Editing System of linear equations (section)

===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.