Editing System of linear equations (section)

===Row reduction===
{{Main|Gaussian elimination}}

In '''row reduction''' (also known as '''Gaussian elimination'''), the linear system is represented as an [[augmented matrix]]{{sfnp|Beauregard|Fraleigh|1973|p=68}}

:<math>\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
3 & 5 & 6 & 7 \\
2 & 4 & 3 & 8
\end{array}\right]\text{.}
</math>
This matrix is then modified using [[elementary row operations]] until it reaches [[reduced row echelon form]]. There are three types of elementary row operations:{{sfnp|Beauregard|Fraleigh|1973|p=68}}
:'''Type 1''': Swap the positions of two rows.
:'''Type 2''': Multiply a row by a nonzero [[scalar (mathematics)|scalar]].
:'''Type 3''': Add to one row a scalar multiple of another.
Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.

There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are [[Gaussian elimination]] and [[Gauss–Jordan elimination]]. The following computation shows Gauss–Jordan elimination applied to the matrix above:
:<math>\begin{align}\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
3 & 5 & 6 & 7 \\
2 & 4 & 3 & 8
\end{array}\right]&\sim
\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
0 & -4 & 12 & -8 \\
2 & 4 & 3 & 8
\end{array}\right]\sim
\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
0 & -4 & 12 & -8 \\
0 & -2 & 7 & -2
\end{array}\right]\sim
\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
0 & 1 & -3 & 2 \\
0 & -2 & 7 & -2
\end{array}\right]
\\
&\sim
\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
0 & 1 & -3 & 2 \\
0 & 0 & 1 & 2
\end{array}\right]\sim
\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
0 & 1 & 0 & 8 \\
0 & 0 & 1 & 2
\end{array}\right]\sim
\left[\begin{array}{rrr|r}
1 & 3 & 0 & 9 \\
0 & 1 & 0 & 8 \\
0 & 0 & 1 & 2
\end{array}\right]\sim
\left[\begin{array}{rrr|r}
1 & 0 & 0 & -15 \\
0 & 1 & 0 & 8 \\
0 & 0 & 1 & 2
\end{array}\right].\end{align}</math>
The last matrix is in reduced row echelon form, and represents the system {{nowrap|''x'' {{=}} &minus;15}}, {{nowrap|''y'' {{=}} 8}}, {{nowrap|''z'' {{=}} 2}}. A comparison with the example in the previous section on the algebraic elimination of variables shows that these two methods are in fact the same; the difference lies in how the computations are written down.