stylistic:style_transfer
More than two variables

A linear equation with more than two variables may always be assumed to have the form

The coefficient b, often denoted a is called the constant term (sometimes the absolute term in old books). Depending on the context, the term coefficient can be reserved for the a with i > 0.

When dealing with n = 3 {\displaystyle n=3} variables, it is common to use x, y {\displaystyle x,\;y} and z {\displaystyle z} instead of indexed variables.

A solution of such an equation is a n -tuple such that substituting each element of the tuple for the corresponding variable transforms the equation into a true equality.

For an equation to be meaningful, the coefficient of at least one variable must be non-zero. If every variable has a zero coefficient, then, as mentioned for one variable, the equation is either inconsistent (for b ≠ 0) as having no solution, or all n -tuples are solutions.

The n -tuples that are solutions of a linear equation in n variables are the Cartesian coordinates of the points of an (n − 1) -dimensional hyperplane in an n -dimensional Euclidean space (or affine space if the coefficients are complex numbers or belong to any field). In the case of three variables, this hyperplane is a plane.

If a linear equation is given with a ≠ 0, then the equation can be solved for x, yielding

If the coefficients are real numbers, this defines a real-valued function of n real variables.