# linear equation

linear equation
1. a first-order equation involving two variables: its graph is a straight line in the Cartesian coordinate system.
2. any equation such that the sum of two solutions is a solution, and a constant multiple of a solution is a solution. Cf. linear operator.
[1810-20]

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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 + … + anxn = c, in which x1, …, xn are variables, the coefficients a0, …, an 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 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. 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 =  kePx, where e is the base of the natural logarithm.

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Universalium. 2010.

### Look at other dictionaries:

• Linear equation — Linear Lin e*ar (l[i^]n [ e]*[ e]r), a. [L. linearis, linearius, fr. linea line: cf. F. lin[ e]aire. See 3d {Line}.] 1. Of or pertaining to a line; consisting of lines; in a straight direction; lineal. [1913 Webster] 2. (Bot.) Like a line;… …   The Collaborative International Dictionary of English

• linear equation — n. an algebraic equation whose variable quantity or quantities are in the first power only and whose graph is a straight line (Ex.: 2x + 3y 5 = 0) …   English World dictionary

• Linear equation — A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.Linear equations can have one, two, three or more variables. Linear equations occur with… …   Wikipedia

• linear equation — tiesinė lygtis statusas T sritis fizika atitikmenys: angl. linear equation vok. lineare Gleichung, f rus. линейное уравнение, n pranc. équation linéaire, f …   Fizikos terminų žodynas

• linear equation — noun a polynomial equation of the first degree • Hypernyms: ↑equation …   Useful english dictionary

• linear equation — noun Date: 1816 an equation of the first degree in any number of variables …   New Collegiate Dictionary

• linear equation — noun A polynomial equation of the first degree (such as x = 2y 7) …   Wiktionary

• linear equation — noun an equation between two variables that gives a straight line when plotted on a graph …   English new terms dictionary

• linear equation — lin′ear equa′tion n. math. a first order equation involving two variables: its graph is a straight line in the Cartesian coordinate system …   From formal English to slang

• linear equation — /lɪniər əˈkweɪʒən/ (say lineeuhr uh kwayzhuhn) noun an equation, all of whose variables are in the first degree …