systems of equations?

4. What ways do you know solutions to the systems of equations?

Work in the group.

Group and individual ..

Moves to the statement of his opinion.

Carry out: self-esteem and comrade evaluation.

7. Camera lesson. Homework.

The ability to relate the goals and results of your own act. Under-holding a healthy spirit of the opponewood for under-holding the motivation of educational activities; Participation in collective discussion of problems.

p p. 4.4 №623

Work in the group.

Frontal-allocation and formulam-peria Positive goal The reflection of methods and conditions of action

Analysis and synthesis of objects

Moves to the statement of his opinion.

Gives a comment k homework; Task search in the text of the features ...

Children participate in the discussion, analyze, pronounce. Comprehensive and fix their achievements.

Today I learned in the lesson ...

Today at the lesson I learned ...

Let's figure it out how to solve the system of equations in the substitution method?

1) Express the unknown system from the first or second equation h. or w. (as it is more convenient);

2) we will substitute to another equation (in whom I did not express the unknown) instead of an unknown h. or w. (If expressed h., we substitute instead h.; If expressed w., we substitute instead w.) the resulting expression;

3) solve the equation that they received. Find h. or y;

4) We substitute the obtained value of the unknown and find the second unknown.

Rule is recorded. Now let's try to apply it when solving the system of equations.

Example 1..

Carefully look at the system of equations. We notice that from the first equation it is easier to express w..

Express w.:

-2u \u003d 11 - 3x

y \u003d (11 - 3x) / (- 2)

y \u003d -5.5 + 1.5x

Now neatly substitute in the second equation instead w. expression -5.5 + 1.5x.

We get: 4x - 5 (-5.5 + 1.5x) \u003d 3

We solve this equation:

4x + 27.5 - 7.5x \u003d 3

-3.5x \u003d 3 - 27.5

-3.5x \u003d -24.5

x \u003d -225 / (- 3.5)

We substitute in the expression y \u003d - 5.5 + 1.5x instead h. The value we found. We get:

y \u003d - 5.5+ 1.5 · 7 \u003d -5.5 + 10.5 \u003d 5.

Answer: (7; 5)

Interesting, and if you express from the first equation not w., but h.whether to change the answer?

Let's try to express h. From the first equation.

x \u003d (11 + 2y) / 3

Substitute instead h. In the second equation, the expression (11 + 2y) / 3, we obtain an equation with one unknown and solve it.

4 (11 + 2y) / 3 - 5y \u003d 3, multiply both parts of the equation by 3, we obtain

4 (11 + 2y) - 15th \u003d 9

44 + 8th - 15th \u003d 9

-7u \u003d 9 - 44

y \u003d -35 / (- 7)

We find the variable x, substituting 5 in the expression x \u003d (11 + 2y) / 3.

x \u003d (11 + 2 · 5) / 3 \u003d (11 + 10) / 3 \u003d 21/3 \u003d 7

Answer: (7; 5)

As you can see The answer turned out the same. If you are attentive and neat, then no matter what variable you express - h. or w., the answer will get the right one.

Quite often, students ask: " Are there any other ways to solve systems, except for addition and substitution?»

There is some modification of the method of substitution - method of comparing unknowns .

1) It is necessary from each equation of the system to express one and the same unknown through the second.

2) The obtained unknowns are compared, the equation with one unknown is obtained.

3) find the value of one unknown.

4) substitute the obtained value of the unknown and find the second unknown.

Example 2.. Solve the system of equations

Of the two equations, we will express the variable h. through w..

We obtain from the first equation X \u003d (13 - 6U) / 5, and from the second x \u003d (-1 - 18th) / 7.

Comparing these expressions, we obtain an equation with one unknown and solve it:

(13 - 6U) / 5 \u003d (-1 - 18U) / 7

7 (13 - 6U) \u003d 5 (-1 - 18U)

91 - 42U \u003d -5 - 90U

-42u + 90u \u003d -5 - 91

y \u003d - 96/48

Unknown H. Find substitution value w. in one of the expressions for h..

(13 – 6(– 2)) / 5= (13+12) / 5 = 25/5 = 5

Answer: (5; -2).

I think that you will succeed. If you have any questions, come to me for lessons.

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Typically, the system equations are recorded in a column one below the other and combine the figure bracket.

System of equations of this species where a, b, c - Numbers, and x, Y. - variables called system of linear equations.

When solving the system of equations, properties are valid for solving equations.

Solution of the system of linear equations by the method of substitution

Consider example

1) Express in one of the equations variable. For example, express y. In the first equation, we get the system:

2) we substitute in the second equation of the system instead y. expression 3x-7.:

3) solve the resulting second equation:

4) The solution obtained is substituted into the first system equation:

The system of equations has a single solution: a couple of numbers x \u003d 1, y \u003d -4. Answer: (1; -4) , written in brackets, in the first position x., On the second - y..

Solving a system of linear equations by the method of addition

I decide the system of equations from the previous example Method of addition.

1) Convert the system so that the coefficients for one of the variables become opposite. Multiply the first system equation on "3".

2) We fold the system equation. The second equation of the system (any) is rewritten unchanged.

3) The solution obtained is substituted into the first system equation:

Solution of the linear equation system graphically

The graphical solution of the system of equations with two variables is reduced to looking for the coordinates of the common points of graphs of equations.

The graph of the linear function is straight. Two straight ones can intersect at one point, be parallel or coincided. Accordingly, the system of equations may: a) have the only solution; b) not to have solutions; c) have infinite set solutions.

2) by solving the system of equations is a point (if equations are linear) intersection of graphs.

Graphic solution system

Method of introducing new variables

Replacing variables can lead to a solution of a simpler system of equations than the initial one.

Consider the solution of the system

We introduce a replacement, then

Go to the initial variable

Special cases

Not solving the system of linear equations, you can determine the number of its solutions by coefficients at the corresponding variables.

We will analyze two types of solutions of the systems of the equation:

1. Solution of the system by substitution.
2. Solution of the system by the method of kindy addition (subtraction) of the system equations.

In order to solve the system of equations for a substitution method You need to follow a simple algorithm:
1. Express. From any equation, we express one variable.
2. Substitute. We substitute to another equation instead of a pronounced variable obtained.
3. We solve the obtained equation with one variable. We find the system solution.

To solve introduction system system (subtraction) need to:
1. To give a variable to which we will do the same coefficients.
2. Wide or subtract the equations, as a result, we obtain an equation with one variable.
3. We solve the resulting linear equation. We find the system solution.

The system solution is the point of intersection of the function graphs.

Consider in detail the examples of systems.

Example number 1:

By deciding by substitution

Solution of the system of equations by substitution

2x + 5y \u003d 1 (1 equation)
x-10y \u003d 3 (2 equation)

1. Express
It can be seen that in the second equation there is a variable x with a coefficient of 1, it turns out that it turns out that the variable x from the second equation is easier to express.
x \u003d 3 + 10y

2. After how expressed substituted in the first equation 3 + 10y instead of a variable x.
2 (3 + 10y) + 5Y \u003d 1

3. Over the resulting equation with one variable.
2 (3 + 10y) + 5y \u003d 1 (reveal brackets)
6 + 20Y + 5Y \u003d 1
25Y \u003d 1-6.
25Y \u003d -5 |: (25)
Y \u003d -5: 25
Y \u003d -0.2

The solution of the system of the equation is the points of intersection of graphs, therefore we need to find x and y, because the intersection point consists of their X and Y.Nad X, in the first paragraph where we expressed there we substitute Y.
x \u003d 3 + 10y
x \u003d 3 + 10 * (- 0.2) \u003d 1

Points are taken to record in the first place we write the variable x, and on the second variable y.
Answer: (1; -0.2)

Example number 2:

By deciding by the method of kindy addition (subtraction).

Solution of the system of equations by addition

3x-2y \u003d 1 (1 equation)
2x-3y \u003d -10 (2 equation)

1. Select the variable, let's say, select x. In the first equation in the variable x coefficient 3, in the second 2. It is necessary to make the coefficients are the same, for this we have the right to multiply equations or divide on any number. The first equation is permiterable by 2, and the second to 3 and we obtain the total coefficient 6.

3x-2y \u003d 1 | * 2
6x-4y \u003d 2

2x-3y \u003d -10 | * 3
6x-9y \u003d -30

2. The first equation will be subtracted second to get rid of the variable x.resh linear equation.
__6x-4y \u003d 2

5Y \u003d 32 | :five
Y \u003d 6,4.

3. Land x. We substitute in any of the equations found y, let's say in the first equation.
3x-2y \u003d 1
3x-2 * 6,4 \u003d 1
3x-12.8 \u003d 1
3x \u003d 1 + 12.8
3x \u003d 13.8 |: 3
x \u003d 4.6

Point of intersection will be x \u003d 4.6; Y \u003d 6,4.
Answer: (4,6; 6.4)

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