LINEAR EQUATION

statement that a first-degree polynomialthat is, the sum of a set of terms, each of which is the product of a constant and the first power of a variableis equal to zero. Specifically, a linear equation in n variables is of the form a0 + a1x1 + + anxn = 0, in which x1, . . . , xn are variables and a0, . . . , an are scalars. If there is more than one variable, the equation may be linear in some 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; in the absence of a constant term, 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 system both 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.