# Equation

The first use of an equals sign, equivalent to 14x + 15 = 71 in modern notation. From The Whetstone of Witte by Robert Recorde (1557).

In mathematics, an equation is a formula of the form A = B, where A and B are expressions containing one or several variables called unknowns, and "=" denotes the equality binary relation. Although written in the form of proposition, an equation is not a statement, but a problem consisting in finding the values, called solutions, that, when substituted to the unknowns, yields equal values of expressions A and B. For example, 2 is the unique solution of the equation x + 2 = 4, in which the unknown is x.[1] Historically, equations arose from the mathematical discipline of algebra, but later become ubiquitous. An equality may not be confused with identities which are presented with the same notation but have a different semantic: for example 2 + 2 = 4 and x + y = y + x are identities (which implies they are necessarily true) in arithmetic, and do not constitute any values-finding problem, even if include variables.

Equation may also refer to a relation between some variables that is expressed by the equality of some expressions of their values. For example the equation of the unit circle is x2 + y2 = 1, which means that a point belongs to the circle if and only if its coordinates are related by this equation. Most physical laws are expressed by equations. One of the most popular ones is Einstein's equation E = mc2.

The = symbol was invented by Robert Recorde (1510–1558), who considered that nothing could be more equal than parallel straight lines with the same length.

Centuries ago, the word "equation" frequently meant what we now usually call "correction" or "adjustment".[original research?] This meaning is still occasionally found, especially in names which were originally given long ago. The "equation of time", for example, is a correction that must be applied to the reading of a sundial in order to obtain mean time, as would be shown by a clock.

## Parameters and unknowns

Equations often contain other variables than the unknowns. These other variables that are supposed to be known are usually called constants, coefficients or parameters. Usually, the unknowns are denoted by letters at the end of the alphabet, x, y, z, w, …, while coefficients 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 one value for each unknown, which is a solution to each equation in the system. For example, the system

\begin{align} 3x+5y&=2\\ 5x+8y&=3 \end{align}

has the unique solution x = -1, y = 1.

## Analogous illustration

Illustration of a simple equation; x, y, z are real numbers, analogous to weights.

The analogy often presented is a weighing scale, balance, seesaw, or the like.

Each side of the balance corresponds to each side of the equation. Different quantities can be placed on each side, if they are equal the balance corresponds to an equality (equation), if not then an inequality.

In the illustration, x, y and z are all different quantities (in this case real numbers) represented as circular weights, each of x, y, z has a different weight. Addition corresponds to adding weight, subtraction corresponds to removing weight from what is already placed on. The total weight on each side is the same.

## Types of equations

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

## Identities

An identity is a statement resembling an equation which is true for all possible values of the variable(s) it contains. Many identities are known, especially in trigonometry. Probably the best known example is: $\sin^2(\theta)+\cos^2(\theta)=1,$, which is true for all values of θ.

In the process of solving an equation, it is often useful to combine it with an identity to produce an equation which is more easily soluble. For example, to solve the equation:

$3\sin(\theta) \cos(\theta)= 1,$ where θ is known to be between zero and 45 degrees,

use the identity: $\sin(2 \theta)=2\sin(\theta) \cos(\theta),$ so the above equation becomes:

$\frac{3}{2}\sin(2 \theta) = 1$

Whence:

$\theta = \frac{1}{2} \arcsin\left(\frac{2}{3}\right),$ which comes to about 20.9 degrees.

## 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 into an equivalent one:

• 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 constant.
• Applying an identity to transform one side of the equation. For example, expanding a product or factoring a sum.
• For a systems: adding to both sides of an equation the corresponding side of another, equation multiplied by the same quantity.

If some functions is be 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. If the function is not defined everywhere, (like 1/x that is not defined for x=0) some solutions may be lost. Thus caution must be exercised when applying such a transformation to an equation. 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 our equation into $x^2=1$, which not only has the previous solution but also introduces the extraneous solution, $x=-1.$

Above transformations are the basis of most elementary methods for equation solving and some less elementary ones, like Gaussian elimination