Linear equation

A linear equation in algebra is an equation which is constructed by equating two linear functions. It may read

3x + y − 5 = −7x + 4y + 3.

A linear equation is an equation containing only functions that are linear in the variables of interest: put more simply, terms such as x² are not allowed.

A very simple example of a linear equation is

y = 3x.

If one plots the graph of this equation it yields a straight line (thus providing the terminology). In fact a general linear equation in variables x and y can be re-arranged to something nearly as simple:

y = ax + b

with constants a and b, unless rearrangement gives one of the forms

x = c or 0 = 0.

The two exceptional cases correspond (respectively) to the equation of a vertical line, and to an equation that is an identity (gives no information).

There may be further variables, not just two; and several simultaneous equations. For more see system of linear equations.

Connection with linear functions and operators

In the example (but not in the exceptions) the variable y is a function of x, and the graph of this function is the graph of the equation.

In general there are linear equations arising in applications written as

y = f(x)

where f has the properties:

f(x + y) = f(x) + f(y)
:f(ax) = af(x)

where a is a scalar.

A function which satisfies these properties is called a linear function, or more generally a linear operator.

Because of the linear property above, the solutions of linear equations of this kind can in general be described as a superposition of other solutions of the same equation. This makes linear equations particularly easy to solve and reason about.

Linear equations occur with great regularity in applied mathematics. Whilst they arise quite naturally when modelling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state.

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Linear equation

Linear equation

Connection with linear functions and operators

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