Editing Linear equation (section)

==More than two variables==
A linear equation with more than two variables may always be assumed to have the form
:<math>a_1 x_1 + a_2 x_2 + \cdots + a_n x_n + b=0.</math>

The coefficient {{mvar|b}}, often denoted {{math|''a''{{sub|0}}}} is called the ''constant term'' (sometimes the ''absolute term'' in old books<ref>{{cite book |title=An Elementary Course in Theory of Equations |author1=Charles Hiram Chapman |edition= |publisher=J. Wiley & sons |year=1892 |isbn= |page=17 |url=https://books.google.com/books?id=2PQGAAAAYAAJ}} [https://books.google.com/books?id=2PQGAAAAYAAJ&pg=PA17 Extract of page 17]</ref><ref>{{cite book |title=Numbers Universalized: An Advanced Algebra |author1=David Martin Sensenig |edition= |publisher=American Book Company |year=1890 |isbn= |page=113 |url=https://books.google.com/books?id=TvMGAAAAYAAJ}} [https://books.google.com/books?id=TvMGAAAAYAAJ&pg=PA113 Extract of page 113]</ref>). Depending on the context, the term ''coefficient'' can be reserved for the {{math|''a''{{sub|''i''}}}} with {{math|''i'' > 0}}.

When dealing with <math>n=3</math> variables, it is common to use <math>x,\; y</math> and <math>z</math> instead of indexed variables.

A solution of such an equation is a {{mvar|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 {{mvar|''b'' ≠ 0}}) as having no solution, or all {{nowrap|{{mvar|n}}-tuples}} are solutions.

The {{mvar|n}}-tuples that are solutions of a linear equation in {{nowrap|{{mvar|n}} variables}} are the [[Cartesian coordinates]] of the points of an {{math|(''n'' − 1)}}-dimensional [[hyperplane]] in an {{nowrap|{{mvar|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 (geometry)|plane]].

If a linear equation is given with {{math|''a''{{sub|''j''}} ≠ 0}}, then the equation can be solved for {{math|''x''{{sub|''j''}}}}, yielding 
:<math>x_j = -\frac b{a_j} -\sum_{i\in \{1,\ldots,n\}, i\ne j} \frac {a_i}{a_j}x_i .</math>
If the coefficients are [[real number]]s, this defines a [[real-valued function|real-valued]] [[function of several real variables|function of {{mvar|n}} real variables]].