Linear Equation

linear equation, statement that a first-degree polynomial—that is, the sum of a set of terms, each of which is the product of a constant and the first power of a variable—is equal to a constant. Specifically, a linear equation in n variables is of the form a₀ + a₁x₁ + … + a_nx_n = c, in which x₁, …, x_n are variables, the coefficients a₀, …, a_n are constants, and c is a constant. If there is more than one variable, the equation may be linear in some variables and not in the others. Thus, the equation x + y = 3 is linear in both x and y, whereas x + y² = 0 is linear in x but not in y. Any equation of two variables, linear in each, represents a straight line in Cartesian coordinates; if the constant term c = 0, the line passes through the origin.

A set of equations that has a common solution is called a system of simultaneous equations. For example, in the systemboth equations are satisfied by the solution x = 2, y = 3. The point (2, 3) is the intersection of the straight lines represented by the two equations. See also Cramer’s rule.

A linear differential equation is of first degree with respect to the dependent variable (or variables) and its (or their) derivatives. As a simple example, note dy/dx + Py = Q, in which P and Q can be constants or may be functions of the independent variable, x, but do not involve the dependent variable, y. In the special case that P is a constant and Q = 0, this represents the very important equation for exponential growth or decay (such as radioactive decay) whose solution is y = ke^−Px, where e is the base of the natural logarithm.