Linear equations. Types of linear equations. Linear equations with one and two variables, linear inequalities How to understand a linear equation with two variables

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

We have often come across equations of the form ax + b = 0, where a, b are numbers, x is a variable. For example, bx - 8 = 0, x + 4 = O, - 7x - 11 = 0, etc. The numbers a, b (equation coefficients) can be any, except for the case when a = 0.

The equation ax + b = 0, where a, is called a linear equation with one variable x (or a linear equation with one unknown x). We can solve it, that is, express x through a and b:

We noted earlier that quite often mathematical model the real situation is a linear equation with one variable or an equation that, after transformations, reduces to a linear one. Now let's look at this real situation.

From cities A and B, the distance between which is 500 km, two trains departed towards each other, each with its own constant speed. It is known that the first train left 2 hours earlier than the second. 3 hours after the second train left, they met. What are the train speeds?

Let's create a mathematical model of the problem. Let x km/h be the speed of the first train, y km/h be the speed of the second train. The first one was on the road for 5 hours and, therefore, covered a distance of bx km. The second train was on the way for 3 hours, i.e. walked a distance of 3 km.

Their meeting took place at point C. Figure 31 shows a geometric model of the situation. In algebraic language it can be described as follows:

5x + Zu = 500

or
5x + Zu - 500 = 0.

This mathematical model is called a linear equation with two variables x, y.
At all,

ax + by + c = 0,

where a, b, c are numbers, and , is linear the equation with two variables x and y (or with two unknowns x and y).

Let's return to the equation 5x + 3 = 500. We note that if x = 40, y = 100, then 5 40 + 3 100 = 500 is a correct equality. This means that the answer to the question of the problem can be as follows: the speed of the first train is 40 km/h, the speed of the second train is 100 km/h. A pair of numbers x = 40, y = 100 is called a solution to the equation 5x + 3 = 500. It is also said that this pair of values ​​(x; y) satisfies the equation 5x + 3 = 500.

Unfortunately, this solution is not the only one (we all love certainty and unambiguity). In fact, the following option is also possible: x = 64, y = 60; indeed, 5 64 + 3 60 = 500 is a correct equality. And this: x = 70, y = 50 (since 5 70 + 3 50 = 500 is a true equality).

But, say, a pair of numbers x = 80, y = 60 is not a solution to the equation, since with these values ​​a true equality does not work:

In general, a solution to the equation ax + by + c = 0 is any pair of numbers (x; y) that satisfies this equation, that is, turns the equality with the variables ax + by + c = 0 into a true numerical equality. There are infinitely many such solutions.

Comment. Let us return once again to the equation 5x + 3 = 500, obtained in the problem discussed above. Among the infinite number of its solutions there are, for example, the following: x = 100, y = 0 (indeed, 5 100 + 3 0 = 500 is a correct numerical equality); x = 118, y = - 30 (since 5,118 + 3 (-30) = 500 is a correct numerical equality). However, being solutions to the equation, these pairs cannot serve as solutions to this problem, because the speed of the train cannot be equal to zero (then it does not move, but stands still); Moreover, the speed of the train cannot be negative (then it does not travel towards another train, as stated in the problem statement, but in the opposite direction).

Example 1. Draw solutions to a linear equation with two variables x + y - 3 = 0 by points in the xOy coordinate plane.

Solution. Let's select several solutions to a given equation, that is, several pairs of numbers that satisfy the equation: (3; 0), (2; 1), (1; 2) (0; 3), (- 2; 5).

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions

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§ 1 Selection of equation roots in real situations

Let's consider this real situation:

The master and apprentice together made 400 custom parts. Moreover, the master worked for 3 days, and the student for 2 days. How many parts did each person make?

Let's create an algebraic model of this situation. Let the master produce parts in 1 day. And the student is at the details. Then the master will make 3 parts in 3 days, and the student will make 2 parts in 2 days. Together they will produce 3 + 2 parts. Since, according to the condition, a total of 400 parts were manufactured, we obtain the equation:

The resulting equation is called a linear equation in two variables. Here we need to find a pair of numbers x and y for which the equation will take the form of a true numerical equality. Note that if x = 90, y = 65, then we get the equality:

3 ∙ 90 + 65 ∙ 2 = 400

Since the correct numerical equality has been obtained, the pair of numbers 90 and 65 will be a solution to this equation. But the solution found is not the only one. If x = 96 and y = 56, then we get the equality:

96 ∙ 3 + 56 ∙ 2 = 400

This is also a true numerical equality, which means that the pair of numbers 96 and 56 is also a solution to this equation. But a pair of numbers x = 73 and y = 23 will not be a solution to this equation. In fact, 3 ∙ 73 + 2 ∙ 23 = 400 will give us the incorrect numerical equality 265 = 400. It should be noted that if we consider the equation in relation to this real situation, then there will be pairs of numbers that, being a solution to this equation, will not be a solution to the problem. For example, a couple of numbers:

x = 200 and y = -100

is a solution to the equation, but the student cannot make -100 parts, and therefore such a pair of numbers cannot be the answer to the question of the problem. Thus, in each specific real situation it is necessary to take a reasonable approach to selecting the roots of the equation.

Let's summarize the first results:

An equation of the form ax + bу + c = 0, where a, b, c are any numbers, is called a linear equation with two variables.

The solution to a linear equation in two variables is a pair of numbers corresponding to x and y, for which the equation turns into a true numerical equality.

§ 2 Graph of a linear equation

The very recording of the pair (x;y) leads us to think about the possibility of depicting it as a point with coordinates xy y on a plane. This means that we can obtain a geometric model of a specific situation. For example, consider the equation:

2x + y - 4 = 0

Let's select several pairs of numbers that will be solutions to this equation and construct points with the found coordinates. Let these be points:

A(0; 4), B(2; 0), C(1; 2), D(-2; 8), E(- 1; 6).

Note that all points lie on the same line. This line is called the graph of a linear equation in two variables. It is a graphical (or geometric) model of a given equation.

If a pair of numbers (x;y) is a solution to the equation

ax + vy + c = 0, then the point M(x;y) belongs to the graph of the equation. We can say the other way around: if the point M(x;y) belongs to the graph of the equation ax + y + c = 0, then the pair of numbers (x;y) is a solution to this equation.

From the geometry course we know:

To construct a straight line, you need 2 points, so to plot a graph of a linear equation with two variables, it is enough to know only 2 pairs of solutions. But guessing the roots is not always a convenient or rational procedure. You can act according to another rule. Since the abscissa of a point (variable x) is an independent variable, you can give it any convenient value. Substituting this number into the equation, we find the value of the variable y.

For example, let the equation be given:

Let x = 0, then we get 0 - y + 1 = 0 or y = 1. This means that if x = 0, then y = 1. A pair of numbers (0;1) is the solution to this equation. Let's set another value for the variable x: x = 2. Then we get 2 - y + 1 = 0 or y = 3. The pair of numbers (2;3) is also a solution to this equation. Using the two points found, it is already possible to construct a graph of the equation x - y + 1 = 0.

You can do this: first assign some specific value to the variable y, and only then calculate the value of x.

§ 3 System of equations

Find two natural numbers whose sum is 11 and difference is 1.

To solve this problem, we first create a mathematical model (namely, an algebraic one). Let the first number be x and the second number y. Then the sum of the numbers x + y = 11 and the difference of the numbers x - y = 1. Since both equations deal with the same numbers, these conditions must be met simultaneously. Usually in such cases a special record is used. The equations are written one below the other and combined with a curly brace.

Such a record is called a system of equations.

Now let’s construct sets of solutions to each equation, i.e. graphs of each of the equations. Let's take the first equation:

If x = 4, then y = 7. If x = 9, then y = 2.

Let's draw a straight line through points (4;7) and (9;2).

Let's take the second equation x - y = 1. If x = 5, then y = 4. If x = 7, then y = 6. We also draw a straight line through the points (5;4) and (7;6). We obtained a geometric model of the problem. The pair of numbers we are interested in (x;y) must be a solution to both equations. In the figure we see a single point that lies on both lines; this is the point of intersection of the lines.

Its coordinates are (6;5). Therefore, the solution to the problem will be: the first required number is 6, the second is 5.

List of used literature:

1. Mordkovich A.G., Algebra 7th grade in 2 parts, Part 1, Textbook for general education institutions / A.G. Mordkovich. – 10th ed., revised – Moscow, “Mnemosyne”, 2007
2. Mordkovich A.G., Algebra 7th grade in 2 parts, Part 2, Problem book for educational institutions / [A.G. Mordkovich and others]; edited by A.G. Mordkovich - 10th edition, revised - Moscow, “Mnemosyne”, 2007
3. HER. Tulchinskaya, Algebra 7th grade. Blitz survey: a manual for students of general education institutions, 4th edition, revised and expanded, Moscow, “Mnemosyne”, 2008
4. Alexandrova L.A., Algebra 7th grade. Thematic test papers in a new form for students of general education institutions, edited by A.G. Mordkovich, Moscow, “Mnemosyne”, 2011
5. Alexandrova L.A. Algebra 7th grade. Independent works for students of general education institutions, edited by A.G. Mordkovich - 6th edition, stereotypical, Moscow, “Mnemosyne”, 2010

Any schoolchild begins to study this topic in the elementary grades, when he goes through the signs “greater than,” “less than,” and “equal to.” This type of inequalities and equations is one of the simplest in the entire curriculum for the entire period of study of a schoolchild and student. The solution to absolutely any equation or inequality comes down to simplifying it to a linear form. What do linear equations and inequalities look like?

In such an equation, the unknown is in the first degree, which allows you to simply and quickly separate variables from constants by placing them on opposite sides of the dividing sign (equality or inequality). What does a method look like that will help you easily and simply solve any linear equation?

Let's say there is an equation 3x - 89 = (5x - 32)/2. The first thing to do is to simplify the fractional part by multiplying the entire equation by 2. Then the result will be that 6x - 178 = 5x - 32. In fact, this is already a linear equation. Now we need to simplify it by moving all the variables to the left side and the constants to the right. The result is that x = 146. If the factor of the variable is greater than one, the entire linear equation should be divided by it, and in this case the required answer will be obtained.

The same applies to inequalities. First, you need to simplify the linear inequality, and then move the variables to its left side, and the constants to the right. After this, the linear inequality is simplified again so that the coefficient of the variable is equal to one. The answer to the inequality is obtained automatically, after which it only needs to be written down in the required form (in the form of an inequality, an interval or a gap on the axis).

As you can understand from the above, linear equations and inequalities are very simple even for elementary school children. However, it is worth remembering that this type of equation has variations.

There is such a type of them as linear equations with two variables. How to solve them? This is a rather labor-intensive process. At school, such cases begin to be encountered, therefore, linear equations with two variables can be classified as more complex topics.

Let's say there is an equation 2x + y = 3x + 17. The first thing to do is to express one unknown quantity in terms of another. This is done quite simply: one variable is moved to the left, all other variables and numbers are moved to the right; all linear equations with two variables are solved in this way. As a result, you will receive an equation of the form y = x + 17. The answer is expressed by plotting this function in a coordinate system and has the form of a straight line. This is how linear equations with two variables are solved.

It is also worth noting that in addition to equations with two variables, there are similar inequalities. Unlike equations in which the answer is the graph of a function, an inequality contains its answer in the plane limited by this graph. It is worth considering: if the inequality is strict, then the graph is not included in the answer!

So now you have an idea of ​​how to solve linear equations and inequalities. Although this topic is quite simple to study, it is worth paying attention to, since some of the subtleties may not be very clear, which can lead to unpleasant mistakes on the control test and a decrease in the final scores. Linear equation - it's simple, the main thing - adhere to the necessary mathematical rules, such as dividing or multiplying the entire equation by any value, transferring elements of a function beyond the equal sign, correctly constructing graphs, and writing the answer correctly.

Knowing how to correctly write and solve linear equations and inequalities will help you understand more complex types of equations and inequalities. That is why this topic is considered so important - almost the cornerstone of mathematics, because the principles for solving such examples underlie the solution of the lion's share of other equations, inequalities and problems.