Equation

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . 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.

''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.

An equation is written as two expressions, connected by an equals sign ("="). 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. Assuming 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 polynomial equation (commonly called also an ''algebraic equation'') in which the two sides are polynomials.
The sides of a polynomial equation contain one or more terms. For example, the equation

:$Ax^2 +Bx + C - y = 0$

has left-hand side $Ax^2 +Bx + C - y$, which has four terms, and right-hand side $0$, consisting of just one term. The names of the variables suggest that and are unknowns, and that , , and are parameters, but this is normally fixed by the context (in some contexts, may be a parameter, or , , and 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.

In Cartesian geometry, equations are used to describe geometric figures. As the equations that are considered, such as implicit equations or parametric equations, have infinitely many solutions, the objective is now different: instead of giving the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of algebraic geometry, an important area of mathematics.

Algebra studies two main families of equations: polynomial equations and, among them, the special case of linear equations. When there is only one variable, polynomial equations have the form ''P''(''x'') = 0, where ''P'' is a polynomial, and linear equations have the form ''ax'' + ''b'' = 0, where ''a'' and ''b'' are parameters. 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. The techniques used are different and come from number theory. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.

Differential equations are equations that involve one or more functions and their derivatives. 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.

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.Recorde, Robert, ''The Whetstone of Witte'' ... (London, England: Kyngstone, 1557)
the third page of the chapter "The rule of equation, commonly called Algebers Rule."
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Introduction

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 real numbers) represented as circular weights, and each of ''x'', ''y'', and ''z'' has a different weight. Addition corresponds to adding weight, while subtraction corresponds to removing weight from what is already there. When equality holds, the total weight on each side is the same.

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

:$x^2 +y^2 = R^2 .$

When ''R ''is chosen to have the value of 2 (''R ''= 2), this equation would be recognized in Cartesian coordinates as the equation for the circle of radius of 2 around the origin. Hence, the equation with ''R'' unspecified is the general equation for the circle.

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 ''ax''² + ''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
:$\begin 3x+5y&=2\\ 5x+8y&=3 \end$
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:

:$x^2 - y^2 = (x+y)(x-y)$

which is true for all ''x'' and ''y''.

Trigonometry is an area where many identities exist; these are useful in manipulating or solving trigonometric equations. Two of many that involve the sine and cosine functions are:

:$\sin^2(\theta)+\cos^2(\theta) = 1$

and

:$\sin(2\theta)=2\sin(\theta) \cos(\theta)$

which are both true for all values of ''θ''.

For example, to solve for the value of ''θ'' that satisfies the equation:

:$3\sin(\theta) \cos(\theta)= 1\,,$

where ''θ'' is limited to between 0 and 45 degrees, one may use the above identity for the product to give:

:$\frac\sin(2 \theta) = 1\,,$

yielding the following solution for ''θ:''

:$\theta = \frac \arcsin\left(\frac\right) \approx 20.9^\circ.$

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.

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 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