Equation

For other uses, see Equation (disambiguation).
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 that may contain 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 that is either true or false, but a problem consisting of finding the values, called solutions, that, when substituted for the unknowns, yield equal values of the 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 became ubiquitous. "Equations" should not be confused with "identities", which are presented with the same notation but have a different meaning: for example 2 + 2 = 4 and x + y = y + x are identities (which implies they are necessarily true) in arithmetic, and do not constitute a values-finding problem, even when variables are present as in the latter example.

The term "equation" may also refer to a relation between some variables that is presented as the equality of some expressions written in terms of those variables' 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 famous 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.

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. 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 values for each of the unknowns, which together form 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.

A weighing scale, balance, or seesaw is often presented as an analogy to an equation.

Each side of the balance corresponds to one side of the equation. Different quantities can be placed on each side: if the weights on the two sides are equal the scale balances, corresponding to an equality represented by an equation; 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.

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:

$\theta = \frac{1}{2} \arcsin\left(\frac{2}{3}\right) \approx 20.9^\circ.$

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 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 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.

For more details on this topic, see Equation solving.