Equation

Equation, statement of an equality between two expressions, used in almost all branches of pure and applied mathematics and in the physical, biological, and social sciences. An equation usually involves one or more unknown quantities, called variables or indeterminates. These are commonly denoted by letters or other symbols, as in the equations x² + x - 4 = 8, y = sin x + x, and 3y = log x. An equation is named for the number of variables it contains, called an equation in one, two, three, or more variables.

An equation is said to be satisfied or to be true for certain values of the variables if, when the variables are replaced by these values, the expression on the left side of the equals sign is equal to that on the right side. For example, the equation 2x + 5 = 13 is satisfied when x = 4. If one or more values of the variable fail to satisfy the equation, the equation is called conditional. The equation in two variables 3x + 4y = 8 is a conditional equation because it is not satisfied when x = 1 and y = 3. An equation is called an identity if it is satisfied by all possible values of the variables. For example, the equations (x + y)² = x² + 2xy + y² and sin²x + cos²x = 1 are identities because they are both true for all possible values of the unknowns. A solution of a conditional equation is a value of the variable, or a set of values of the variables, that satisfies the equation; thus, 3 is a solution of the equation x² - 2x = 3; and x = 2, y = 4 is a solution of the equation 3x² + 4y = 28. A solution of an equation in one variable is commonly called a root of the equation.

A polynomial equation has the form

a0xn + a1xn-1 + a 2xn-2 + ... + an-2x2 + an-1x + an = 0

in which the coefficients a₀, a₁, ..., a_n are constants, the leading coefficient a₀ is not equal to zero, and n is a positive integer. The greatest exponent n is the degree of the equation. Equations of the first, second, third, fourth, and fifth degrees are often called, respectively, linear, quadratic, cubic, biquadratic or quartic, and quintic equations. Other important types of equations are algebraic, as in = = 7; trigonometric, as in sin x + cos 2x = y; logarithmic, as in log x + 2 log (x + 1) = 8; and exponential, as in 3^x + 2x - 5 = 0. Diophantine equations are equations in one or more unknowns, usually with integral coefficients, for which integral solutions are sought. Differential and integral equations, which involve derivatives or differentials, and integrals, occur in calculus and its applications.

A system of simultaneous equations is a set of two or more equations in two or more unknowns. A solution of such a system is a set of values of the unknowns that satisfies every equation of the set simultaneously.