Editing System of linear equations (section)

==Properties==

===Independence===
The equations of a linear system are '''independent''' if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as [[linear independence]].

[[File:Three Intersecting Lines.svg|thumb|The equations {{nowrap|''x'' &minus; 2''y'' {{=}} &minus;1}}, {{nowrap|3''x'' + 5''y'' {{=}} 8}}, and {{nowrap|4''x'' + 3''y'' {{=}} 7}} are linearly dependent.]]
For example, the equations
: <math>3x+2y=6\;\;\;\;\text{and}\;\;\;\;6x+4y=12</math>
are not independent — they are the same equation when scaled by a factor of two, and they would produce identical graphs. This is an example of equivalence in a system of linear equations.

For a more complicated example, the equations
: <math>\begin{alignat}{5}
 x &&\; - \;&& 2y &&\; = \;&& -1 & \\
 3x &&\; + \;&& 5y &&\; = \;&& 8 & \\
 4x &&\; + \;&& 3y &&\; = \;&& 7 &
\end{alignat}</math>
are not independent, because the third equation is the sum of the other two. Indeed, any one of these equations can be derived from the other two, and any one of the equations can be removed without affecting the solution set. The graphs of these equations are three lines that intersect at a single point.

===Consistency===
{{See also|Consistent and inconsistent equations}}
[[File:Parallel Lines.svg|thumb|The equations {{nowrap|3''x'' + 2''y'' {{=}} 6}} and {{nowrap|3''x'' + 2''y'' {{=}} 12}} are inconsistent.]]

A linear system is '''inconsistent''' if it has no solution, and otherwise, it is said to be '''consistent'''.{{sfnp|Whitelaw|1991|p=[https://books.google.com/books?id=6M_kDzA7-qIC&pg=PA70 70]}} When the system is inconsistent, it is possible to derive a [[contradiction]] from the equations, that may always be rewritten as the statement {{nowrap|0 {{=}} 1}}.

For example, the equations
: <math>3x+2y=6\;\;\;\;\text{and}\;\;\;\;3x+2y=12</math>
are inconsistent. In fact, by subtracting the first equation from the second one and multiplying both sides of the result by 1/6, we get {{nowrap|0 {{=}} 1}}. The graphs of these equations on the ''xy''-plane are a pair of [[parallel (geometry)|parallel]] lines.

It is possible for three linear equations to be inconsistent, even though any two of them are consistent together. For example, the equations
: <math>\begin{alignat}{7}
 x &&\; + \;&& y &&\; = \;&& 1 & \\
2x &&\; + \;&& y &&\; = \;&& 1 & \\
3x &&\; + \;&& 2y &&\; = \;&& 3 &
\end{alignat}</math>
are inconsistent. Adding the first two equations together gives {{nowrap|3''x'' + 2''y'' {{=}} 2}}, which can be subtracted from the third equation to yield {{nowrap|0 {{=}} 1}}. Any two of these equations have a common solution. The same phenomenon can occur for any number of equations.

In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent.

Putting it another way, according to the [[Rouché–Capelli theorem]], any system of equations (overdetermined or otherwise) is inconsistent if the [[rank (linear algebra)|rank]] of the [[augmented matrix]] is greater than the rank of the [[coefficient matrix]]. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has ''k'' free parameters where ''k'' is the difference between the number of variables and the rank; hence in such a case there is an infinitude of solutions. The rank of a system of equations (that is, the rank of the augmented matrix) can never be higher than [the number of variables] + 1, which means that a system with any number of equations can always be reduced to a system that has a number of [[independent equation]]s that is at most equal to [the number of variables] + 1.

===Equivalence===
Two linear systems using the same set of variables are '''equivalent''' if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one. It follows that two linear systems are equivalent if and only if they have the same solution set.