Editing System of linear equations (section)

===Cramer's rule===
{{Main|Cramer's rule}}

'''Cramer's rule''' is an explicit formula for the solution of a system of linear equations, with each variable given by a quotient of two [[determinant]]s.{{sfnp|Sterling|2009|p=[https://books.google.com/books?id=PsNJ1alC-bsC&pg=PA235 235]}} For example, the solution to the system
:<math>\begin{alignat}{7}
 x &\; + &\; 3y &\; - &\; 2z &\; = &\; 5 \\
3x &\; + &\; 5y &\; + &\; 6z &\; = &\; 7 \\
2x &\; + &\; 4y &\; + &\; 3z &\; = &\; 8 
\end{alignat}</math>
is given by
:<math>
x=\frac
{\, \begin{vmatrix}5&3&-2\\7&5&6\\8&4&3\end{vmatrix} \,}
{\, \begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix} \,}
,\;\;\;\;
y=\frac
{\, \begin{vmatrix}1&5&-2\\3&7&6\\2&8&3\end{vmatrix} \,}
{\, \begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix} \,}
,\;\;\;\;
z=\frac
{\, \begin{vmatrix}1&3&5\\3&5&7\\2&4&8\end{vmatrix} \,}
{\, \begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix} \,}.
</math>
For each variable, the denominator is the determinant of the [[matrix of coefficients]], while the numerator is the determinant of a matrix in which one column has been replaced by the vector of constant terms.

Though Cramer's rule is important theoretically, it has little practical value for large matrices, since the computation of large determinants is somewhat cumbersome. (Indeed, large determinants are most easily computed using row reduction.)
Further, Cramer's rule has very poor numerical properties, making it unsuitable for solving even small systems reliably, unless the operations are performed in rational arithmetic with unbounded precision.{{Citation needed|date=March 2017}}