Linear equation

A Linear equation is an equation which is the graph of a linear function.

An 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 this example the variable y is a function of x, and the graph of this function is the graph of the equation. Let's call this function f(x). Then f has the following banal, but incredibly useful properties:

f(x+y) = f(x)+f(y)

f(ax) = af(x)

where a is a scalar.

We call a function which satisfies these properties a linear function. Such an equation has certain properties which are also present in more complicated equations, and which are often exploited in the solution of such equations. In these more general contexts a linear function is often referred to as a linear operator.

A linear equation is an equation containing only functions that are linear in the variables of interest.

Because of the linear property above, the solutions of linear equations 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're 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.