What is called a linear equation. Linear equations

1. The concept of an equation with one variable

2. The equivalent equations. Theorems on equality of equations

3. Solution of equations with one variable

Equations with one variable

Take two expressions with a variable: 4 H.and 5. H. + 2. By connecting them with the sign of equality, we get a sentence 4x= 5 H. + 2. It contains a variable and when substituting the values \u200b\u200bof the variable draws to the statement. For example, for x \u003d-2 Offer 4x= 5 H. + 2 turns to true numerical equality 4 · (-2) \u003d 5 · (-2) + 2, and when x \u003d1 - in false 4 · 1 \u003d 5 · 1 + 2. Therefore, the proposal 4x \u003d 5x + 2there is a utteral form. It is called equation with one variable.

IN general The equation with one variable can be determined as follows:

Definition. Let F (x) and G (x) be two expressions from the variable x and the field of definition X. Then the utteral form of the form f (x) \u003d g (x) is called the equation with one variable.

Variable value h.from set Xin which the equation refers to true numerical equality is called the root of the equation(or his decision). Solve equation -this means finding a lot of its roots.

So, the root of the equation 4x \u003d 5x+ 2, if we consider it on the set R. Actual numbers, is the number -2. This equation does not have other roots. So there are many roots (-2).

Suppose on the set of valid numbers the equation is set ( h. - 1) (x+ 2) \u003d 0. It has two roots - numbers 1 and -2. Therefore, many roots of this equation Such: (-2, -1).

The equation (3x +.1)-2 = 6 H. + 2, specified on the set of valid numbers, refers to true numerical equality with all valid values \u200b\u200bof the variable h.: If you reveal the brackets in the left side, then we get 6x + 2 \u003d 6x + 2.In this case, it is said that its root is any valid number, and many roots are many values \u200b\u200bof all valid numbers.

The equation (3x+ 1) · 2 \u003d 6 H. + 1, predetermined on the set of valid numbers, does not refund on true numerical equality at no one valid x:after the disclosures of the brackets in the left part we get that 6 H. + 2 = 6x +.1, which is impossible at no one x.In this case, it is said that this equation has no own and that many of its roots are empty.

To solve any equation, it is first transformed by replacing other, simpler; The resulting equation is again transformed by replacing easier, etc. This process is continued until the equation is obtained, the roots of which can be found in a known manner. But that these roots have the roots of a given equation, it is necessary that the equations, many roots of which coincide in the process of transformations. Such equations are called equivalent.

Class: 7

Lesson number 1.

Type of lesson: fixing the material passed.

Objectives lesson:

Educational:

skill formation solutions equation With one unknown note of it to a linear equation using the properties of equivalence.

Developing:

formation of clarity and accuracy of thought, logical thinking, elements of algorithmic culture;
development of mathematical speech;
development of attention, memory;
formation of skills of self and mutual test.

Educational:

formation of volitional qualities;
formation Communicability;
developing an objective assessment of their achievements;
formation of responsibility.

Equipment: Interactive board, board for markers, cards with tasks for independent work, cards for correcting knowledge for poorly speaking students, textbook, workbook, notebook for homework, notebook for independent work.

During the classes

2. Check homework - 4 min.

Students check homework, the solution of which is derived from the back of the board with one of the students.

3. Oral work - 6 min.

(1) While an oral account is underway, weakly speaking students get knowledge Correction Card and perform 1), 2), 4) and 6) specification tasks. (Cm. Attachment 1.)

Card for knowledge correction.

(2) For other students, tasks are projected on an interactive board: (see Presentation: Slide 2.)

Instead of an asterisk, put the sign "+" or "-", and instead of the points - numbers:
a) (* 5) + (* 7) \u003d 2;
b) (* 8) - (* 8) \u003d (* 4) -12;
c) (* 9) + (* 4) \u003d -5;
d) (-15) - (* ...) \u003d 0;
e) (* 8) + (* ...) \u003d -12;
e) (* 10) - (* ...) \u003d 12.
Composition equations equivalent to equation:
but) x - 7 \u003d 5;
b) 2x - 4 \u003d 0;
c) x -11 \u003d x - 7;
d) 2 (x -12) \u003d 2x - 24.

3. Logic task:Vika, Natasha and Lena in the store bought cabbage, apples and carrots. All bought different products. Vika bought a vegetable, Natasha - apples or carrots, Lena bought no vegetable. Who bought what? (One of the students who completed the task goes to the board and fills in the table.) (Slide 3)

	Vika	Natasha	Lena
TO
I
M.

Fill out the table

	Vika	Natasha	Lena
TO	+	–	–
I	–	–	+
M.	–	+	–

4. Generalization of the ability to solve the equation with information with the linear equation -9 min.

Collective work with class. (Slide 4)

Resolving equation

12 - (4x - 18) \u003d (36 + 5x) + (28 - 6x). (1)

to do this, perform the following transformations:

1. Recall brackets. If there is a sign "plus" in front of the brackets, then the brackets can be omitted by retaining the sign of each compound enclosed in brackets. If there is a "minus" sign in front of the brackets, then the brackets can be omitted by changing the sign of each compound enclosed in the bracket:

12 - 4x + 18 \u003d 36 + 5x + 28 - 6x. (2)

Equations (2) and (1) are equivalent:

2. We suffer with opposite signs unknown members so that they are only in one part of the equation (or in the left or right). At the same time, we will transfer well-known members with opposite signs so that they are only in another part of the equation.

For example, we transfer unknown members with opposite signs to the left, and known to the right of the equation, then we obtain the equation

- 4x - 5x + 6x \u003d 36 + 28 - 18 - 12, (3)

equivalent equation (2) , consequently, the equation (1) .

3. We give similar terms:

-3x \u003d 34. (4)

The equation (4) equivalent to equation (3) , consequently, the equation (1) .

4. We split both parts of the equation (4) on the coefficient at unknown.

The resulting equation x \u003d It will be equivalent to equation (4), and consequently, equations (3), (2), (1)

Therefore, the root of equation (1) will be the number

According to this scheme (algorithm), we solve the equation at today's lesson:

Remove brackets.
Collect members containing unknowns in one part of the equation, and the rest of the members in the other.
Certify such members.
Divide both parts of the coefficient equation at unknown.

Note: It should be noted that the shown scheme is not mandatory, since there are often equations that some of these stages are unnecessary. When solving other equations it is easier to retreat from this scheme, such as, for example, in equation:

7 (x - 2) \u003d 42.

5. Training exercises - 8 min.

No. 132 (A, D), 135 (A, D), 138 (B, D) - With a comment and writing on the board.

6. Independent work - 14 min. (performed in notebooks for independent work, followed by a mutually verification; Answers will be displayed on an interactive chalkboard)

Before independent work Students will be offered assurance Task - 2 min.

Without taking a pencil from paper and without passing twice to the same line of the line, draw the printed letter. (Slide 5)

(Students use plastic sheets and markers.)

1. Solve equations (on cards) (see Appendix 2.)

Additional task number135 (B, B).

7. Summing up the lesson - 1 min.

Algorithm Details of the equation to the linear equation.

8. Message of homework - 2 min.

p.6, No. 136 (AA-D), 240 (a), 243 (A, B), 224 (Clarify the content of the homework).

Lesson number 2.

Objectives lesson:

Educational:

repetition of rules, systematization, deepening and expansion of students' zins by solving linear equations;
the formation of the ability to apply the knowledge gained in solving equations in various ways.

Developing:

development of intellectual skills: analysis of an algorithm for solving an equation, logical thinking when constructing an algorithm for solving the equation, variations in the choice of a solution method, systematization of equations for solutions methods;
development of mathematical speech;
development of visual memory.

Educational:

education cognitive activity;
formation of self-control skills, interconnection and self-esteem;
education of the sense of responsibility, mutual assistance;
prolongation of accuracy, mathematical literacy;
upbringing a sense of partnership, courtesy, discipline, responsibility;
Health.

a) educational: repetition of rules, systematization, deepening and expansion of students' zins by solving linear equations;

b) developing: the development of flexibility of thinking, memory, attention and intelligence;

c) Educational: Increasing interest in the subject and to the history of the native land.

Equipment: Interactive board, signal cards (green and red), sheets with test work, textbook, workbook, home workbook, notebook for independent work.

Form of work: Individual, collective.

During the classes

1. Organizing time - 1 min.

Greeting students, check their readiness for the lesson, declare theme of the lesson and the purpose of the lesson.

2. Oral work - 10 min.

(Tasks for oral account Displays an interactive board.) (Slide 6)

1) Solve the tasks:

a) Mom older than his daughter for 22 years. How old mom, if they are 46 years old
b) In the family, the three brothers and every next younger than the previous one twice. Together with all brothers 21 years old. How old is everyone?

2) Decide equations:(Clarify)

4) Explain tasks from homeworkwho caused the difficulty.

3. Exercise exercises - 10 min. (Slide 8)

(1) What inequality is satisfied with the root of the equation:

a) x\u003e 1;
b) X.< 0;
c) x\u003e 0;
d) X.< –1.

(2) What is the value of expression w. The value of the expression 2y - 4. at 5 times less value Expressions 5th - 10?

(3) With what value k. the equation kX - 9 \u003d 0 Does the root of equal - 2?

Look and remember (7 seconds). (Slide 9)

After 30 seconds, students reproduce the drawing on plastic sheets.

4. Fizkultminutka - 1.5 min.

Exercise for eyes and hands

(Students look and repeat the exercises that are projected on an interactive board.)

5. Independent test work - 15 min.

(Students perform test work in notebooks for independent work, duplicating responses in workbooks. By passing tests, students are responsible for answers displayed on the board)

Students who coped with the work before all helps weakly speaking students.

6. Summing up the lesson - 2 min.

- What equation with one variable is called linear?

- What is called the root of the equation?

- What does it mean to "solve the equation"?

- How many roots may have an equation?

7. Message of homework. - 1 min.

p.6, number 294 (A, B), 244, 241 (A, B), 240 (g) - level A, in

p.6, No. 244, 241 (B, B), 243 (B), 239, 237- level with

(Clarify the content of homework.)

8. Reflection - 0.5 min.

- Are you satisfied with your work at the lesson?

- What kind of activity you liked most in the lesson.

Literature:

Algebra 7. / Yu.N. Makarychev, N.G. Mindyuk, k.I. Peshkov, S.V. Suvorov. Edited by S.A. Velyakovsky. / M.: Enlightenment, 1989 - 2006.
Collection test tasks For thematic and outcome control. Algebra Grade 7 / Guseva I.L., Pushkin S.A., Rybakova N.V.. General Ed.: Tatur A.O. - M.: "Intellect-Center" 2009 - 160 p.
Purchasing planning By algebra. / T.N. Eine. Manual for teachers / M: ed. "Exam", 2008. - 302, p.
Cards for the correction of knowledge in mathematics for grade 7. / Levitas G.G. / M.: Ilex, 2000. - 56 p.

Equality with a variable f (x) \u003d g (x) called equation from one variable x. Any value of the variable in which f (x) and g (x) take equal numeric values, is called the root of such an equation. Therefore, solve the equation - it means to find all the roots of the equation or prove that they are not.

Equation x 2 + 1 \u003d 0 does not have valid roots, but it has an imaginary roots: in this case, these are roots x 1 \u003d i, x 2 \u003d -i. In the future, we will also be interested in the valid roots of the equation.

If the equations have the same roots, then they are called equivalent. Those equations that do not have roots relate to equivalent.

We define whether equations are equivalent:

a) x + 2 \u003d 5 and x + 5 \u003d 8

1. Resister the first equation

2. Resister the second equation

The roots of the equations coincide, therefore x + 2 \u003d 5 and x + 5 \u003d 8 are equivalent.

b) x 2 + 1 \u003d 0 and 2x 2 + 5 \u003d 0

Both data equations do not have valid roots, therefore are equivalent.

c) x - 5 \u003d 1 and x 2 \u003d 36

1. Find the first equation roots

2. Find the roots of the second equation

x 1 \u003d 6, x 2 \u003d -6

The roots of the equations do not coincide, therefore x - 5 \u003d 1 and x 2 \u003d 36 are not fearful.

When solving the equation, it is trying to replace with an equivalent, but simpler equation. Therefore, it is important to know, as a result of which transformations, this equation passes into equations is equivalent to it.

Theorem 1. If in the equation from one part to another to transfer any term, changing the sign, then the equation is equivalent to this.

For example, the equation x 2 + 2 \u003d 3x is equivalent to the equation x 2 + 2 - 3x \u003d 0.

Theorem 2. If both parts of the equation are multiplying or divided into one and the same number (not equal to zero), the equation is equivalent to this.

For example, equation (x 2 - 1) / 3 \u003d 2x is equivalent to equation x 2 - 1 \u003d 6x. Both parts of the first equation we multiplied by 3.

The linear equation with one variable is called the equation of the form ah \u003d b, where a and b is valid numbers, and the coefficient is called the coefficient with a variable, and B is a free member.

Consider three cases for the linear equation ah \u003d b.

1. A ≠ 0. In this case, x \u003d b / A (because but different from zero).

2. a \u003d 0, b \u003d 0. The equation will take the form: 0 ∙ x \u003d 0. This equation is true for any x, i.e. The root of the equation is any valid number.

3. a \u003d 0, b ≠ 0. In this case, the equation will not have roots, because The division to zero is prohibited (0 ∙ x \u003d b).

As a result of transformations, many equations are reduced to linear.

Resolving equations

a) (1/5) x + 2/15 \u003d 0

1. We transfer the 2/15 component from the left part of the equation to the right with the opposite sign. Such a transformation is regulated by theorem 1. So, the equation will take the form: (1/5) x \u003d -2/15.

2. To get rid of the denominator, the dominaries of both parts of the equation for 15. Make it allows us theorem 2. So, the equation will take the form:

(1/5) x ∙ 15 \u003d - 2/15 ∙ 15

Thus, the root of the equation is -2/3.

b) 2/3 + x / 4 + (1 - x) / 6 \u003d 5x / 12 - 1

1. To get rid of the denominator, the dominaries of both parts equation at 12 (by Theorem 2). The equation will take the form:

12 (2/3 + x / 4 + (1 - x) / 6) \u003d 12 (5x / 12 - 1)

8 + 3x + 2 - 2x \u003d 5x - 12

10 + x \u003d 5x - 12

2. Using Theorem 1, "Gather" all numbers on the right, and components with x - left. The equation will take the form:

10 +12 \u003d 5x - x

Thus, the equation root is 5.5.

the site, with full or partial copying of the material reference to the original source is required.

Equality with variable is called the equation.
Solve equation - it means to find a lot of its roots. The equation may have one, two, several, many roots or not have them at all.
Each value of the variable in which this equation turns into faithful equality is called the root of the equation.
Equations having the same roots are called equivalent equations.
Any characteristic equation can be transferred from one part of the equality to another, when you change the sign of the allegiated to the opposite.
If both parts of the equation are multiplied or divided into one and the same different number from zero, the equation is equivalent to this equation.

Examples. Solve equation.

1. 1.5x + 4 \u003d 0.3x-2.

1.5x-0.3x \u003d -2-4. Collected terms containing a variable in the left part of equality, and free members - in the right part of equality. At the same time used property:

1.2x \u003d -6. Led similar terms according to rule:

x \u003d -6. : 1.2. Both parts of equality were divided into a coefficient with a variable, since

x \u003d -5. Divided the decimal fraction on the division rule on decimal fraction:

to divide the number for a decimal fraction, you need to transfer commas in divide and divider to as many digits to the right, how many of them are after the comma in the divider, and then perform division to the natural number:

6 : 1,2 = 60 : 12 = 5.

Answer: 5.

2. 3∙ (2x-9) \u003d 4 ∙ (X-4).

6x-27 \u003d 4x-16. Opened brackets using the distribution law of multiplication relative to subtraction: (A-B) ∙ C \u003d A. ∙ C-B. ∙ c.

6x-4x \u003d -16 + 27. Collected terms containing a variable in the left part of equality, and free members - in the right part of equality. At the same time used property: any characteristic equation can be transferred from one part of the equality to another, when you change the sign of the allegiated to the opposite.

2x \u003d 11. Led similar terms according to the rule: to bring similar terms, it is necessary to fold their coefficients and the resulting result multiply to their general lettering part (i.e., to the resulting result to attribute them to the overall lettering).

x \u003d 11. : 2. Both parts of equality were divided into a coefficient with a variable, since if both parts of the equation are multiplied or divided into one and the same different number from zero, the equation is equivalent to this equation.

Answer: 5,5.

3. 7x- (3 + 2x) \u003d x-9.

7x-3-2x \u003d x-9. Opened brackets according to the rules of disclosing brackets, in front of which there is a sign "-": if the "-" sign stands in front of the brackets, then we remove the brackets, the sign "-" and write the terms stood in brackets, with opposite signs.

7x-2x-x \u003d -9 + 3. Collected terms containing a variable in the left part of equality, and free members - in the right part of equality. At the same time used property: any characteristic equation can be transferred from one part of the equality to another, when you change the sign of the allegiated to the opposite.

4x \u003d -6. Led similar terms according to rule: to bring similar terms, it is necessary to fold their coefficients and the resulting result multiply to their general lettering part (i.e., to the resulting result to attribute them to the overall lettering).

x \u003d -6. : 4. Both parts of equality were divided into a coefficient with a variable, since if both parts of the equation are multiplied or divided into one and the same different number from zero, the equation is equivalent to this equation.

Answer: -1,5.

3 ∙ (x-5) \u003d 7 ∙ 12 — 4 ∙ (2x-11). Multiple both parts of equality at 12 - the smallest common denominator for the data denominers.

3x-15 \u003d 84-8x + 44. Opened brackets using the distribution law of multiplication relative to subtraction: to multiply the difference between the two numbers to multiply by the third number, it is possible to separately be reduced and separately subtracted to multiply by the third number, and then from the first result, the second result is from the first result, i.e. (A-B) ∙ C \u003d A. ∙ C-B. ∙ c.

3x + 8x \u003d 84 + 44 + 15. Collected terms containing a variable in the left part of equality, and free members - in the right part of equality. At the same time used property: any characteristic equation can be transferred from one part of the equality to another, when you change the sign of the allegiated to the opposite.

Linear equation - This is an algebraic equation. In this equation, the total degree of components of its polynomials is equal to one.

Linear equations are presented in this form:

In general form: a. 1 x. 1 + a. 2 x. 2 + … + a n x n + b. = 0

In canonical form: a 1 x 1 + a 2 x 2 + ... + a n x n \u003d b.

Linear equation with one variable.

The linear equation with the 1st variable is given to the form:

aX.+ b.=0.

For example:

2x + 7 \u003d 0. Where a \u003d 2, b \u003d 7;

0.1x - 2.3 \u003d 0. Where a \u003d 0.1, b \u003d -2.3;

12x + 1/2 \u003d 0. Where a \u003d 12, B \u003d 1/2.

The number of roots depending on a. and b.:

When a.= b.=0 So, the equation has an unlimited number of solutions, since.

When a.=0 , b.≠ 0 It means that the equation has no roots, because.

When a. ≠ 0 It means that the equation has only one root.

Linear equation with two variables.

By the variable equation x. It is equality type A (x) \u003d b (x)where A (X) and B (x) - Expressions OT x.. When substituting the set T. Values x.the equation is obtained by true numerical equality called a lot of truth this equation either solution of a given equation, and all the values \u200b\u200bof the variable - root equations.

Linear equations of 2 variables are presented in this form:

In general form: aX + BY + C \u003d 0,

In canonical form: aX + BY \u003d -C,

In the form of a linear function: y \u003d kx + mwhere .

The solution or roots of this equation is such a pair of variable values (x; y)which turns it into identity. These solutions (roots) in a linear equation with 2 variables unlimited quantity. The geometric model (graph) of this equation is straight y \u003d kx + m.

If there is an X-in equation in a square, then such an equation is called