A line on a plane is a collection of points on this plane that have certain properties, while points that do not lie on a given line do not have these properties. The equation of a line defines an analytically expressed relationship between the coordinates of points lying on this line. Let this relationship be given by the equation

F( x,y)=0. (2.1)

A pair of numbers satisfying (2.1) is not arbitrary: if X given, then at cannot be anything, meaning at associated with X. When changing X changes at, and a point with coordinates ( x,y) describes this line. If the coordinates of point M 0 ( X 0 ,at 0) satisfy equation (2.1), i.e. F( X 0 ,at 0)=0 is a true equality, then point M 0 lies on this line. The converse is also true.

Definition. An equation of a line on a plane is an equation that is satisfied by the coordinates of any point lying on this line, and not satisfied by the coordinates of points not lying on this line.

If the equation of a certain line is known, then the study of the geometric properties of this line can be reduced to the study of its equation - this is one of the main ideas of analytical geometry. To study equations, there are well-developed methods of mathematical analysis that simplify the study of the properties of lines.

When considering lines the term is used current point line – variable point M( x,y), moving along this line. Coordinates X And at current point are called current coordinates line points.

If from equation (2.1) we can express explicitly at
through X, that is, write equation (2.1) in the form , then the curve defined by such an equation is called schedule functions f(x).

1. The equation is given: , or . If X takes arbitrary values, then at takes values ​​equal to X. Consequently, the line defined by this equation consists of points equidistant from the coordinate axes Ox and Oy - this is the bisector of the I–III coordinate angles (straight line in Fig. 2.1).

The equation, or, determines the bisector of the II–IV coordinate angles (straight line in Fig. 2.1).

0 x 0 x C 0 x

rice. 2.1 fig. 2.2 fig. 2.3

2. The equation is given: , where C is some constant. This equation can be written differently: . This equation is satisfied by those and only those points, ordinates at which are equal to C for any abscissa value X. These points lie on a straight line parallel to the Ox axis (Fig. 2.2). Similarly, the equation defines a straight line parallel to the Oy axis (Fig. 2.3).

Not every equation of the form F( x,y)=0 defines a line on the plane: the equation is satisfied by a single point – O(0,0), and the equation is not satisfied by any point on the plane.

In the examples given, we used a given equation to construct a line determined by this equation. Let's consider the inverse problem: construct its equation using a given line.

3. Create an equation for a circle with center at point P( a,b) And
radius R .

○ A circle with a center at point P and radius R is a set of points located at a distance R from point P. This means that for any point M lying on the circle, MP = R, but if point M does not lie on the circle, then MP ≠ R.. ●

Let's review * Which equation is called quadratic? * What equations are called incomplete quadratic equations? * Which quadratic equation is called reduced? * What is called the root of a quadratic equation? * What does it mean to solve a quadratic equation? Which equation is called quadratic? What equations are called incomplete quadratic equations? Which quadratic equation is called reduced? What is the root of a quadratic equation? What does it mean to solve a quadratic equation? Which equation is called quadratic? What equations are called incomplete quadratic equations? Which quadratic equation is called reduced? What is the root of a quadratic equation? What does it mean to solve a quadratic equation?

Algorithm for solving a quadratic equation: 1. Determine the most rational way to solve a quadratic equation 2. Choose the most rational way to solve 3. Determining the number of roots of a quadratic equation 4. Finding the roots of a quadratic equation For better memorization, fill out the table... For better memorization, fill out the table... For better memorization, fill out table...

Additional condition Equation Roots Examples 1. b = c = 0, a 0 ax 2 = 0 x 1 = 0 2. c = 0, a 0, b 0 ax 2 + bx = 0 x 1 = 0, x 2 = -b /a 3. c = 0, a 0, c 0 ax 2 + c = 0 a) x 1.2 = ±(c/a), where c/a 0. b) if c/a 0, then there are no solutions 4. a 0 ax 2 + bx + c = 0 x 1.2 =(-b±D)/2 a, where D = b 2 – 4 ac, D0 5. c – even number (b = 2k), a 0, in 0, c 0 х 2 + 2kx + c = 0 x 1.2 =(-b±D)/а, D 1 = k 2 – ac, where k = 6. The inverse theorem to Vieta’s theorem x 2 + px + q = 0x 1 + x 2 = - p x 1 x 2 = q

II. Special methods 7. Method of isolating the square of a binomial. Goal: Reduce a general equation to an incomplete quadratic equation. Note: the method is applicable to any quadratic equations, but is not always convenient to use. Used to prove the formula for the roots of a quadratic equation. Example: solve the equation x 2 -6 x+8=0 8. Method of “transferring” the highest coefficient. The roots of the quadratic equations ax 2 + bx + c = 0 and y 2 +by+ac=0 are related by the relations: and Note: the method is good for quadratic equations with “convenient” coefficients. In some cases, it allows you to solve a quadratic equation orally. Example: solve the equation 2 x 2 -9 x-5=0 Based on theorems: Example: solve the equation 157 x x-177=0 9. If in a quadratic equation a+b+c=0, then one of the roots is 1, and the second, according to Vieta’s theorem, is equal to c / a 10. If in a quadratic equation a + c = b, then one of the roots is equal to -1, and the second, according to Vieta’s theorem, is equal to –c / a Example: solve the equation 203 x x + 17 = 0 x 1 =y 1 /a, x 2 =y 2 /a

III. General methods for solving equations 11. Factorization method. Goal: Reduce a general quadratic equation to the form A(x)·B(x)=0, where A(x) and B(x) are polynomials with respect to x. Methods: Taking the common factor out of brackets; Using abbreviated multiplication formulas; Grouping method. Example: solve the equation 3 x 2 +2 x-1=0 12. Method of introducing a new variable. Good choice of a new variable makes the structure of the equation more transparent Example: solve the equation (x 2 +3 x-25) 2 -6(x 2 +3 x-25) = - 8

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.

1. Equation of a line on a plane.
2. Equation of a straight line on a plane. Types of equations.
3. 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 direct.

A line on a plane can be defined using two equations

Where X And y - coordinates of an arbitrary point M(X; at), lying on this line, and t- a variable called parameter.

Parameter t determines the position of the point ( X; at) on a plane.

So, if

then the parameter value t= 2 corresponds to point (4; 1) on the plane, because X = 2 + 2 = 4, y= 2 2 – 3 = 1.

If the parameter t changes, then the point on the plane moves, describing this line. This method of defining a curve is called parametric, and equations (1) - parametric line equations.

Let's consider examples of well-known curves specified in parametric form.

1) Astroid:

Where A> 0 – constant value.

At A= 2 has the form:

Fig.4. Astroid

2) Cycloid: Where A> 0 – constant.

At A= 2 has the form:

Fig.5. Cycloid

Vector line equation

A line on a plane can be specified vector equation

Where t– scalar variable parameter.

Each parameter value t 0 corresponds to a certain plane vector. When changing a parameter t the end of the vector will describe a certain line (Fig. 6).

Vector equation of a line in a coordinate system Ohoo

correspond to two scalar equations (4), i.e. projection equations

on the coordinate axis of the vector equation of a line there are its parametric equations.

Fig.6. Vector line equation

The vector equation and the parametric line equations have a mechanical meaning. If a point moves on a plane, then the indicated equations are called equations of motion, line – trajectory points, parameter t- time.

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

See also

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 a high school algebra course / L. A. Markushevich, R. S. Cherkasov. / Mathematics at school. - 2004. - No. 1.
• Kaplan Y. V. Rivnyannya. - Kyiv: Radyanska School, 1968.
• Equation- article from the Great Soviet Encyclopedia
• Equations// Collier's Encyclopedia. - Open society. 2000.
• Equation// Encyclopedia Around the World
• 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.

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