Editing System of linear equations (section)

==Solution set==
[[File:Intersecting Lines.svg|thumb|The solution set for the equations {{nowrap|''x'' &minus; ''y'' {{=}} &minus;1}} and {{nowrap|3''x'' + ''y'' {{=}} 9}} is the single point (2,&nbsp;3).]]

A ''[[Solution (mathematics)|solution]]'' of a linear system is an assignment of values to the variables 
<math>x_1, x_2,\dots,x_n</math> such that each of the equations is satisfied. The [[Set (mathematics)|set]] of all possible solutions is called the ''[[solution set]]''.<ref>{{Cite web |title=Systems of Linear Equations |url=https://math.berkeley.edu/~arash/54/notes/01_01.pdf |website=math.berkeley.edu |access-date=February 3, 2025 }}</ref>

A linear system may behave in any one of three possible ways:
# The system has ''infinitely many solutions''.
# The system has a ''unique solution''.
# The system has ''no solution''.

===Geometric interpretation===
For a system involving two variables (''x'' and ''y''), each linear equation determines a [[line (mathematics)|line]] on the ''xy''-[[Cartesian coordinate system|plane]]. Because a solution to a linear system must satisfy all of the equations, the solution set is the [[intersection (set theory)|intersection]] of these lines, and is hence either a line, a single point, or the [[empty set]].

For three variables, each linear equation determines a [[plane (mathematics)|plane]] in [[three-dimensional space]], and the solution set is the intersection of these planes. Thus the solution set may be a plane, a line, a single point, or the empty set. For example, as three parallel planes do not have a common point, the solution set of their equations is empty; the solution set of the equations of three planes intersecting at a point is single point; if three planes pass through two points, their equations have at least two common solutions; in fact the solution set is infinite and consists in all the line passing through these points.{{sfnp|Cullen|1990|p=3}}

For ''n'' variables, each linear equation determines a [[hyperplane]] in [[n-dimensional space|''n''-dimensional space]]. The solution set is the intersection of these hyperplanes, and is a [[flat (geometry)|flat]], which may have any dimension lower than ''n''.

===General behavior===
[[File:Intersecting Planes 2.svg|thumb|The solution set for two equations in three variables is, in general, a line.]]

In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns. Here, "in general" means that a different behavior may occur for specific values of the coefficients of the equations.
* In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an [[underdetermined system]].
* In general, a system with the same number of equations and unknowns has a single unique solution.
* In general, a system with more equations than unknowns has no solution. Such a system is also known as an [[overdetermined system]].
In the first case, the [[dimension]] of the solution set is, in general, equal to {{nowrap|''n'' &minus; ''m''}}, where ''n'' is the number of variables and ''m'' is the number of equations.

The following pictures illustrate this trichotomy in the case of two variables:
:{| class="wikitable"
|-
| width="150" align="center" | [[File:One Line.svg|120px]]
| width="150" align="center" | [[File:Two Lines.svg|120px]]
| width="150" align="center" | [[File:Three Lines.svg|120px]]
|-
| align="center" | One equation
| align="center" | Two equations
| align="center" | Three equations
|}
The first system has infinitely many solutions, namely all of the points on the blue line. The second system has a single unique solution, namely the intersection of the two lines. The third system has no solutions, since the three lines share no common point.

It must be kept in mind that the pictures above show only the most common case (the general case). It is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines intersect at a single point).

A system of linear equations behave differently from the general case if the equations are ''[[linear independence|linearly dependent]]'', or if it is ''[[#Consistency|inconsistent]]'' and has no more equations than unknowns.