### Cramer's Rule

For a three by three system:

${\displaystyle {\begin{array}{c}ax+by+cz=j\\dx+ey+fz=k\\gx+hy+iz=l\\\end{array}}}$
In matrix form:
${\displaystyle {\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}j\\k\\l\end{bmatrix}}}$

${\displaystyle x={\frac {\begin{vmatrix}j&b&c\\k&e&f\\l&h&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}}$ ${\displaystyle y={\frac {\begin{vmatrix}a&j&c\\d&k&f\\g&l&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}}$ ${\displaystyle z={\frac {\begin{vmatrix}a&b&j\\d&e&k\\g&h&l\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}}$

In general: ${\displaystyle Ax=b}$
${\displaystyle x_{i}={\frac {|A_{i}|}{|A|}}\;\forall i=\{1,\dots ,n\}}$ Where ${\displaystyle A_{i}}$ is formed by replacing the vector associated with ${\displaystyle x_{i}}$ by the column vector b.