Examples of solving systems of linear equations using the substitution method. Lesson topic: "Substitution method" Solving systems using the substitution method

2. Method of algebraic addition.
3. Method of introducing a new variable (variable replacement method).

Definition: A system of equations refers to several equations for one or more variables that must be executed simultaneously, i.e. with the same values ​​of the variables for all equations. Equations in the system are combined with a system symbol – a curly brace.
Example 1:

— a system of two equations with two variables x And y.
The solution to the system is the roots. When these values ​​are substituted, the equations become true identities:

Solving systems of linear equations.

The most common method for solving a system is the substitution method.

Substitution method.

The substitution method for solving systems of equations is to express a variable from one equation of the system in terms of others, and substitute this expression into the remaining equations of the system instead of the expressed variable.
Example 2:
Solve the system of equations:

Solution:
A system of equations is given and it needs to be solved by the substitution method.
Let's express the variable y from the second equation of the system.
Comment:“Expressing a variable” means transforming the equality so that this variable remains to the left of the equal sign with a coefficient of 1, and all other terms move to the right side of the equality.
Second equation of the system:

Let's leave only on the left y:

And let’s substitute (that’s where the name of the method comes from) into the first equation instead of at the expression to which it is equal, i.e. .
First equation:

Let's substitute:

Let's solve this banal quadratic equation. For those who have forgotten how to do this, there is an article Solving quadratic equations. .

So the variable values x found.
Let's substitute these values ​​into the expression for the variable y. There are two meanings here x, i.e. for each of them you should find a value y .
1) Let
We substitute it into the expression.

2) Let
We substitute it into the expression.

Everything can be answered:
Comment: In this case, the answer should be written in pairs so as not to confuse which value of the variable y corresponds to which value of the variable x.
Answer:
Comment: In example 1, only one pair is indicated as a solution to the system, i.e. this pair is a solution to the system, but not a complete one. Therefore, how to solve an equation or system means indicating the solution and showing that there are no other solutions. And here's another couple.

Let’s formalize the solution to this system in a school style:

Comment: The sign “” means “equivalently”, i.e. the next system or expression is equivalent to the previous one.

Usually the equations of the system are written in a column one below the other and combined with a curly brace

A system of equations of this type, where a, b, c- numbers, and x, y- variables are called system of linear equations.

When solving a system of equations, properties that are valid for solving equations are used.

Solving a system of linear equations using the substitution method

Let's look at an example

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

2) Substitute into the second equation of the system instead of y expression 3x-7:

3) Solve the resulting second equation:

4) We substitute the resulting solution into the first equation of the system:

A system of equations has a unique solution: a pair of numbers x=1, y=-4. Answer: (1; -4) , written in brackets, in the first position the value x, On the second - y.

Solving a system of linear equations by addition

Let's solve the system of equations from the previous example addition method.

1) Transform the system so that the coefficients for one of the variables become opposite. Let's multiply the first equation of the system by "3".

2) Add the equations of the system term by term. We rewrite the second equation of the system (any) without changes.

3) We substitute the resulting solution into the first equation of the system:

Solving a system of linear equations graphically

The graphical solution of a system of equations with two variables comes down to finding the coordinates of the common points of the graphs of the equations.

The graph of a linear function is a straight line. Two straight lines on a plane can intersect at one point, be parallel, or coincide. Accordingly, a system of equations can: a) have a unique solution; b) have no solutions; c) have an infinite number of solutions.

2) The solution to the system of equations is the point (if the equations are linear) of the intersection of the graphs.

Graphic solution of the system

Method for introducing new variables

Changing variables can lead to solving a simpler system of equations than the original one.

Consider the solution of the system

Let's introduce the replacement , then

Let's move on to the initial variables

Special cases

Without solving a system of linear equations, you can determine the number of its solutions from the coefficients of the corresponding variables.

Let us analyze two types of solutions to systems of equations:

1. Solving the system using the substitution method.
2. Solving the system by term-by-term addition (subtraction) of the system equations.

In order to solve the system of equations by substitution method you need to follow a simple algorithm:
1. Express. From any equation we express one variable.
2. Substitute. We substitute the resulting value into another equation instead of the expressed variable.
3. Solve the resulting equation with one variable. We find a solution to the system.

To solve system by term-by-term addition (subtraction) method need to:
1. Select a variable for which we will make identical coefficients.
2. We add or subtract equations, resulting in an equation with one variable.
3. Solve the resulting linear equation. We find a solution to the system.

The solution to the system is the intersection points of the function graphs.

Let us consider in detail the solution of systems using examples.

Example #1:

Let's solve by substitution method

Solving a system of equations using the substitution method

2x+5y=1 (1 equation)
x-10y=3 (2nd equation)

1. Express
It can be seen that in the second equation there is a variable x with a coefficient of 1, which means that it is easiest to express the variable x from the second equation.
x=3+10y

2.After we have expressed it, we substitute 3+10y into the first equation instead of the variable x.
2(3+10y)+5y=1

3. Solve the resulting equation with one variable.
2(3+10y)+5y=1 (open the brackets)
6+20y+5y=1
25y=1-6
25y=-5 |: (25)
y=-5:25
y=-0.2

The solution to the equation system is the intersection points of the graphs, therefore we need to find x and y, because the intersection point consists of x and y. Let's find x, in the first paragraph where we expressed it we substitute y.
x=3+10y
x=3+10*(-0.2)=1

It is customary to write points in the first place we write the variable x, and in the second place the variable y.
Answer: (1; -0.2)

Example #2:

Let's solve using the term-by-term addition (subtraction) method.

Solving a system of equations using the addition method

3x-2y=1 (1 equation)
2x-3y=-10 (2nd equation)

1. We choose a variable, let’s say we choose x. In the first equation, the variable x has a coefficient of 3, in the second - 2. We need to make the coefficients the same, for this we have the right to multiply the equations or divide by any number. We multiply the first equation by 2, and the second by 3 and get a total coefficient of 6.

3x-2y=1 |*2
6x-4y=2

2x-3y=-10 |*3
6x-9y=-30

2. Subtract the second from the first equation to get rid of the variable x. Solve the linear equation.
__6x-4y=2

5y=32 | :5
y=6.4

3. Find x. We substitute the found y into any of the equations, let’s say into the first equation.
3x-2y=1
3x-2*6.4=1
3x-12.8=1
3x=1+12.8
3x=13.8 |:3
x=4.6

The intersection point will be x=4.6; y=6.4
Answer: (4.6; 6.4)

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Let's figure it out How to solve systems of equations using the substitution method?

1) Express the unknown from the first or second equation of the system X or at(whatever is more convenient for us);

2) Substitute into another equation (into the one from which the unknown was not expressed) instead of the unknown X or at(if expressed X, substitute instead X; if expressed at, substitute instead at) the resulting expression;

3) Solve the equation that we received. We find X or y;

4) Substitute the resulting value of the unknown and find the second unknown.

The rule is written. Now let's try to apply it when solving a system of equations.

Example 1.

Let's look carefully at the system of equations. We note that from the first equation it is easier to express at.

We express at:

–2у = 11 – 3х

y = (11 – 3x)/(–2)

y = –5.5 + 1.5x

Now let’s carefully substitute into the second equation instead at expression –5.5 + 1.5x.

We get: 4x – 5(–5.5 + 1.5x) = 3

Let's solve this equation:

4x + 27.5 – 7.5x = 3

–3.5x = 3 – 27.5

–3.5x = –24.5

x = –24.5/(–3.5)

We substitute y = – 5.5 + 1.5x into the expression instead X the value we found. We get:

y = – 5.5+ 1.5 7 = –5.5 + 10.5 = 5.

Answer: (7; 5)

It’s interesting, but if we express from the first equation not at, A X, will the answer change?

Let's try to express X from the first equation.

x = (11 + 2y)/3

Let's substitute instead X into the second equation the expression (11 +2у)/3, we get an equation with one unknown and solve it.

4(11 + 2у)/3 – 5у = ​​3, multiply both sides of the equation by 3, we get

4(11 + 2y) – 15y=9

44 + 8у – 15у = 9

–7у = 9 – 44

y = –35/(–7)

We find the variable x by substituting 5 into the expression x = (11 +2y)/3.

x = (11 +2 5)/3 = (11+10)/3 = 21/3 = 7

Answer: (7; 5)

As you can see, the answer was the same. If you are careful and careful, then no matter what variable you express - X or at, you will get the correct answer.

Quite often students ask: “ Are there other ways to solve systems besides addition and substitution?»

There is some modification of the substitution method - way to compare unknowns .

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

2) The resulting unknowns are compared and an equation with one unknown is obtained.

3) Find the value of one unknown.

4) Substitute the resulting value of the unknown and find the second unknown.

Example 2. Solve system of equations

From two equations we express the variable X through at.

From the first equation we obtain x = (13 – 6y) / 5, and from the second equation x = (–1 – 18y) / 7.

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

(13 – 6y) / 5 = (–1 – 18y) / 7

7 (13 – 6y) = 5 (–1 – 18y)

91 – 42у = –5 – 90у

–42у + 90у = –5 – 91

y = – 96 / 48

Unknown X let's find by substituting the value at into one of the expressions for X.

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

Answer: (5; –2).

I think that you will succeed too. If you have any questions, come to my lessons.

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1 . FULL NAME. teachers: ____Tkachuk Natalya Petrovna _________________________________________________________________________________________________

2. Class: _8 Date: .11.03________Subject_-mathematics, lesson No. 71 according to the schedule:

3. Lesson topic Solving systems by substitution 4 . The place and role of the lesson in the topic being studied :. Lesson to consolidate knowledge. The purpose of the lesson :

Educational: develop knowledge of solving systems of equations using the substitution method. Know/understand: if the graphs have common points, then the system has solutions; if the graphs do not have common points, then the system has no solutions; algorithm for solving systems of equations.Be able to solve systems by substitution Promote the development of skills to apply acquired knowledge in non-standard (standard) conditionsDevelopmental: To promote the development of students’ skills to generalize acquired knowledge, conduct analysis, synthesis, comparisons, and draw the necessary conclusions. To promote the development of skills to apply acquired knowledge in non-standard and standard conditions.Educational: Promote the development of a creative attitude towards learning activities

Characteristics of the lesson stages

Activity

students

Self-determination.

Activate cognitive activity

Solve the system

verbal

Frontal

Greeting students. carrying out. Creating a situation of readiness for the lesson, success in the upcoming lesson.

Check readiness for the lesson.

2. Updating knowledge.

Identify the quality and level of mastery of knowledge and skills acquired in previous lessons on the topic

Find out whether a pair of numbers is a solution to the system. x=5 y=9

What operations can be performed with equations?

(multiply both sides of the equation by the same number, divide by a number not equal to zero....)

Group work

Frontal. Guppovaya - analysis of algorithms for solving problems;

Asks leading questions when necessary.

They answer the questions asked.

3. Statement of the educational task, lesson goals.

Formation

and skill development

define and formulate

problem, goal and topic

to study lines

How to solve a system of equations by addition, by substitution.

Which method is appropriate to use when solving. this system?

Group work.

Individual.

Frontal.

What steps did we take to find out the purchase price?

What topic will we study?

They speak out.

4. Stage of updating knowledge on the topic

To promote the development of skills to distinguish and compare lines. Provide conditions for the development of skills to express one’s thoughts competently, clearly and accurately.

№ 621

Find out the relative positions of the lines

2x+0.5y= 1.2 and x- 4y=0

Is it possible to determine whether lines intersect or not by their coefficients?

2. create equations of lines that are parallel to each other.

Working with a student

Work in pairs with self-test

Frontal, individual. problem solving workshop

Asks leading questions when necessary. Draws parallels with previously studied material.

Provides motivation to complete proposed tasks.

Leads students to the conclusion about the existence of formulas.

Solve problems, answer teacher questions if necessary. Do the exercise in a notebook.

Take turns commenting, analyzing, identifying reasons and solutions.

5.Work independently

application of acquired knowledge. Updating knowledge and skills in problem solving.

Formation and development of number reading skills. Planning your activities to solve a given problem, monitoring the result obtained, correcting the result obtained, self-regulation

1 var –

2 var

Independent work. Checking your neighbor.

"brainstorm",

Monitors the execution of work.

Provides: individual control; selective control.

Encourages you to express your opinion.

Solve problems. Carry out: self-assessment; mutual verification; provide a preliminary assessment.

6. Lesson assessment, self-assessment.

Formation and development of the ability to analyze and comprehend one’s achievements.

The ability to determine the level of mastery of educational material.

Assessment of intermediate results and self-regulation to increase motivation for educational activities

Assessment at every stage

1. Can you graph linear equations?

2.Can you determine whether they intersect or not?

3. Do you know an algorithm for solving systems of equations?

4. what methods do you know for solving systems of equations?

Group work.

Group and individual...

Encourages you to express your opinion.

Carry out: self-assessment and assessment of a friend.

7. Lesson summary. Homework.

The ability to correlate goals and results of one’s own activities. Maintaining a healthy spirit of competition to maintain motivation for educational activities; participation in collective discussion of problems.

p. 4.4 No. 623

Group work.

Frontal - Identification and formulation of a cognitive goal, reflection on methods and conditions of action

Analysis and synthesis of objects

Encourages you to express your opinion.

Gives comments on homework; task to search for features in the text...

Children participate in the discussion, analyze, talk. Reflect and record their achievements.

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Today in class I learned...