Which equation is called the equation of a line. Definition of the equation of a line, examples of a line on a plane. Parametric line equations

An equality of the form F(x, y) = 0 is called an equation with two variables x, y if it is not true for all pairs of numbers x, y. They say that two numbers x = x 0, y = y 0 satisfy some equation of the form F(x, y) = 0 if, when substituting these numbers instead of the variables x and y into the equation, its left side becomes zero.

The equation of a given line (in a designated coordinate system) is an equation with two variables that is satisfied by the coordinates of every point lying on this line and not satisfied by the coordinates of every point not lying on it.

In what follows, instead of the expression “given the equation of the line F(x, y) = 0,” we will often say more briefly: given the line F(x, y) = 0.

If the equations of two lines are given: F(x, y) = 0 and Ф(x, y) = 0, then the joint solution of the system

F(x,y) = 0, Ф(x, y) = 0

gives all their intersection points. More precisely, each pair of numbers that is a joint solution of this system determines one of the intersection points,

157. Given points *) M 1 (2; -2), M 2 (2; 2), M 3 (2; - 1), M 4 (3; -3), M 5 (5; -5), M 6 (3; -2). Determine which of the given points lie on the line defined by the equation x + y = 0 and which do not lie on it. Which line is defined by this equation? (Draw it on the drawing.)

158. On the line defined by the equation x 2 + y 2 = 25, find points whose abscissas are equal to the following numbers: 1) 0, 2) -3, 3) 5, 4) 7; on the same line find points whose ordinates are equal to the following numbers: 5) 3, 6) -5, 7) -8. Which line is defined by this equation? (Draw it on the drawing.)

159. Determine which lines are determined by the following equations (construct them on the drawing): 1)x - y = 0; 2) x + y = 0; 3) x - 2 = 0; 4)x + 3 = 0; 5) y - 5 = 0; 6) y + 2 = 0; 7) x = 0; 8) y = 0; 9) x 2 - xy = 0; 10) xy + y 2 = 0; 11) x 2 - y 2 = 0; 12) xy = 0; 13) y 2 - 9 = 0; 14) x 2 - 8x + 15 = 0; 15) y 2 + by + 4 = 0; 16) x 2 y - 7xy + 10y = 0; 17) y - |x|; 18) x - |y|; 19) y + |x| = 0; 20) x + |y| = 0; 21) y = |x - 1|; 22) y = |x + 2|; 23) x 2 + y 2 = 16; 24) (x - 2) 2 + (y - 1) 2 = 16; 25 (x + 5) 2 + (y-1) 2 = 9; 26) (x - 1) 2 + y 2 = 4; 27) x 2 + (y + 3) 2 = 1; 28) (x - 3) 2 + y 2 = 0; 29) x 2 + 2y 2 = 0; 30) 2x 2 + 3y 2 + 5 = 0; 31) (x - 2) 2 + (y + 3) 2 + 1 = 0.

160. Given lines: l)x + y = 0; 2)x - y = 0; 3)x 2 + y 2 - 36 = 0; 4) x 2 + y 2 - 2x + y = 0; 5) x 2 + y 2 + 4x - 6y - 1 = 0. Determine which of them pass through the origin.

161. Given lines: 1) x 2 + y 2 = 49; 2) (x - 3) 2 + (y + 4) 2 = 25; 3) (x + 6) 2 + (y - Z) 2 = 25; 4) (x + 5) 2 + (y - 4) 2 = 9; 5) x 2 + y 2 - 12x + 16y - 0; 6) x 2 + y 2 - 2x + 8y + 7 = 0; 7) x 2 + y 2 - 6x + 4y + 12 = 0. Find their points of intersection: a) with the Ox axis; b) with the Oy axis.

162. Find the intersection points of two lines:

1) x 2 + y 2 - 8; x - y =0;

2) x 2 + y 2 - 16x + 4y + 18 = 0; x + y = 0;

3) x 2 + y 2 - 2x + 4y - 3 = 0; x 2 + y 2 = 25;

4) x 2 + y 2 - 8y + 10y + 40 = 0; x 2 + y 2 = 4.

163. In the polar coordinate system, the points M 1 (l; π/3), M 2 (2; 0), M 3 (2; π/4), M 4 (√3; π/6) and M 5 ( 1; 2/3π). Determine which of these points lie on the line defined in polar coordinates by the equation p = 2cosΘ, and which do not lie on it. Which line is determined by this equation? (Draw it on the drawing.)

164. On the line defined by the equation p = 3/cosΘ, find points whose polar angles are equal to the following numbers: a) π/3, b) - π/3, c) 0, d) π/6. Which line is defined by this equation? (Build it on the drawing.)

165. On the line defined by the equation p = 1/sinΘ, find points whose polar radii are equal to the following numbers: a) 1 6) 2, c) √2. Which line is defined by this equation? (Build it on the drawing.)

166. Establish which lines are determined in polar coordinates by the following equations (construct them on the drawing): 1) p = 5; 2) Θ = π/2; 3) Θ = - π/4; 4) p cosΘ = 2; 5) p sinΘ = 1; 6.) p = 6cosΘ; 7) p = 10 sinΘ; 8) sinΘ = 1/2; 9) sinp = 1/2.

167. Construct the following Archimedes spirals on the drawing: 1) p = 20; 2) p = 50; 3) p = Θ/π; 4) p = -Θ/π.

168. Construct the following hyperbolic spirals on the drawing: 1) p = 1/Θ; 2) p = 5/Θ; 3) p = π/Θ; 4) p= - π/Θ

169. Construct the following logarithmic spirals on the drawing: 1) p = 2 Θ; 2) p = (1/2) Θ.

170. Determine the lengths of the segments into which the Archimedes spiral p = 3Θ is cut by a beam emerging from the pole and inclined to the polar axis at an angle Θ = π/6. Make a drawing.

171. On the Archimedes spiral p = 5/πΘ, point C is taken, the polar radius of which is 47. Determine how many parts this spiral cuts the polar radius of point C. Make a drawing.

172. On the hyperbolic spiral P = 6/Θ, find a point P whose polar radius is 12. Make a drawing.

173. On a logarithmic spiral p = 3 Θ, find a point P whose polar radius is 81. Make a drawing.

Solving the equation

Illustration of a graphical method for finding the roots of an equation

Solving an equation is the task of finding such values of the arguments at which this equality is achieved. Additional conditions (integer, real, etc.) can be imposed on the possible values of the arguments.

Substituting another root produces an incorrect statement:

Thus, the second root must be discarded as extraneous.

Types of equations

There are algebraic, parametric, transcendental, functional, differential and other types of equations.

Some classes of equations have analytical solutions, which are convenient because they not only give the exact value of the root, but also allow you to write the solution in the form of a formula, which can include parameters. Analytical expressions allow not only to calculate the roots, but also to analyze their existence and their quantity depending on the parameter values, which is often even more important for practical use than the specific values of the roots.

Equations for which analytical solutions are known include algebraic equations of no higher than the fourth degree: linear equation, quadratic equation, cubic equation and fourth degree equation. Algebraic equations of higher degrees in the general case do not have an analytical solution, although some of them can be reduced to equations of lower degrees.

An equation that includes transcendental functions is called transcendental. Among them, analytical solutions are known for some trigonometric equations, since the zeros of trigonometric functions are well known.

In the general case, when an analytical solution cannot be found, numerical methods are used. Numerical methods do not provide an exact solution, but only allow one to narrow the interval in which the root lies to a certain predetermined value.

Examples of equations

Literature

Bekarevich, A. B. Equations in a school mathematics course / A. B. Bekarevich. - M., 1968.
Markushevich, L. A. Equations and inequalities in the final repetition of the high school algebra course / L. A. Markushevich, R. S. Cherkasov. / Mathematics at school. - 2004. - No. 1.
Kaplan Y. V. Rivnyannya. - Kyiv: Radyanska School, 1968.
The equation- article from the Great Soviet Encyclopedia
Equations// Collier's Encyclopedia. - Open society. 2000.
The equation// Encyclopedia Around the World
The equation// Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

Links

EqWorld - World of Mathematical Equations - contains extensive information about mathematical equations and systems of equations.

Wikimedia Foundation. 2010.

Synonyms:

Antonyms:

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See what “Equation” is in other dictionaries:

THE EQUATION- (1) a mathematical representation of the problem of finding such values of the arguments (see (2)), for which the values of two data (see) are equal. The arguments on which these functions depend are called unknowns, and the values of the unknowns at which the values ... ... Big Polytechnic Encyclopedia

THE EQUATION- EQUATION, equations, cf. 1. Action under Ch. equalize equalize and condition according to ch. equalize equalize. Equal rights. Equation of time (translation of true solar time into mean solar time, accepted in society and in science;... ... Ushakov's Explanatory Dictionary

THE EQUATION- (equation) The requirement that a mathematical expression take on a specific value. For example, a quadratic equation is written as: ax2+bx+c=0. The solution is the value of x at which the given equation becomes an identity. IN… … Economic dictionary

THE EQUATION- a mathematical representation of the problem of finding the values of the arguments for which the values of two given functions are equal. The arguments on which these functions depend are called unknowns, and the values of the unknowns at which the function values are equal... ... Big Encyclopedic Dictionary

THE EQUATION- EQUATION, two expressions connected by an equal sign; these expressions involve one or more variables called unknowns. To solve an equation means to find all the values of the unknowns at which it becomes an identity, or to establish... Modern encyclopedia

Target: Consider the concept of a line on a plane, give examples. Based on the definition of a line, introduce the concept of an equation of a line on a plane. Consider the types of straight lines, give examples and methods of defining a straight line. Strengthen the ability to translate the equation of a straight line from a general form into an equation of a straight line “in segments”, with an angular coefficient.

Equation of a line on a plane.
Equation of a straight line on a plane. Types of equations.
Methods for specifying a straight line.

1. Let x and y be two arbitrary variables.

Definition: A relation of the form F(x,y)=0 is called equation , if it is not true for any pairs of numbers x and y.

Example: 2x + 7y – 1 = 0, x 2 + y 2 – 25 = 0.

If the equality F(x,y)=0 holds for any x, y, then, therefore, F(x,y) = 0 is an identity.

Example: (x + y) 2 - x 2 - 2xy - y 2 = 0

They say that the numbers x are 0 and y are 0 satisfy the equation , if when substituting them into this equation it turns into a true equality.

The most important concept of analytical geometry is the concept of the equation of a line.

Definition: The equation of a given line is the equation F(x,y)=0, which is satisfied by the coordinates of all points lying on this line, and not satisfied by the coordinates of any of the points not lying on this line.

The line defined by the equation y = f(x) is called the graph of f(x). The variables x and y are called current coordinates, because they are the coordinates of a variable point.

Some examples line definitions.

1) x – y = 0 => x = y. This equation defines a straight line:

2) x 2 - y 2 = 0 => (x-y)(x+y) = 0 => points must satisfy either the equation x - y = 0, or the equation x + y = 0, which corresponds on the plane to a pair of intersecting straight lines that are bisectors of coordinate angles:

3) x 2 + y 2 = 0. This equation is satisfied by only one point O(0,0).

2. Definition: Any straight line on the plane can be specified by a first-order equation

Ax + Wu + C = 0,

Moreover, the constants A and B are not equal to zero at the same time, i.e. A 2 + B 2 ¹ 0. This first order equation is called general equation of a straight line.

Depending on the values of constants A, B and C, the following special cases are possible:

C = 0, A ¹ 0, B ¹ 0 – the straight line passes through the origin

A = 0, B ¹ 0, C ¹ 0 (By + C = 0) - straight line parallel to the Ox axis

B = 0, A ¹ 0, C ¹ 0 (Ax + C = 0) – straight line parallel to the Oy axis

B = C = 0, A ¹ 0 – the straight line coincides with the Oy axis

A = C = 0, B ¹ 0 – the straight line coincides with the Ox axis

The equation of a straight line can be presented in different forms depending on any given initial conditions.

Equation of a straight line with an angular coefficient.

If the general equation of the straight line Ax + By + C = 0 is reduced to the form:

and denote , then the resulting equation is called equation of a straight line with slope k.

Equation of a straight line in segments.

If in the general equation of the straight line Ах + Ву + С = 0 С ¹ 0, then, dividing by –С, we get: or , where

The geometric meaning of the coefficients is that the coefficient A is the coordinate of the point of intersection of the line with the Ox axis, and b– the coordinate of the point of intersection of the straight line with the Oy axis.

Normal equation of a line.

If both sides of the equation Ax + By + C = 0 are divided by a number called normalizing factor, then we get

xcosj + ysinj - p = 0 – normal equation of a straight line.

The sign ± of the normalizing factor must be chosen so that m×С< 0.

p is the length of the perpendicular dropped from the origin to the straight line, and j is the angle formed by this perpendicular with the positive direction of the Ox axis.

3. Equation of a straight line using a point and slope.

Let the angular coefficient of the line be equal to k, the line passes through the point M(x 0, y 0). Then the equation of the straight line is found by the formula: y – y 0 = k(x – x 0)

Equation of a line passing through two points.

Let two points M 1 (x 1, y 1, z 1) and M 2 (x 2, y 2, z 2) be given in space, then the equation of the line passing through these points is:

If any of the denominators is zero, the corresponding numerator should be set equal to zero.

On the plane, the equation of the straight line written above is simplified:

if x 1 ¹ x 2 and x = x 1, if x 1 = x 2.

The fraction = k is called slope straight.

Equality of the form F (x, y) = 0 called an equation in two variables x, y, if it is not true for all pairs of numbers x, y. They say two numbers x = x 0 , y=y 0, satisfy some equation of the form F(x, y)=0, if when substituting these numbers instead of variables X And at in the equation, its left side vanishes.

In what follows, instead of the expression “the equation of the line is given F(x, y) = 0" we will often say in short: given a line F (x, y) = 0.

If the equations of two lines are given F(x, y) = 0 And Ф(x, y) = Q, then the joint solution of the system

gives all their intersection points. More precisely, each pair of numbers that is a joint solution of this system determines one of the intersection points.

*) In cases where the coordinate system is not named, it is assumed that it is Cartesian rectangular.

157. Points are given *) M 1 (2; - 2), M 2 (2; 2), M 3 (2; - 1), M 4 (3; -3), M 5 (5; -5), M 6 (3; -2). Determine which published points lie on the line defined by the equation X+ y = 0, and which ones do not lie on it. Which line is defined by this equation? (Draw it on the drawing.)

158. On the line defined by the equation X 2 +y 2 =25, find the points whose abscissas are equal to the following numbers: a) 0, b) - 3, c) 5, d) 7; on the same line find points whose ordinates are equal to the following numbers: e) 3, f) - 5, g) - 8. Which line is determined by this equation? (Draw it on the drawing.)

159. Determine which lines are determined by the following equations (construct them on the drawing):

1) x - y = 0; 2) x + y = 0; 3) x- 2 = 0; 4) x+ 3 = 0;

5) y - 5 = 0; 6) y+ 2 = 0; 7) x = 0; 8) y = 0;

9) x 2 - xy = 0; 10) xy+ y 2 = 0; eleven) x 2 - y 2 = 0; 12) xy= 0;

13) y 2 - 9 = 0; 14) xy 2 - 8xy+15 = 0; 15) y 2 +5y+4 = 0;

16) X 2 y - 7xy + 10y = 0; 17) y =|x|; 18) x =|at|; 19)y + |x|=0;

20) x +|at|= 0; 21)y =|X- 1|; 22) y = |x+ 2|; 23) X 2 + at 2 = 16;

24) (x-2) 2 +(y-1) 2 =16; 25) (x+ 5) 2 +(y- 1) 2 = 9;

26) (X - 1) 2 + y 2 = 4; 27) x 2 +(y + 3) 2 = 1; 28) (x -3) 2 + y 2 = 0;

29) X 2 + 2y 2 = 0; 30) 2X 2 + 3y 2 + 5 = 0

31) (x- 2) 2 + (y + 3) 2 + 1=0.

160.Given lines:

1)X+ y = 0; 2)x - y = 0; 3) x 2 + y 2 - 36 = 0;

4) x 2 +y 2 -2x==0; 5) x 2 +y 2 + 4x-6y-1 =0.

Determine which of them pass through the origin.

161.Lines given:

1) x 2 + y 2 = 49; 2) (x- 3) 2 + (y+ 4) 2 = 25;

3) (x+ 6) 2 + (y - 3) 2 = 25; 4) ( x + 5) 2 + (y - 4) 2 = 9;

5) x 2 +y 2 - 12x + 16y = 0; 6) x 2 +y 2 - 2x + 8at+ 7 = 0;

7) x 2 +y 2 - 6x + 4y + 12 = 0.

Find their points of intersection: a) with the axis Oh; b) with an axis OU.

162.Find the intersection points of two lines;

1)X 2 +y 2 = 8, x-y = 0;

2) X 2 +y 2 -16x+4at+18 = 0, x + y= 0;

3) X 2 +y 2 -2x+4at -3 = 0, X 2 + y 2 = 25;

4) X 2 +y 2 -8x+10у+40 = 0, X 2 + y 2 = 4.

163. Points are given in the polar coordinate system

M 1 (1; ), M 2 (2; 0), M 3 (2; )

M 4 (
;) And M 5 (1; )

Determine which of these points lie on the line defined by the equation in polar coordinates  = 2 cos , and which do not lie on it. Which line is determined by this equation? (Draw it on the drawing:)

164. On the line defined by the equation  = , find points whose polar angles are equal to the following numbers: a) ,b) - , c) 0, d) . Which line is defined by this equation?

(Build it on the drawing.)

165.On the line defined by the equation  = , find points whose polar radii are equal to the following numbers: a) 1, b) 2, c)
. Which line is defined by this equation? (Build it on the drawing.)

166. Establish which lines are determined in polar coordinates by the following equations (construct them on the drawing):

1)  = 5; 2)  = ; 3)  = ; 4)  cos  = 2; 5)  sin  = 1;

6)  = 6 cos ; 7)  = 10 sin ; 8) sin  = 9) sin  =

167. Construct the following Archimedes spirals on the drawing:

1)  = 5, 2)  = 5; 3)  = ; 4)р = -1.

168. Construct the following hyperbolic spirals on the drawing:

1)  = ; 2) = ; 3) = ; 4) = - .

169. Construct the following logarithmic spirals on the drawing:

,
.

170. Determine the lengths of the segments into which the Archimedes spiral cuts

ray emerging from the pole and inclined to the polar axis at an angle
. Make a drawing.

171. On the Archimedes spiral
point taken WITH, whose polar radius is 47. Determine how many parts this spiral cuts the polar radius of the point WITH, Make a drawing.

172. On a hyperbolic spiral
find a point R, whose polar radius is 12. Make a drawing.

173. On a logarithmic spiral
find point Q whose polar radius is 81. Make a drawing.

1. Which statement is called a corollary? Prove that a line intersecting one of two parallel lines also intersects the other. 2. Prove that

If two lines are parallel to a third line, then they are parallel.3. What theorem is called the converse of this theorem? Give examples of theorems converse to these data. 4. Prove that when two parallel lines intersect with a transversal, the angles are equal. 5. Prove that if a line is perpendicular to one of two parallel lines, then it is also perpendicular to another.6.Prove that when two parallel lines intersect with a transversal: a) the corresponding angles are equal; b) the sum of one-sided angles is 180°.

Please help me with questions on geometry (grade 9)! 2) What does it mean to decompose a vector into two

to these vectors. 9) What is the radius vector of a point? Prove that the coordinates of the point are equal to the corresponding coordinates of the vectors. 10) Derive formulas for calculating the coordinates of a vector from the coordinates of its beginning and end. 11) Derive formulas for calculating the coordinates of a vector from the coordinates of its ends. 12) Derive a formula for calculating the length of a vector from its coordinates. 13) Derive a formula for calculating the distance between two points based on their coordinates. 15) What equation is called the equation of this line? Give an example. 16) Derive the equation of a circle of a given radius with a center at a given point.

1) State and prove the lemma about collinear vectors.

3)Formulate and prove a theorem about the decomposition of a vector into two non-collinear vectors.
4) Explain how a rectangular coordinate system is introduced.
5) What are coordinate vectors?
6)Formulate and prove a statement about the decomposition of an arbitrary vector into coordinate vectors.
7) What are vector coordinates?
8) Formulate and prove the rules for finding the coordinates of the sum and difference of vectors, as well as the product of a vector and a number at given vector coordinates.
10) Derive formulas for calculating the coordinates of a vector from the coordinates of its beginning and end.
11) Derive formulas for calculating the coordinates of a vector from the coordinates of its ends.
12) Derive a formula for calculating the length of a vector from its coordinates.
13) Derive a formula for calculating the distance between two points based on their coordinates.
14) Give an example of solving a geometric problem using the coordinate method.
16) Derive the equation of a circle of a given radius with a center at a given point.
17) Write the equation of a circle of given radius with center at the origin.
18) Derive the equation of this line in a rectangular coordinate system.
19) Write the equation of lines passing through a given point M0 (X0: Y0) and parallel to the coordinate axes.
20) Write the equation of the coordinate axes.
21) Give examples of using the equations of a circle and a line when solving geometric problems.

Please, I really need it! Preferably with drawings (where necessary)!

GEOMETRY 9TH GRADE.

1) State and prove the lemma about collinear vectors.
2) What does it mean to decompose a vector into two given vectors.
3)Formulate and prove a theorem about the decomposition of a vector into two non-collinear vectors.
4) Explain how a rectangular coordinate system is introduced.
5) What are coordinate vectors?
6)Formulate and prove a statement about the decomposition of an arbitrary vector into coordinate vectors.
7) What are vector coordinates?
8) Formulate and prove the rules for finding the coordinates of the sum and difference of vectors, as well as the product of a vector and a number at given vector coordinates.
9) What is the radius vector of a point? Prove that the coordinates of a point are equal to the corresponding coordinates of the vectors.
14) Give an example of solving a geometric problem using the coordinate method.
15)What equation is called the equation of this line? Give an example.
17) Write the equation of a circle of given radius with center at the origin.
18) Derive the equation of this line in a rectangular coordinate system.
19) Write the equation of lines passing through a given point M0 (X0: Y0) and parallel to the coordinate axes.
20) Write the equation of the coordinate axes.
21) Give examples of using the equations of a circle and a line when solving geometric problems.