Equation

This article is about equations in mathematics. For the chemistry term, see chemical 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 an expression of the shape A = B, where A and B are expressions containing one or several variables called unknowns. An equation looks like an equality, but has a very different meaning: An equality is a mathematical statement that asserts that the left-hand side and the right-hand side of the equals sign (=) are the same or represent the same mathematical object; for example 2 + 2 = 4; is an equality. On the other hand, an equation is not a statement, but a problem consisting in finding the values, called solutions, that, when substituted to the unknowns, transform the equation into an equality. For example, 2 is the unique solution of the equation x + 2 = 4, in which the unknown is x.

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". 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

See also: Expression (mathematics)

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

Multiplication, division, collecting like terms.

Addition, subtraction, negation, collecting like terms.

Animations of the equation balance analogy.

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:

• An algebraic equation or polynomial equation is an equation in which both sides are polynomials (see also system of polynomial equations). These are further classified by degree:
  • linear equation for degree one
  • quadratic equation for degree two
  • cubic equation for degree three
  • quartic equation for degree four
  • quintic equation for degree five
• A Diophantine equation is an equation where the unknowns are required to be integers
• A transcendental equation is an equation involving a transcendental function of its unknowns
• A parametric equation is an equation for which the solutions are sought as functions of some other variables, called parameters appearing in the equations
• A functional equation is an equation in which the unknowns are functions rather than simple quantities
• A differential equation is a functional equation involving derivatives of the unknown functions
• An integral equation is a functional equation involving the antiderivatives of the unknown functions
• An integro-differential equation is a functional equation involving both the derivatives and the antiderivatives of the unknown functions
• A difference equation is an equation where the unknown is a function f which occurs in the equation through f(x), f(x-1), ..., f(x-k), for some whole integer called the order of the equation. If x is restricted to be an integer, a difference equation is the same as a recurrence relation

Identities

Main articles: Identity (mathematics) and List of trigonometric 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

For more details on this topic, see Equation solving.

See also

• Term (mathematics)
• Expression
• Formula
• Formula editor
• Functional equation
• History of elementary algebra
• Inequality
• Inequation
• List of equations
• Scientific equations named after people
• Theory of equations
• Algebra

External links

• Winplot: General Purpose plotter which can draw and animate 2D and 3D mathematical equations.
• Mathematical equation plotter: Plots 2D mathematical equations, computes integrals, and finds solutions online.
• Equation plotter: 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).
• EqWorld—contains information on solutions to many different classes of mathematical equations.
• EquationSolver: A webpage that can solve single equations and linear equation systems.