Equation
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Description
An equation is written as two expressions, connected by an equals sign ("=").[2] The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation. Very often the right-hand side of an equation is assumed to be zero. This does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides.
The most common type of equation is a
has left-hand side , which has four terms, and right-hand side , consisting of just one term. The names of the variables suggest that x and y are unknowns, and that A, B, and C are parameters, but this is normally fixed by the context (in some contexts, y may be a parameter, or A, B, and C may be ordinary variables).
An equation is analogous to a scale into which weights are placed. When equal weights of something (e.g., grain) are placed into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. More generally, an equation remains in balance if the same operation is performed on its both sides.
Properties
Two equations or two systems of equations are equivalent, if they have the same set of solutions. The following operations transform an equation or a system of equations into an equivalent one – provided that the operations are meaningful for the expressions they are applied to:
- Adding or subtracting the same quantity to both sides of an equation. This shows that every equation is equivalent to an equation in which the right-hand side is zero.
- Multiplying or dividing both sides of an equation by a non-zero quantity.
- Applying an identity to transform one side of the equation. For example, expanding a product or factoring a sum.
- For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity.
If some
The above transformations are the basis of most elementary methods for equation solving, as well as some less elementary ones, like Gaussian elimination.
Examples
Analogous illustration
An equation is analogous to a weighing scale, balance, or seesaw.
Each side of the equation corresponds to one side of the balance. Different quantities can be placed on each side: if the weights on the two sides are equal, the scale balances, and in analogy, the equality that represents the balance is also balanced (if not, then the lack of balance corresponds to an inequality represented by an inequation).
In the illustration, x, y and z are all different quantities (in this case
Parameters and unknowns
Equations often contain terms other than the unknowns. These other terms, which are assumed to be known, are usually called constants, coefficients or parameters.
An example of an equation involving x and y as unknowns and the parameter R is
When R is chosen to have the value of 2 (R = 2), this equation would be recognized in
Usually, the unknowns are denoted by letters at the end of the alphabet, x, y, z, w, ..., while coefficients (parameters) are denoted by letters at the beginning, a, b, c, d, ... . For example, the general quadratic equation is usually written ax2 + bx + c = 0.
The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called solving the equation. Such expressions of the solutions in terms of the parameters are also called solutions.
A system of equations is a set of simultaneous equations, usually in several unknowns for which the common solutions are sought. Thus, a solution to the system is a set of values for each of the unknowns, which together form a solution to each equation in the system. For example, the system
has the unique solution x = −1, y = 1.
Identities
An identity is an equation that is true for all possible values of the variable(s) it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation, making it more easily solvable.
In algebra, an example of an identity is the difference of two squares:
which is true for all x and y.
and
which are both true for all values of θ.
For example, to solve for the value of θ that satisfies the equation:
where θ is limited to between 0 and 45 degrees, one may use the above identity for the product to give:
yielding the following solution for θ:
Since the sine function is a periodic function, there are infinitely many solutions if there are no restrictions on θ. In this example, restricting θ to be between 0 and 45 degrees would restrict the solution to only one number.
Algebra
Polynomial equations
In general, an algebraic equation or
- , or
where P and Q are polynomials with coefficients in some field (e.g., rational numbers, real numbers, complex numbers). An algebraic equation is univariate if it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate (multiple variables, x, y, z, etc.).
For example,
is a univariate algebraic (polynomial) equation with integer coefficients and
is a multivariate polynomial equation over the rational numbers.
Some polynomial equations with
A large amount of research has been devoted to compute efficiently accurate approximations of the
Systems of linear equations
A
is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by
since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually.
In mathematics, the theory of linear systems is a fundamental part of linear algebra, a subject which is used in many parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in physics, engineering, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.
Geometry
Analytic geometry
In Euclidean geometry, it is possible to associate a set of coordinates to each point in space, for example by an orthogonal grid. This method allows one to characterize geometric figures by equations. A plane in three-dimensional space can be expressed as the solution set of an equation of the form , where and are real numbers and are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values are the coordinates of a vector perpendicular to the plane defined by the equation. A line is expressed as the intersection of two planes, that is as the solution set of a single linear equation with values in or as the solution set of two linear equations with values in
A conic section is the intersection of a cone with equation and a plane. In other words, in space, all conics are defined as the solution set of an equation of a plane and of the equation of a cone just given. This formalism allows one to determine the positions and the properties of the focuses of a conic.
The use of equations allows one to call on a large area of mathematics to solve geometric questions. The
Currently, analytic geometry designates an active branch of mathematics. Although it still uses equations to characterize figures, it also uses other sophisticated techniques such as functional analysis and linear algebra.
Cartesian equations
One can use the same principle to specify the position of any point in three-dimensional space by the use of three Cartesian coordinates, which are the signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).
The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane, centered on a particular point called the origin, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.
Parametric equations
A
For example,are parametric equations for the unit circle, where t is the parameter. Together, these equations are called a parametric representation of the curve.
The notion of parametric equation has been generalized to
Number theory
Diophantine equations
A Diophantine equation is a
Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an
The word Diophantine refers to the
Algebraic and transcendental numbers
An
Algebraic geometry
The fundamental objects of study in algebraic geometry are
Differential equations
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics.
In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.
If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of
Ordinary differential equations
An
Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by
Partial differential equations
A
PDEs can be used to describe a wide variety of phenomena such as
Types of equations
Equations can be classified according to the types of operations and quantities involved. Important types include:
- An algebraic equation or polynomial equation is an equation in which both sides are polynomials (see also system of polynomial equations). These are further classified by degree:
- linear equation for degree one
- quadratic equation for degree two
- cubic equation for degree three
- quartic equation for degree four
- quintic equationfor degree five
- sextic equation for degree six
- septic equation for degree seven
- octic equationfor degree eight
- A Diophantine equation is an equation where the unknowns are required to be integers
- A transcendental equation is an equation involving a transcendental function of its unknowns
- A parametric equation is an equation in which the solutions for the variables are expressed as functions of some other variables, called parameters appearing in the equations
- A functional equation is an equation in which the unknowns are functions rather than simple quantities
- Equations involving derivatives, integrals and finite differences:
- A differential equation is a functional equation involving derivatives of the unknown functions, where the function and its derivatives are evaluated at the same point, such as . Differential equations are subdivided into ordinary differential equations for functions of a single variable and partial differential equations for functions of multiple variables
- An integral equation is a functional equation involving the antiderivatives of the unknown functions. For functions of one variable, such an equation differs from a differential equation primarily through a change of variable substituting the function by its derivative, however this is not the case when the integral is taken over an open surface
- An integro-differential equation is a functional equation involving both the derivatives and the antiderivatives of the unknown functions. For functions of one variable, such an equation differs from integral and differential equations through a similar change of variable.
- A functional differential equation of delay differential equation is a function equation involving derivatives of the unknown functions, evaluated at multiple points, such as
- A difference equation is an equation where the unknown is a function f that occurs in the equation through f(x), f(x−1), ..., f(x−k), for some whole integer k called the order of the equation. If x is restricted to be an integer, a difference equation is the same as a recurrence relation
- A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process
See also
Notes
References
- ^ a b Recorde, Robert, The Whetstone of Witte ... (London, England: Jhon Kyngstone, 1557), the third page of the chapter "The rule of equation, commonly called Algebers Rule."
- ^ a b "Equation - Math Open Reference". www.mathopenref.com. Retrieved 2020-09-01.
- ^ "Equations and Formulas". www.mathsisfun.com. Retrieved 2020-09-01.
- ^ Marcus, Solomon; Watt, Stephen M. "What is an Equation?". Retrieved 2019-02-27.
- ^ Lachaud, Gilles. "Équation, mathématique". Encyclopædia Universalis (in French).
- ^ "A statement of equality between two expressions. Equations are of two types, identities and conditional equations (or usually simply "equations")". « Equation », in Mathematics Dictionary, Glenn James et Robert C. James (éd.), Van Nostrand, 1968, 3 ed. 1st ed. 1948, p. 131.
- ^ Thomas, George B., and Finney, Ross L., Calculus and Analytic Geometry, Addison Wesley Publishing Co., fifth edition, 1979, p. 91.
- ^ Weisstein, Eric W. "Parametric Equations." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ParametricEquations.html
External links
- Winplot: General Purpose plotter that can draw and animate 2D and 3D mathematical equations.
- Equation plotter: A web page for producing and downloading pdf or postscript plots of the solution sets to equations and inequations in two variables (x and y).