Equation

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In mathematics, the equation is an equivalence statement containing one or more variables. Complete the equation consists of determining which value of the variable makes the equation true. The variable is also called unknown and the unknown values ​​that satisfy the equation are called solutions of the equation. There are two types of equations: identity and conditional equations. Identity is true for all values ​​of variables. The conditional equation is true only for the values ​​of a particular variable.

The equation is written as two expressions, connected by an equal sign ("="). Expressions on both sides of the sign are equal to "left side" and "right side" of the equation.

Jenis persamaan yang paling umum adalah persamaan aljabar, di mana kedua belah pihak adalah emissar aljabar. Setiap sisi persamaan aljabar akan berisi satu atau beberapa istilah. Misalnya, persamaannya

${\ displaystyle Axe2 Bx C = y} Â Â$

memiliki sisi kiri ${\ displaystyle Ax2 Bx C} Â Â$ , yang memiliki tiga istilah, dan sisi kanan ${\ displaystyle y} Â Â$ , hanya terdiri dari satu istilah. Yang tidak diketahui adalah x dan y dan parameternya adalah A , B , dan C .

The equation is analogous to the scale at which the weights are placed. When the same weight of something (eg wheat) is placed into two pans, two weights cause the scale to be balanced and said to be the same. If the amount of grain is removed from a balance pan, the same amount of grain must be removed from another pan to keep the balance balanced. Likewise, to keep equations in equilibrium, the same addition, subtraction, multiplication and division operations must be performed on both sides of the equation in order to remain true.

In geometry, equations are used to describe geometric numbers. Since the equations considered, such as implicit equations or parametric equations, have many solutions, the goal is now different: rather than explicitly providing solutions or computing it, which is impossible, one uses equations to study character properties. This is the initial idea of ​​algebraic geometry, an important field of mathematics.

Algebra studies the two main families of equations: polynomial equations and, among them, special cases of linear equations. When there is only one variable, the polynomial equation has the form P ( x ) Ã, = Ã, 0, where P is polynomial, and linear the equation has the shape ax Ã, b Ã, = Ã, 0, where a and b are parameters. To solve the equations of one family, one uses an algorithmic or geometric technique derived from linear algebra or mathematical analysis. Algebra also studies the Diophantine equation where the coefficients and their solutions are integers. The technique used is different and comes from number theory. This equation is difficult in general; someone often searches only to find the presence or absence of a solution, and, if they exist, to calculate the number of solutions.

The differential equation is an equation involving one or more functions and their derivatives. They are solved by finding an expression for a function that does not involve a derivative. Differential equations are used to model processes involving variable rate changes, and are used in fields such as physics, chemistry, biology, and economics.

The symbol "=", which appears in any equation, was discovered in 1557 by Robert Recorde, who assumes that nothing is more equal than a parallel straight line of the same length.

Video Equation

Introduction

Analogue illustration

The equation is analogous to the scale of the weigh, balance, or seesaw.

Each side of the equation corresponds to one side of the equilibrium. Different numbers can be placed on each side: if the weights on both sides are equal, the balance scale, and in analogy equations representing equilibrium are also balanced (otherwise the lack of equilibrium corresponding to the inequality is represented by an inequality).

In the illustrations, x , y and z are all different quantities (in this case real numbers) represented as circular weights, and each x , y , and z have different weights. Additions correspond to weight gain, while reductions are related to removing the weight of what is already there. When equality holds, the total weight on each side is the same.

Parameter and unknown

Equations often contain terms other than unknown. These other terms, which are considered known , are usually called constants , coefficients or parameters >.

Contoh persamaan yang melibatkan x dan y sebagai tidak diketahui dan parameter R adalah

${\ displaystyle x2 and 2 = R2.} Â Â$

When R is selected to have value 2 ( R = 2), this equation will be recognized, when sketched in cartesian coordinates, as an equation for a given circle with radius 2. Because that, the equation with R is not specific is the general equation for the circle.

Usually, the unknown is denoted by the letters at the end of the alphabet, x , y , z , w ,.. , while the coefficient (parameter) is denoted by the letters at the beginning, a , b , c , d ,.... For example, the general quadratic equation is usually written ax ² Ã, bx Ã, c Ã, = Ã, 0. The process of finding a solution, or, in terms of parameters, declaring the unknown in terms of parameters is called completion of the equation. The expression of such a solution in terms of parameters is also called solution .

A system of equations is a set of simultaneous equations , usually in some unknown, in which general solutions are sought. So the solution for the system is a set of values ​​for each unknown, which together form the solution for each equation in the system. For example, the system

${\ displaystyle {\ begin {aligned} 3x 5y & amp; = 2 \\ 5x 8y & amp; = 3 \ end {aligned}}}$

has unique solution x Ã, = Ã, -1, y Ã, = Ã, 1.

Identity

An identity is the correct equation for all possible values ​​of the variable in it. Many identities are known in algebra and calculus. In the process of settling an equation, identity is often used to simplify the equations so that it is easier to solve.

Dalam aljabar, contoh identitas adalah perbedaan dua kotak:

${\ displaystyle x2 -y2 = = (x and) (x-y)} Â Â$

correct for all x and y .

Trigonometri adalah area di mana banyak identitas ada; Ini berguna dalam memanipulasi atau memecahkan persamaan trigonometri. Second dari banyak yang melibatkan fungsi sinus dan kosinus adalah:

${\ displaystyle \ sin2 (\ theta) \ cos2 (\ theta) = 1} Â Â$

dan

${\ displaystyle \ sin (2 \ theta) = 2 \ sin (\ theta) \ cos (\ theta)} Â Â$

which both apply to all values? .

Misalnya, untuk memecahkan nilai ? yang memenuhi persamaan:

${\ displaystyle 3 \ sin (\ theta) \ cos (\ theta) = 1 \ ,,} Â Â$

di mana ? diketahui terbatas antara 0 dan 45 derajat, kami dapat menggunakan identitas di atas untuk produk yang akan diberikan:

${\ displaystyle {\ frac {3} {2}} \ sin (2 \ theta) = 1 \ ,,} Â Â$

menghasilkan solusi untuk ?

${\ displaystyle \ theta = {\ frac {1} {2}} \ arcsin \ left ({\ frac {2} {3}} \ right) \ kira-kira 20,9 ^ {\ circ}.} Â Â$

Since the sine function is a periodic function, there are many solutions if there are no restrictions on ? . In this example, the restrictions are ? between 0 and 45 degrees implies there is only one solution.

Maps Equation

Properties

Two equations or two systems of equations are equivalent if they have the same set of solutions. The following operations change the equation or system of equations into equivalents - provided the operation is meaningful for the expression applied to:

• Increase or decrease the same amount for both sides of the equation. This shows that each equation is equivalent to the equation in which the right side is zero.
• Multiplies or divides both sides of the equation with non-zero quantities.
• Applying identity to change one side of the equation. For example, expanding the product or factoring the amount.
• For the system: adding to both sides the equation of the side corresponding to another equation, multiplied by the same quantity.

If some function is applied to both sides of the equation, the resulting equation has a solution of the initial equation among the solutions, but may have a further solution called a foreign solution. For example, ${\ displaystyle x = 1}$ has a solution ${\ displaystyle x = 1.}$ Increase both sides to exponent 2 (which means implementing the ${\ displaystyle f (s) = s ^ {2}}$ to both sides of the equation) change the equation to ${\ displaystyle x ^ {2} = 1}$ , which not only has the previous solution but also introduced a foreign solution, ${\ displaystyle x = -1.}$ x , which is not defined for x = 0), the solutions that exist on those values ​​may be lost. Thus, caution must be exercised when applying such a transformation to the equation.

The above transformation is the basis of most basic methods for solving equations as well as some less basic ones, such as Gauss elimination.

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Algebra

The polynomial equation

Secara umum, persamaan aljabar atau persamaan polinomial adalah persamaan bentuk

${\ displaystyle P = 0} Â$ , atau

${\ displaystyle P = Q} Â Â$

where P and Q are polynomials with coefficients in several fields (real numbers, complex numbers, etc.), which are often rational numbers. The algebraic equation is univariate if it involves only one variable. On the other hand, the polynomial equation may involve several variables, in this case called multivariate (some variables, x, y, z, etc.). The term polynomial equations is usually preferred over algebraic equations .

Sebagai contoh,

${\ displaystyle x ^ 5 -3x 1 = 0} Â Â$

adalah persamaan aljabar (polinomial) univariat dengan koefisien bilangan bulat dan

${\ displaystyle and {4} {\ frac {xy} {2}} = {\ frac {x3}} {3}} - xy2 and 2 - {\ frac {1} {7}}} Â Â$

is a multivariate polynomial equation over rational numbers.

Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression with a limited number of operations involving only those coefficients (ie, can be solved algebraically). This can be done for all equations of one, two, three, or four degrees; but for a five or more degree, it can be solved for some equations but, as the Abel-Ruffini theorem shows, not for all. A large number of studies have been devoted to calculating accurate accurate estimates of real or complex solutions of univariate algebraic equations (see root search algorithm) and general solutions of multivariate polynomial equations (see System polynomial equations).

System of linear equations

adalah sistem tiga persamaan dalam tiga variabel x , y , z . Solusi ke sistem linear adalah penugasan angka-angka ke variabel-variabel seperti semua persamaan secara bersamaan dipenuhi. Solusi untuk sistem di atas diberikan oleh

${\ displaystyle {\ begin {alignedat} {2} x & amp; \, = \, & amp; 1 \\ y & amp; \, = \, & amp; -2 \\ z & amp ; \, = \, & amp; -2 \ end {alignedat}}}  ÂÂ$

because it makes the three equations apply. The word " system " indicates that equations should be considered collectively, not individually.

In mathematics, the theory of linear systems is the fundamental and fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms to find solutions are an important part of numerical linear algebra, and play an important role in physics, engineering, chemistry, computer science, and economics. A system of nonlinear equations can often be approached by a linear system (see linearisation), a technique that helps when making mathematical models or computer simulations of relatively complex systems.

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Geometry

Analytic geometry

Dalam geometri Euclidean, adalah mungkin untuk mengasosiasikan satu set koordinat ke setiap titik dalam ruang, misalnya dengan grid ortogonal. Metode ini memungkinkan seseorang untuk mengkarakterisasi Angka geometric denounces persuasion. Pesawat dalam ruang tiga dimensi dapat dinyatakan sebagai solusi dari persamaan bentuk ${\ displaystyle ax cz d = 0} Â Â$ , di mana ${\ displaystyle a, b, c} Â$ dan ${\ displaystyle d} Â Â$ adalah bilangan real dan ${\ displaystyle x, y, z} Â Â$ adalah tidak diketahui yang sesuai dengan koordinat titik dalam sistem yang diberikan oleh grid ortogonal. Nilai-nilai ${\ displaystyle a, b, c} Â$ adalah koordinat dari sebuah vektor yang tegak lurus terhadap bidang yang ditentukan oleh persamaan. Garis dinyatakan sebagai perpotongan dua bidang, yaitu sebagai solusi dari persamaan linier tunggal dengan nilai dalam ${\ displaystyle \ mathbb {R} 2}} Â$ atau sebagai solusi dari dua persamaan linear dengan nilai dalam ${\ displaystyle \ mathbb {R} ^ 3}} Â Â$

Bagian berbentuk kerucut adalah perpotongan dari sebuah kerucut dengan persamaan ${\ displaystyle x2 and 2 = z2} Â Â$ dan sebuah pesawat. Denounce kata lain, dalam ruang, semua konik didefinisikan sebagai solusi dari persamaan bidang dan persamaan kerucut yang baru saja diberikan. Formalisme ini memungkinkan seseorang untuk menentukan posisi dan sifat-sifat fokus dari sebuah konik.

The use of equations allows one to call a large area of ​​mathematics to solve geometric questions. Cartesian coordinate systems convert geometric problems into analytical problems, once numbers are converted into equations; so the name of analytic geometry. This viewpoint, outlined by Descartes, enriched and modified the type of geometry conceived by ancient Greek mathematicians.

Today, analytic geometry refers to the active branch of mathematics. While still using equations to characterize numbers, it also uses other advanced techniques such as functional analysis and linear algebra.

Cartesian Equations

Kartesius coordinate system is a coordinate system that assigns each dot uniquely in a plane by a pair of coordinates of the number, which is the distance marked from the second to the perpendicular straight line, characterized by the same unit of length.

One can use the same principle to determine the position of any point in three-dimensional space by using three Cartesian coordinates, which are the signed distances to three perpendicular plane (or, equivalently, by projection perpendicular to three vertical lines straight).

The discovery of Cartesian coordinates in the 17th century by Renà © ¨ Descartes (Latinized name: Cartesius ) revolutionized mathematics by providing the first systematic relationship between Euclidean and algebraic geometry. Using a Cartesian coordinate system, geometric shapes (such as curves) can be explained by a Cartesian equation : an algebraic equation involving the coordinates of points located in the form. For example, a circle with radius 2 in a plane, centered on a certain point called origin, can be described as the set of all points whose coordinates x and y satisfy the equation < i> x ² y ² = 4 .

Parametric equations

Persamaan parametrik untuk kurva menyatakan koordinat titik-titik kurva sebagai fungsi dari suatu variabel, yang disebut parameter. Sebagai contoh,

${\ displaystyle {\ begin {aligned} x & amp; = \ cos t \\y & amp; = \ sin t \ end {aligned}}}  ÂÂ$

is the parametric equation for the unit circle, where t is the parameter. Together, this equation is called the parametric representation of the curve.

The idea of ​​the parametric equation has been generalized to the surface, manifold and algebraic variations of the higher dimension, with the same number of parameters as the manifold or variation dimensions, and the number of equations equal to the dimensions of space in which various or variations are considered for the dimension curve is one and one parameter is used, for the surface dimension two and two parameters, etc.).

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Number theory

The diophantine equation

A Diophantine Equation is a polynomial equation in two or more unknowns that only an integer solution is sought for (integer solution is the solution so that all unknowns take an integer value). A Linear Diophantine equation is the equation between two monomial numbers of degrees zero or one. An example of a linear Diophantine equation is ax by = c where a , b , and c are constants. An exponential Diophantine equation is one that the exponent of the equation term can be unknown.

The diophantine problem has fewer equations than an unknown variable and involves searching for the integers that work correctly for all equations. In more technical language, they define the algebraic curve, algebraic surface, or more general object, and ask about the lattice point above it.

The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made the study of the equation and was one of the first mathematicians to introduce symbolism into algebra. The Diophantine Diophantus Diagnostic Study begins now called Diophantine Analysis .

Algebra and transcendental numbers

The algebraic number is a number that is the solution of a non-zero polynomial equation in one variable with a rational coefficient (or the equivalent - by emptying the denominator - by integer coefficients). Figures like ? that is not algebra is said to be transcendental. Almost all real and complex numbers are transcendental.

Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying the solution of polynomial equations. Modern algebraic geometry is based on abstract techniques that are more abstract algebra, especially commutative algebra, with language and geometry problems.

The basic object of study in algebraic geometry is algebraic varieties, which are the geometric manifestations of polynomial equation system solutions. Examples of the most learned classes of algebraic varieties are: algebraic curves of the plane, which include lines, circles, parabolas, ellipses, hyperboles, cubic curves such as ellipsic curves and quartic curves such as lemniskats and Cassini ovals. A field point belongs to the algebraic curve if its coordinates satisfy the given polynomial equation. The basic question involves the study of special interest points such as single points, inflection points and points in infinity. The more advanced question involves the topology of the curve and the relationship between the curves given by different equations.

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Differential equations

Differential equations are mathematical equations that connect multiple functions with their derivatives. In applications, functions typically represent physical quantities, derivatives representing their rate of change, and the equations define the relationship between the two. Because such relationships are so common, differential equations play an important role in many disciplines including physics, engineering, economics, and biology.

In pure mathematics, differential equations are studied from several different perspectives, mostly related to their solution - sets of functions that satisfy the equation. Only the simplest differential equations can be solved by explicit formulas; however, some solution properties of certain differential equations can be determined without finding the exact form.

If the complete formula for the solution is not available, the solution can be approximated numerically using the computer. The theory of dynamic systems places an emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a certain degree of accuracy.

Ordinary differential equations

The usual differential equation or ODE is an equation that contains the function of one independent variable and its derivative. The term " ordinary " is used in contrast to the term partial differential equations, which may be related to more than one independent variable.

Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well defined and understood, and the appropriate closed form solution is obtained. In contrast, ODEs that do not have additive solutions are nonlinear, and resolving them is much more complicated, since they can seldom represent them by basic functions in a closed form: In contrast, the precise and analytical solutions of ODEs are in series or integral. Graphical and numerical methods, applied by hand or by computer, can approach the ODE solution and may yield useful information, often sufficient in the absence of appropriate analytical solutions.

Partial differential equations

The partial differential equation ( PDE ) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which relate to the functions of a single variable and its derivatives.) PDE is used to formulate problems involving functions of several variables, and solved by hand, or used to create relevant computer models..

PDEs can be used to describe various phenomena such as sound, heat, electrostatic, electrodynamics, fluid flow, elasticity, or quantum mechanics. This different-looking physical phenomenon can be formalized in terms of PDE. Just like the ordinary differential equations that often model a one-dimensional dynamic system, partial differential equations often model multidimensional systems. PDEs find their generalizations in stochastic partial differential equations.

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Equation type

Equations can be classified according to the type of operation and the number involved. Important types include:

• The algebraic equation or polynomial equation is an equation in which both sides are polynomials (see also system of polynomial equations). This is further classified by the degree:
  • linear equations for degrees one
  • quadratic equation for degree two
  • cubic equations for third degree
  • quartic equations for degrees four
  • quintik equations for degree five
  • gender equations for six degrees
  • septic equations for seventh degree
  • octic equations for eight degrees
• The Diophantine equation is an equation where the unknown must be an integer
• The transcendental equation is an equation involving the transcendental function of the unknown
• The parametric equation is an equation in which the solution for the variable is expressed as a function of some other variable, called the parameter that appears in the equation
• Functional equations are equations in which unknown functions rather than simple quantities
• The differential equation is a functional equation involving a derivative of an unknown function
• The integral equation is a functional equation involving antiderivatives of an unknown function
• The integro-differential equation is a functional equation involving the derivative and antiderivative of an unknown function
• The difference equation is an equation where the unknown is the function f that occurs in the equation through f ( x ), f ( x -1),..., f ( x - k ), for some integers k are called orders of the equation. If x is limited to an integer, the difference equation is equal to the recurrent relation

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See also

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References

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External links

• Winplot: General Purpose plotter that can draw and animate 2D and 3D math equations.
• Compiler of mathematical equations: Plot 2D mathematical equations, calculate integrals, and find solutions online.
• Equation plotter: Web page for producing and downloading pdf or postscript plots from solution set to equations and inequalities in two variables ( x and y ).
• EqWorld - contains information about solutions for different classes of mathematical equations.
• fxSolver: Online formula database and graphing calculator for math, natural sciences, and engineering.
• EquationSolver: A web page that can solve single equations and systems of linear equations.
• vCalc: A web page with a widely modified user equations library.

Source of the article : Wikipedia