Editing Equation (section)

==Examples==
===Analogous illustration===

[[File:Equation illustration colour.svg|thumb|Illustration of a simple equation; ''x'', ''y'', ''z'' are real numbers, analogous to weights.]]

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 (mathematics)|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===
{{see also|Expression (mathematics)}}
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

:<math> x^2 +y^2 = R^2 .</math>

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''<sup>2</sup>&nbsp;+&nbsp;''bx''&nbsp;+&nbsp;''c''&nbsp;=&nbsp;0.

The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called [[Equation solving|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
:<math>\begin{align}
3x+5y&=2\\
5x+8y&=3
\end{align}
</math>
has the unique solution ''x''&nbsp;=&nbsp;−1, ''y''&nbsp;=&nbsp;1.

===Identities===

{{main|Identity (mathematics)|List of trigonometric 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]]:

:<math>x^2 - y^2 = (x+y)(x-y) </math>

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

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

:<math>\sin^2(\theta)+\cos^2(\theta) = 1 </math>

and

:<math>\sin(2\theta)=2\sin(\theta) \cos(\theta) </math>

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

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

:<math>3\sin(\theta) \cos(\theta)= 1\,, </math>

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

:<math>\frac{3}{2}\sin(2 \theta) = 1\,,</math>

yielding the following solution for ''θ:''

:<math>\theta = \frac{1}{2} \arcsin\left(\frac{2}{3}\right) \approx 20.9^\circ.</math>

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.