Editing System of linear equations (section)

===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.