equation (mathematics)

In mathematics, an expression that represents the equality of two expressions involving constants and/or variables, and thus usually includes an equals (=) sign. For example, the equation A = πr² equates the area A of a circle of radius r to the product πr². This is also known as the formula for the area of a circle. The algebraic equation y = mx + c is the general one in coordinate geometry for a straight line and is known as a linear equation. See also algebra, quadratic equations, simultaneous equations, inequations or inequalities, and graphs.

Solving an equation
To solve an equation means to find the value or values of the unknown quantity that satisfy the equation; for example,

x + 4 = 7 is true when x is 3

The values of the unknown that make an equation true are called its solutions or roots.

In general, solving an equation depends on transforming it into a simple standard form. This can be achieved by using the following processes:

(i) adding the same quantity to each side of the equation

(ii) subtracting the same quantity from each side of the equation

(iii) multiplying each side of the equation by the same quantity (so long as it is not zero)

These processes can be used to change an equation into a simpler form but they will not alter its solution. For example, to solve the equation 7x - 4 = 3x + 8:

subtract 3x from each side in order to collect the xs on the left-hand side:

7x - 4 - 3x = 3x + 8 - 3x so 4x - 4 = 8

add 4 to each side in order to collect the numbers on the right-hand side:

4x - 4 + 4 = 8 + 4 so 4x = 12

divide both sides by 4 to obtain the solution:

4x ÷ 4 = 12 ÷ 4 so x = 3

Polynomials
A type of equation that has been studied particularly intensively is where there is one unknown and the expression involving it is a polynomial. A polynomial equation has the form:

f(x) = a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x + a₀

where a_n, a_{n - 1}, …, a₀ are all constants, n is a positive integer, and a_n ≠ 0.

The ‘degree’ of a polynomial equation is simply the degree of the polynomial involved. A polynomial of degree one, that is, whose highest power of x is 1, as in 2x + 1, is called a linear polynomial;

3x² + 2x + 1 is quadratic;

4x³ + 3x² + 2x + 1 is cubic.

Indeterminate equations
An indeterminate equation is an equation for which there is an infinite set of solutions – for example, 2x = y. A diophantine equation is an indeterminate equation in which both the solution and the terms must be whole numbers (after Diophantus of Alexandria, c. AD 250).

Identity
An equation that is true for all values of the unknown is called an identity, for example x + x = 2x. It is denoted by ≡.

Thus (x + y)² ≡ x² + 2xy + y² for all real numbers x, y.