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.
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 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:
- Khadzhimba, Raul Dzhumkovich
- ES COMPUTER
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
If a rule is specified according to which a certain number u is associated with each point M of the plane (or some part of the plane), then they say that on the plane (or on part of the plane) “a point function is given”; the specification of the function is symbolically expressed by an equality of the form u=f(M). The number u associated with point M is called the value of this function at point M. For example, if A is a fixed point on the plane, M is an arbitrary point, then the distance from A to M is a function of point M. In this case, f(m)=AM .
Let some function u=f(M) be given and at the same time a coordinate system be introduced. Then an arbitrary point M is determined by the coordinates x, y. Accordingly, the value of this function at point M is determined by the coordinates x, y, or, as they also say, u=f(M) is function of two variables x and y. A function of two variables x and y is denoted by the symbol f(x; y): if f(M)=f(x;y), then the formula u=f(x; y) is called the expression of this function in the selected coordinate system. So, in the previous example f(M)=AM; if we introduce a Cartesian rectangular coordinate system with the origin at point A, we obtain the expression for this function:
u=sqrt(x^2 + y^2)
PROBLEM 3688 Given a function f (x, y)=x^2–y^2–16.
Given the function f (x, y)=x^2–y^2–16. Determine the expression of this function in the new coordinate system if the coordinate axes are rotated by an angle of –45 degrees.Parametric line equations
Let us denote the coordinates of a certain point M by the letters x and y; Let's consider two functions of the argument t:
x=φ(t), y=ψ(t) (1)
When t changes, the values x and y will, generally speaking, change, therefore, point M will move. Equalities (1) are called parametric line equations, which is the trajectory of point M; the argument t is called a parameter. If the parameter t can be excluded from equalities (1), then we obtain the equation of the trajectory of point M in the form
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. Determine 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) р= - 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 a 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.
The equation of a line on the XOY plane is an equation that is satisfied by the x and y coordinates of every point on that line and not satisfied by the coordinates of any point not on that line. In general, the equation of a line can be written as 0), (yx. F or)(xfy
Let a straight line be given that intersects the y-axis at point B (0, b) and forms an angle α with the x-axis. Let us choose an arbitrary point M(x, y) on the straight line.
x y M N
Coordinates of point N (x, in). From the triangle BMN: k is the angular coefficient of the line. k x by NB MN tg bkxy
Let's consider special cases: - equation of a straight line passing through the origin of coordinates. 10 bkxy 2 bytg 00 - equation of a straight line parallel to the x axis.
that is, a vertical line has no slope. 3 22 tg - does not exist The equation of a straight line parallel to the y-axis, in this case has the form ax where a is the segment cut off by the straight line on the x-axis.
Let a straight line be given, passing through a given point2 and forming an angle α with the x-axis), (111 yx. M
Since point M 1 lies on a straight line, its coordinates must satisfy equation (1): Subtract this equation from equation (1): bkxy 11)(11 xxkyy
If the angular coefficient is not defined in this equation, then it specifies a bundle of straight lines passing through a given point, except for the straight line parallel to the y-axis, which does not have an angular coefficient. xy
Let a line passing through two points be given: Let us write the equation of a bundle of lines passing through point M 1:), (111 yx. M), (222 yx. M)(11 xxkyy
Since point M 2 lies on this line, we substitute its coordinates into the equation of a pencil of lines:)(1212 xxkyy 12 12 xx yy k We substitute k into the equation of a pencil of lines. Thus, we select from this pencil a line passing through two given points :
1 12 12 1 xx xx yy yy or 12 1 xx xx yy yy
SOLUTION. We substitute the coordinates of the points into the equation of a line passing through two points. 53 5 42 4 xy)5(8 6 4 xy 4 1 4 3 xy
Let a straight line be given that cuts off segments equal to a and b on the coordinate axes. This means that it passes through the points)0, (a. A), 0(b. B) Let's find the equation of this line.
xy 0 ab
Let's substitute the coordinates of points A and B into the equation of a straight line passing through two points (3): a ax b y 00 0 a ax b y 1 ax b y 1 b y a x
EXAMPLE. Write an equation for a straight line passing through point A(2, -1) if it cuts off from the positive semi-axis y a segment twice as large as on the positive semi-axis x.
SOLUTION. According to the conditions of the problem, ab 2 Substitute into equation (4): 1 2 a y a x Point A(2, -1) lies on this line, therefore its coordinates satisfy this equation: 1 2 12 aa 1 2 41 a 23 a 1 35. 1 yx
Let's consider the equation: Let's consider special cases of this equation and show that for any values of the coefficients A, B (not equal to zero at the same time) and C, this equation is the equation of a straight line on a plane. 0 CBy. Ax
Then equation (5) can be represented as: Then we obtain equation (1): Let us denote: 10 B B C x B A y k B A b B C bkxy
Then the equation has the form: We obtain the equation: - the equation of a straight line passing through the origin. 2000 CAB x B A y 3 000 CAB BC y is the equation of a line parallel to the x-axis.
Then the equation has the form: We get the equation: - equation of the x axis. 40 y 5 000 CAB is the equation of a line parallel to the y axis. 000 CAB A C x
Then the equation has the form: - equation of the y-axis. 60 x 000 CAB Thus, for any values of the coefficients A, B (not equal to zero at the same time) and C, equation (5) is the equation of a straight line on a plane. This