What is a linear equation with 2 variables. Lesson summary on the topic "linear equations in two variables"

Learning to solve equations is one of the main tasks that algebra poses for students. Starting with the simplest, when it consists of one unknown, and moving on to more and more complex ones. If you have not mastered the actions that need to be performed with equations from the first group, it will be difficult to understand the others.

To continue the conversation, you need to agree on notation.

General form of a linear equation with one unknown and the principle of its solution

Any equation that can be written like this:

a * x = b,

called linear. This is the general formula. But often in assignments linear equations are written in implicit form. Then it is necessary to perform identical transformations to obtain a generally accepted notation. These actions include:

  • opening parentheses;
  • moving all terms with a variable value to the left side of the equality, and the rest to the right;
  • reduction of similar terms.

In the case where an unknown quantity is in the denominator of a fraction, you need to determine its values ​​at which the expression will not make sense. In other words, you need to know the domain of definition of the equation.

The principle by which all linear equations are solved comes down to dividing the value on the right side of the equation by the coefficient in front of the variable. That is, “x” will be equal to b/a.

Special cases of linear equations and their solutions

During reasoning, moments may arise when linear equations take on one of the special forms. Each of them has a specific solution.

In the first situation:

a * x = 0, and a ≠ 0.

The solution to such an equation will always be x = 0.

In the second case, “a” takes the value equal to zero:

0 * x = 0.

The answer to such an equation will be any number. That is, it has an infinite number of roots.

The third situation looks like this:

0 * x = in, where in ≠ 0.

This equation doesn't make sense. Because there are no roots that satisfy it.

General view of a linear equation with two variables

From its name it becomes clear that there are already two unknown quantities in it. Linear equations in two variables look like this:

a * x + b * y = c.

Since there are two unknowns in the record, the answer will look like a pair of numbers. That is, it is not enough to specify only one value. This will be an incomplete answer. A pair of quantities for which the equation becomes an identity is a solution to the equation. Moreover, in the answer, the variable that comes first in the alphabet is always written down first. Sometimes they say that these numbers satisfy him. Moreover, there can be an infinite number of such pairs.

How to solve a linear equation with two unknowns?

To do this, you just need to select any pair of numbers that turns out to be correct. For simplicity, you can take one of the unknowns equal to some prime number, and then find the second.

When solving, you often have to perform steps to simplify the equation. They are called identity transformations. Moreover, the following properties are always true for equations:

  • each term can be moved to the opposite part of the equality by replacing its sign with the opposite one;
  • The left and right sides of any equation are allowed to be divided by the same number, as long as it is not equal to zero.

Examples of tasks with linear equations

First task. Solve linear equations: 4x = 20, 8(x - 1) + 2x = 2(4 - 2x); (5x + 15) / (x + 4) = 4; (5x + 15) / (x + 3) = 4.

In the equation that comes first on this list, simply divide 20 by 4. The result will be 5. This is the answer: x = 5.

The third equation requires that an identity transformation be performed. It will consist of opening the brackets and bringing similar terms. After the first step, the equation will take the form: 8x - 8 + 2x = 8 - 4x. Then you need to move all the unknowns to the left side of the equation, and the rest to the right. The equation will look like this: 8x + 2x + 4x = 8 + 8. After adding similar terms: 14x = 16. Now it looks the same as the first one, and its solution is easy to find. The answer will be x=8/7. But in mathematics you are supposed to isolate the whole part from an improper fraction. Then the result will be transformed, and “x” will be equal to one whole and one seventh.

In the remaining examples, the variables are in the denominator. This means that you first need to find out at what values ​​the equations are defined. To do this, you need to exclude numbers at which the denominators go to zero. In the first example it is “-4”, in the second it is “-3”. That is, these values ​​​​need to be excluded from the answer. After this, you need to multiply both sides of the equality by the expressions in the denominator.

Opening the brackets and bringing similar terms, in the first of these equations we get: 5x + 15 = 4x + 16, and in the second 5x + 15 = 4x + 12. After transformations, the solution to the first equation will be x = -1. The second turns out to be equal to “-3”, which means that the latter has no solutions.

Second task. Solve the equation: -7x + 2y = 5.

Suppose that the first unknown x = 1, then the equation will take the form -7 * 1 + 2y = 5. Moving the factor “-7” to the right side of the equality and changing its sign to plus, it turns out that 2y = 12. This means y =6. Answer: one of the solutions to the equation x = 1, y = 6.

General form of inequality with one variable

All possible situations for inequalities are presented here:

  • a * x > b;
  • a * x< в;
  • a * x ≥b;
  • a * x ≤в.

In general, it looks like a simple linear equation, only the equal sign is replaced by an inequality.

Rules for identity transformations of inequalities

Just like linear equations, inequalities can be modified according to certain laws. They boil down to the following:

  1. any alphabetic or numerical expression can be added to the left and right sides of the inequality, and the sign of the inequality remains the same;
  2. you can also multiply or divide by the same positive number, this again does not change the sign;
  3. When multiplying or dividing by the same negative number, the equality will remain true provided that the inequality sign is reversed.

General view of double inequalities

The following inequalities can be presented in problems:

  • V< а * х < с;
  • c ≤ a * x< с;
  • V< а * х ≤ с;
  • c ≤ a * x ≤ c.

It is called double because it is limited by inequality signs on both sides. It is solved using the same rules as ordinary inequalities. And finding the answer comes down to a series of identical transformations. Until the simplest is obtained.

Features of solving double inequalities

The first of them is its image on the coordinate axis. There is no need to use this method for simple inequalities. But in difficult cases it may simply be necessary.

To depict an inequality, you need to mark on the axis all the points that were obtained during the reasoning. These are invalid values, which are indicated by punctured dots, and values ​​from inequalities obtained after transformations. Here, too, it is important to draw the dots correctly. If the inequality is strict, that is< или >, then these values ​​are punched out. In non-strict inequalities, the points must be shaded.

Then it is necessary to indicate the meaning of the inequalities. This can be done using shading or arcs. Their intersection will indicate the answer.

The second feature is related to its recording. There are two options offered here. The first is ultimate inequality. The second is in the form of intervals. It happens with him that difficulties arise. The answer in spaces always looks like a variable with a membership sign and parentheses with numbers. Sometimes there are several spaces, then you need to write the “and” symbol between the brackets. These signs look like this: ∈ and ∩. Spacing brackets also play a role. The round one is placed when the point is excluded from the answer, and the rectangular one includes this value. The infinity sign is always in parentheses.

Examples of solving inequalities

1. Solve the inequality 7 - 5x ≥ 37.

After simple transformations, we get: -5x ≥ 30. Dividing by “-5” we can get the following expression: x ≤ -6. This is already the answer, but it can be written in another way: x ∈ (-∞; -6].

2. Solve double inequality -4< 2x + 6 ≤ 8.

First you need to subtract 6 everywhere. You get: -10< 2x ≤ 2. Теперь нужно разделить на 2. Неравенство примет вид: -5 < x ≤ 1. Изобразив ответ на числовой оси, сразу можно понять, что результатом будет промежуток от -5 до 1. Причем первая точка исключена, а вторая включена. То есть ответ у неравенства такой: х ∈ (-5; 1].

Lesson objectives:

  • Educational:
    • repeat the topic: “Equations. Linear equations. Equivalent equations and their properties”;
    • ensure that students understand the concept of linear equations with two variables and their solution.
  • Developmental:
    • to form intellectual abilities:
    • the ability to compare, build analogues, highlight the main thing;
    • the ability to generalize and systematize the material covered;
    • develop logical thinking, memory, imagination, mathematical speech;
    • develop active cognitive activity.
  • Educational:
    • to cultivate independence, activity, and interest in students at all stages of the lesson;
    • to form such character qualities as perseverance, perseverance, determination.

Tasks that the teacher must solve in the lesson:

  • learn to highlight the main idea in the text;
  • learn to ask questions to the teacher, yourself or students;
  • learn to use acquired knowledge to solve non-standard problems;
  • teach the ability to express your thoughts mathematically correctly.

Problems that students must solve in this lesson:

  • know the definition of a linear equation with two variables;
  • be able to write simple linear equations;
  • be able to correctly find the values ​​of variables a, b and c;
  • be able to identify linear equations with two variables among equations;
  • answer the question: what is the solution to a linear equation in two variables?
  • How do you know if a pair of numbers is a solution to an equation?
  • be able to express one variable in terms of another.

Lesson type: lesson in learning new material.

DURING THE CLASSES

I. Organizational moment

II. Repetition of covered material

1) On the board: 2x, 2x + 5, 2x + 5 = 17.

2) Questions for the class:

– Define these expressions. (Expected answers: product, monomial, sum, polynomial, equation.)
-What is an equation called?
– Do you need an equation...? (Decide)
– What does it mean to “solve an equation”?
– What is the root of the equation?
– Which equations are equivalent?
– What properties of equivalence of equations do you know?

III. Updating students' knowledge

3) Assignment to the whole class:

– Convert expressions :(two people work at the board).

a) 2(x + 8) + 4(2x – 4) = b) 4(x – 2) + 2(3y + 4) =

After the transformation we got: a) 10x; b) 4x + 6y:

– Use them to create equations (students suggest - teacher writes equations on the board): 10x = 30; 4x + 6y = 28.

Questions:

– What is the name of the first equation?
– Why linear?
– Compare the second equation with the first. Try to formulate the definition of the second equation (Expected answer: an equation with two variables; students’ attention is focused on the type of equation – linear).

IV. Learning new material

1) The topic of the lesson is announced. Recording the topic in notebooks. Students’ independent formulation of the definition of an equation with two variables, a linear equation with two variables (by analogy with the definition of a linear equation with one variable), examples of equations with two variables. The discussion takes place in the form of a frontal conversation, dialogue - reasoning.

2) Class assignment:

a) Write two linear equations with two variables (the teacher and students listen to the answers of several students; at the teacher’s choice, one of them writes his equations on the board).

b) Together with the students, tasks and questions are determined that they should receive an answer to in this lesson. Each student receives cards with these questions.

c) Working with students to solve these issues and tasks:

– Determine which of these equations are linear equations with two variables a) 6x 2 = 36; b) 2x – 5y = 9: c) 7x + 3y 3; d) 1/2x + 1/3y = 6, etc. A problem may arise with the equation x: 5 – y: 4 = 3 (the division sign must be written as a fraction). What properties of equivalence of equations need to be applied? (Students' answers) Determine the coefficient values A, V And With.

– Linear equations with two variables, like all equations, need to be solved. What is the solution to linear equations in two variables? (Children give a definition).

Example: Find solutions to the equation: a) x – y = 12, write the answers in the form (x; y) or x = ...; y = .... How many solutions does the equation have?

Examples: Find solutions to the following equations a) 2x + y = 7; b) 5x – y = 4. How did you find the solutions to these equations? (Picked up).

– How do you know if a pair of numbers is a solution to a linear equation in two variables?

3) Working with the textbook.

– Find in the textbook those places where the main idea of ​​the topic of this lesson is highlighted

a) Oral performance of tasks: No. 1092, No. 1094.

b) Solving examples No. 1096 (for weak students), No. 1097 (for strong students).

c) Repeat the properties of equivalence of equations.

Exercise: Using the properties of equivalence of equations, express the variable Y through the variable X in the equation 5x + 2y = 12 (“a minute” to solve independently, then a general overview of the solution on the board, followed by an explanation).

d) Execution of example No. 1099 (one of the students completes the task at the board).

Historical reference

1. Guys, the equations that we met in class today are called Diophantine linear equations with two variables, named after the ancient Greek scientist and mathematician Diophantus, who lived about 3.5 thousand years ago. Ancient mathematicians first composed problems and then worked to solve them. Thus, many problems were compiled, which we become familiar with and learn to solve.

2. And also these equations are called indefinite equations. Many mathematicians worked on solving such equations. One of them is Pierre Fermat, a French mathematician. He studied the theory of solving indeterminate equations.

V. Lesson summary

1) Summarizing the material covered in the lesson. Answers to all questions posed to students at the beginning of the lesson:

– What equations are called linear with two variables?
– What is called solving a linear equation in two variables?
– How is this decision recorded?
– What equations are called equivalent?
– What are the properties of equivalence of equations?
– What problems did we solve in class, what questions did we answer?

2) Doing independent work.

For the weak:

– Find the values ​​of the variables a, b and c in the equation –1.1x + 3.6y = – 34?
– Find at least one solution to the equation x – y = 35?
– Are the pair of numbers (3; 2) a solution to a given linear equation with two variables 2x – y = 4?

For the strong:

– Write a linear equation with two variables for Diophantus’s problem: There are pheasants and rabbits walking in the yard of the house. The number of all legs turned out to be 26.
– Express the variable y in terms of x in the equation 3x – 5y = 8.

VI. Homework message

View all tasks in the textbook, a quick analysis of each task, select a task.

  • For weak students: No. 1093, No. 1095b).
  • For the strong: 1) No. 1101, No. 1104 (a). 2) solve Diophantus’ problem, find all natural solutions to this equation.

Additionally, at the request of students - No. 1105.

Instead of conclusion: I have been a mathematics teacher for over 40 years. And I want to note that an open lesson is not always the best lesson. It often happens that sometimes ordinary lessons bring more joy and satisfaction to the teacher. And then you think with regret that no one saw this lesson - the creation of the teacher and students.

A lesson is a single organism, a single whole; it is in the lesson that personal and moral experience of education is acquired for both students and teachers. 45 minutes of a lesson is so much and so little. A lot - because during this time you can “look” into the depths of centuries with your students and, “returning” from there, learn a lot of new, interesting things, and still have time to study new material.

Every student must be brought to the understanding that mathematics is the basis of human intellectual development. And the basis for this is the development of logical thinking. Therefore, before each lesson, I set a goal for myself and my students: to teach students to successfully work with definitions, skillfully distinguish the unknown from the known, proven from unproven, analyze, compare, classify, pose questions and learn to skillfully solve them. Use analogies, but if you can’t get out on your own, then next to you is not only a teacher, but your main assistant - a book.

Of course, an open lesson is some result of the teacher’s creative work. And teachers present at this lesson should pay attention to the main thing: the system of work, novelty, idea. Here, I think, it is not particularly important what teaching methodology the teacher uses in the lesson: old, modern or new innovative technologies, the main thing is that its use is appropriate and effective for the teacher and students.

I am very glad that in my life I have school, children, lessons and such kind colleagues. Thank you all!

A linear equation with two variables has the general form ax + by + c = 0. In it, a, b and c are coefficients - some numbers; and x and y are variables - unknown numbers that need to be found.

The solution to a linear equation with two variables is a pair of numbers x and y, for which ax + by + c = 0 is a true equality.

A given linear equation in two variables (for example, 3x + 2y – 1 = 0) has a set of solutions, that is, a set of pairs of numbers for which the equation is true. A linear equation with two variables is transformed into a linear function of the form y = kx + m, which is a straight line on the coordinate plane. The coordinates of all points lying on this line are solutions to a linear equation in two variables.

If two linear equations of the form ax + by + c = 0 are given and it is required to find values ​​of x and y for which both of them will have solutions, then we say that we must solve system of equations. A system of equations is written under a common curly brace. Example:

A system of equations can have no solution if the lines that are the graphs of the corresponding linear functions do not intersect (that is, parallel to each other). To conclude that there is no solution, it is enough to transform both linear equations with two variables to the form y = kx + m. If k is the same number in both equations, then the system has no solutions.

If a system of equations turns out to consist of two identical equations (which may not be obvious immediately, but after transformations), then it has an infinite number of solutions. In this case we talk about uncertainty.

In all other cases, the system has one solution. This conclusion can be drawn from the fact that any two non-parallel lines can intersect only at one point. It is this intersection point that will lie on both the first line and the second, that is, it will be a solution to both the first equation and the second. Therefore, it is a solution to a system of equations. However, it is necessary to stipulate situations when certain restrictions are imposed on the values ​​of x and y (usually according to the conditions of the problem). For example, x > 0, y > 0. In this case, even if the system of equations has a solution, but it does not satisfy the condition, then the conclusion is drawn that the system of equations has no solutions under the given conditions.

There are three ways to solve a system of equations:

  1. By selection method. Most often this is very difficult to do.
  2. Graphic method. When two straight lines (graphs of functions of the corresponding equations) are drawn on the coordinate plane and their point of intersection is found. This method may not give accurate results if the coordinates of the intersection point are fractional numbers.
  3. Algebraic methods. They are versatile and reliable.

Instructions

Given a system of two linear equations, solve it as follows. Choose one of the equations in which the coefficients in front of the variables are smaller and express one of the variables, for example, x. Then substitute this value containing y into the second equation. In the resulting equation there will be only one variable y, move all parts with y to the left side, and free ones to the right. Find y and substitute into any of the original equations to find x.

There is another way to solve a system of two equations. Multiply one of the equations by a number so that the coefficient of one of the variables, such as x, is the same in both equations. Then subtract one of the equations from the other (if the right-hand side is not equal to 0, remember to subtract the right-hand sides in the same way). You will see that the x variable has disappeared and only one y variable remains. Solve the resulting equation, and substitute the found value of y into any of the original equalities. Find x.

The third way to solve a system of two linear equations is graphical. Draw a coordinate system and graph two straight lines whose equations are given in your system. To do this, substitute any two x values ​​into the equation and find the corresponding y - these will be the coordinates of the points belonging to the line. The most convenient way to find the intersection with the coordinate axes is to simply substitute the values ​​x=0 and y=0. The coordinates of the point of intersection of these two lines will be the tasks.

If there is only one linear equation in the problem conditions, then you have been given additional conditions through which you can find a solution. Read the problem carefully to find these conditions. If the variables x and y denote distance, speed, weight, feel free to set the limit x≥0 and y≥0. It is quite possible that x or y hides the number of apples, trees, etc. – then the values ​​can only be integers. If x is the son’s age, it is clear that he cannot be older than his father, so indicate this in the conditions of the problem.

Construct a line graph corresponding to the linear equation. Look at the graph, there may be only a few solutions that satisfy all the conditions - for example, integers and positive numbers. They will be the solutions to your equation.

Sources:

  • how to solve an equation with one variable

One of the main problems of mathematics is solving a system of equations with several unknowns. This is a very practical problem: there are several unknown parameters, several conditions are imposed on them, and it is necessary to find their most optimal combination. Such tasks are commonplace in economics, construction, design of complex mechanical systems, and in general wherever optimization of the costs of material and human resources is required. In this regard, the question arises: how to solve such systems?

Instructions

Mathematics gives us two ways to solve such systems: graphical and analytical. These methods are equivalent, and it cannot be said that any of them is better or worse. In each situation, when optimizing a solution, you need to choose which method gives a simpler solution. But there are also some typical situations. Thus, a system of plane equations, i.e. when two graphs have the form y=ax+b, is easier to solve graphically. Everything is done very simply: two straight lines are constructed: graphs of linear functions, then their intersection point is found. The coordinates of this point (abscissa and ordinate) will be the solution to this equation. Note also that two lines can be parallel. Then the system of equations has no solution, and the functions are called linearly dependent.

The opposite situation can also happen. If we need to find a third unknown, given two linearly independent equations, then the system will be underdetermined and have an infinite number of solutions. In the theory of linear algebra, it is proven that a system has a unique solution if and only if the number of equations coincides with the number of unknowns.

LESSON SUMMARY

Class: 7

UMK: Algebra 7th grade: textbook. for general education organizations / [Yu. N. Makarychev, N.G. Mindyuk et al.]; edited by S.A. Telyakovsky. – 2nd ed. – M.: Education, 2014

Subject: Linear equations in two variables

Goals: Introduce students to the concepts of a linear equation with two variables and its solution, teach how to express from the equationX throughat orat throughX .

Formed UUD:

Cognitive: put forward and justify hypotheses, suggest ways to test them

Regulatory: compare the method and result of one’s actions with a given standard, detect deviations and differences from the standard; draw up a plan and sequence of actions.

Communicative: establish working relationships; collaborate effectively and promote productive cooperation.

Personal: fdeveloping skills for organizing analysis of one’s activities

Equipment:computer, multimedia projector, screen

During the classes:

I Organizing time

Listen to the fairy tale about Grandfather Equally and guess what we will talk about today

Fairy tale "Grandfather-Equal"

A grandfather nicknamed Ravnyalo lived in a hut on the edge of a forest. He loved to joke with numbers. The grandfather will take the numbers on both sides of himself, connect them with signs, and put the fastest ones in brackets, but make sure that one part is equal to the other. And then he will hide some number under the mask of “X” and ask his grandson, little Ravnyalka, to find it. Even though Ravnyalka is small, he knows his stuff: he will quickly move all the numbers except “X” to the other side and will not forget to change their signs to the opposite. And the numbers obey him, quickly carry out all actions on his orders, and “X” is known. The grandfather looks at how cleverly his granddaughter does everything and rejoices: a good replacement for him is growing up.

So, what is this tale about?(about equations)

II . Let's remember everything we know about linear equations and try to draw a parallel between the material we know and the new material.

    What type of equation do we know?(linear equation with one variable)

    Let's recall the definition of a linear equation with one variable.

    What is the root of a linear equation in one variable?

    Let us formulate all the properties of a linear equation with one variable.

1 part of the table is filled in

ax = b, where x is a variable, a, b are numbers.

Example: 3x = 6

The value of x at which the equation becomes true

1) transferring terms from one part of the equation to another, changing their sign to the opposite.

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

Linear equation with two variables.

ax + vy = c, where x, y are variables, a, b.c are numbers.

Example:

x – y = 5

x + y = 56

2x + 6y =68

The values ​​of x, y that make the equation true.

x=8; y=3 (8;3)

x=60; y = - 4 (60;-4)

Properties 1 and 2 are true.

3) equivalent equations:

x-y=5 and y=x-5

(8;3) (8;3)

After we have filled out the first part of the table, based on analogy, we begin to fill out the second row of the table, thereby learning new material.

III . Let's get back to the topic:linear equation in two variables . The very title of the topic suggests that you need to introduce a new variable, for example y.

There are two numbers x and y, one greater than the other by 5. How to write the relationship between them? (x – y = 5) this is a linear equation with two variables. Let us formulate, by analogy with the definition of a linear equation with one variable, the definition of a linear equation with two variables (A linear equation in two variables is an equation of the formax + by = c , Wherea,b Andc - some numbers, andx Andy -variables).

The equation x – y= 5 with x = 8, y = 3 turns into the correct equality 8 – 3 = 5. They say that the pair of values ​​of the variables x = 8, y = 3 is a solution to this equation.

Formulate the definition of a solution to an equation with two variables (A solution to an equation with two variables is a pair of values ​​of variables that turns this equation into a true equality)

Pairs of variable values ​​are sometimes written shorter: (8;3). In such a notation, the value x is written in the first place and the value y in the second.

Equations with two variables that have the same solutions (or no solutions) are called equivalent.

Equations with two variables have the same properties as equations with one variable:

    If you move any term in an equation from one part to another, changing its sign, you will get an equation equivalent to the given one.

    If both sides of the equation are multiplied or divided by the same number (not equal to zero), you get an equation equivalent to the given one.

Example 1. Consider the equation 10x + 5y = 15. Using the properties of the equations, we express one variable in terms of another.

To do this, first move 10x from the left side to the right, changing its sign. We get the equivalent equation 5y = 15 - 10x.

Dividing each part of this equation by the number 5, we get the equivalent equation

y = 3 - 2x. Thus, we expressed one variable in terms of another. Using this equality, for each value of x we ​​can calculate the value of y.

If x = 2, then y = 3 - 2 2 = -1.

If x = -2, then y = 3 - 2· (-2) = 7. Pairs of numbers (2; -1), (-2; 7) are solutions to this equation. Thus, this equation has infinitely many solutions.

From the history. The problem of solving equations in natural numbers was considered in detail in the works of the famous Greek mathematician Diophantus (III century). His treatise “Arithmetic” contains ingenious solutions in natural numbers to a wide variety of equations. In this regard, equations with several variables that require solutions in natural numbers or integers are called Diophantine equations.

Example 2. Flour is packaged in bags of 3 kg and 2 kg. How many bags of each type should you take to make 20 kg of flour?

Let's say that we need to take x bags of 3 kg and y bags of 2 kg. Then 3x + 2y = 20. It is required to find all pairs of natural values ​​of the variables x and y that satisfy this equation. We get:

2y = 20 - 3x

y =

Substituting into this equality instead of x successively all the numbers 1,2,3, etc., we find for which values ​​of x, the values ​​of y are natural numbers.

We get: (2;7), (4;4), (6;1). There are no other pairs that satisfy this equation. This means you need to take either 2 and 7, or 4 and 4, or 6 and 1 packages, respectively.

IV . Work from the textbook (orally) No. 1025, No. 1027 (a)

Independent work with testing in class.

1. Write a linear equation with two variables.

a) 3x + 6y = 5 c) xy = 11 b) x – 2y = 5

2. Is a pair of numbers a solution to an equation?

2x + y = -5 (-4;3), (-1;-3), (0;5).

3. Express from linear equation

4x – 3y = 12 a) x through y b) y through x

4. Find three solutions to the equation.

x + y = 27

V . So, to summarize:

Define a linear equation with two variables.

What is called the solution (root) of a linear equation with two variables.

State the properties of a linear equation with two variables.

Grading.

Homework: paragraph 40, No. 1028, No. 1032