Equation

[2][3] The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.

[5][6] The "=" symbol, which appears in every equation, was invented in 1557 by Robert Recorde, who considered that nothing could be more equal than parallel straight lines with the same length.

More generally, an equation remains balanced if the same operation is performed on each side.

Two of many that involve the sine and cosine functions are: and which are both true for all values of θ.

To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis.

Algebra also studies Diophantine equations where the coefficients and solutions are integers.

A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root finding of polynomials) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).

A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied.

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.

In Euclidean geometry, it is possible to associate a set of coordinates to each point in space, for example by an orthogonal grid.

A plane in three-dimensional space can be expressed as the solution set of an equation of the form

are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid.

A line is expressed as the intersection of two planes, that is as the solution set of a single linear equation with values in

The use of equations allows one to call on a large area of mathematics to solve geometric questions.

This point of view, outlined by Descartes, enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.

Although it still uses equations to characterize figures, it also uses other sophisticated techniques such as functional analysis and linear algebra.

This is the starting idea of algebraic geometry, an important area of mathematics.

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.

Algebraic geometry is a branch of mathematics, classically studying solutions of polynomial equations.

A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation.

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.

The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form.

Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.

(This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.)

PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model.

PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics.

These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs.

PDEs find their generalisation in stochastic partial differential equations.