Equation

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =.^[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.^[4]

For other uses, see Equation (disambiguation).

Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables.^[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.^[1]

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.

Adding

or dividing both sides of an equation by a non-zero quantity.

Multiplying

Applying an to transform one side of the equation. For example, expanding a product or factoring a sum.

identity

For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity.

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:

If some function is applied to both sides of an equation, the resulting equation has the solutions of the initial equation among its solutions, but may have further solutions called extraneous solutions. For example, the equation $x=1$ has the solution $x=1.$ Raising both sides to the exponent of 2 (which means applying the function $f(s)=s^{2}$ to both sides of the equation) changes the equation to $x^{2}=1$ , which not only has the previous solution but also introduces the extraneous solution, $x=-1.$ Moreover, if the function is not defined at some values (such as 1/x, which is not defined for x = 0), solutions existing at those values may be lost. Thus, caution must be exercised when applying such a transformation to an equation.

The above transformations are the basis of most elementary methods for equation solving, as well as some less elementary ones, like Gaussian elimination.

linear equation

A is an equation where the unknowns are required to be integers

Diophantine equation

A is an equation involving a transcendental function of its unknowns

transcendental equation

A is an equation in which the solutions for the variables are expressed as functions of some other variables, called parameters appearing in the equations

parametric equation

A is an equation in which the unknowns are functions rather than simple quantities

functional equation

differential equation

Equations can be classified according to the types of operations and quantities involved. Important types include:

: General Purpose plotter that can draw and animate 2D and 3D mathematical equations.

Winplot

: 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).