Problem:
Solve the following system of three equations: 
$-23 x+15 y+12 z+7=0$, $-6 x+24 y-13 z+22=0$, $-10 x+8 y+4 z-21=0$
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Solve the following system}: \\
 \{
\begin{array}{l}
 7-23 x+15 y+12 z=0 \\
 22-6 x+24 y-13 z=0 \\
 -21-10 x+8 y+4 z=0 \\
\end{array}
 \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Express }\text{the }\text{system }\text{in }\text{standard }\text{form}: \\
 \{
\begin{array}{l}
 -23 x+15 y+12 z=-7 \\
 -6 x+24 y-13 z=-22 \\
 -10 x+8 y+4 z=21 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Express }\text{the }\text{system }\text{in }\text{matrix }\text{form}: \\
 \left(
\begin{array}{ccc}
 -23 & 15 & 12 \\
 -6 & 24 & -13 \\
 -10 & 8 & 4 \\
\end{array}
\right)\left(
\begin{array}{c}
 x \\
 y \\
 z \\
\end{array}
\right) \text{= }\left(
\begin{array}{c}
 -7 \\
 -22 \\
 21 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Solve }\text{the }\text{system }\text{with }\text{Cramer's }\text{rule}: \\
 x=\frac{
\begin{array}{|c|c|c}
 -7 & 15 & 12 \\
 -22 & 24 & -13 \\
 21 & 8 & 4 \\
\end{array}
}{
\begin{array}{|c|c|c}
 -23 & 15 & 12 \\
 -6 & 24 & -13 \\
 -10 & 8 & 4 \\
\end{array}
}\text{ and }y=\frac{
\begin{array}{|c|c|c}
 -23 & -7 & 12 \\
 -6 & -22 & -13 \\
 -10 & 21 & 4 \\
\end{array}
}{
\begin{array}{|c|c|c}
 -23 & 15 & 12 \\
 -6 & 24 & -13 \\
 -10 & 8 & 4 \\
\end{array}
}\text{ and }z=\frac{
\begin{array}{|c|c|c}
 -23 & 15 & -7 \\
 -6 & 24 & -22 \\
 -10 & 8 & 21 \\
\end{array}
}{
\begin{array}{|c|c|c}
 -23 & 15 & 12 \\
 -6 & 24 & -13 \\
 -10 & 8 & 4 \\
\end{array}
} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Evaluate }\text{the }\text{determinant }
\begin{array}{|c|c|c}
 -23 & 15 & 12 \\
 -6 & 24 & -13 \\
 -10 & 8 & 4 \\
\end{array}
=14: \\
 x=\frac{
\begin{array}{|c|c|c}
 -7 & 15 & 12 \\
 -22 & 24 & -13 \\
 21 & 8 & 4 \\
\end{array}
}{\fbox{$14$}}\text{ and }y=\frac{
\begin{array}{|c|c|c}
 -23 & -7 & 12 \\
 -6 & -22 & -13 \\
 -10 & 21 & 4 \\
\end{array}
}{\fbox{$14$}}\text{ and }z=\frac{
\begin{array}{|c|c|c}
 -23 & 15 & -7 \\
 -6 & 24 & -22 \\
 -10 & 8 & 21 \\
\end{array}
}{\fbox{$14$}} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Evaluate }\text{the }\text{determinant }
\begin{array}{|c|c|c}
 -7 & 15 & 12 \\
 -22 & 24 & -13 \\
 21 & 8 & 4 \\
\end{array}
=-12335: \\
 x=\frac{\fbox{$-12335$}}{14}\text{ and }y=\frac{
\begin{array}{|c|c|c}
 -23 & -7 & 12 \\
 -6 & -22 & -13 \\
 -10 & 21 & 4 \\
\end{array}
}{14}\text{ and }z=\frac{
\begin{array}{|c|c|c}
 -23 & 15 & -7 \\
 -6 & 24 & -22 \\
 -10 & 8 & 21 \\
\end{array}
}{14} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Evaluate }\text{the }\text{determinant }
\begin{array}{|c|c|c}
 -23 & -7 & 12 \\
 -6 & -22 & -13 \\
 -10 & 21 & 4 \\
\end{array}
=-9485: \\
 x=\frac{-12335}{14}\text{ and }y=\frac{\fbox{$-9485$}}{14}\text{ and }z=\frac{
\begin{array}{|c|c|c}
 -23 & 15 & -7 \\
 -6 & 24 & -22 \\
 -10 & 8 & 21 \\
\end{array}
}{14} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{gcd }\text{of }-9485 \text{and }\text{14 }\text{is }7, \text{so }\frac{-9485}{14}=\frac{-1355\times 7}{2\times 7}=-\frac{1355}{2}: \\
 x=\frac{-12335}{14}\text{ and }y=\fbox{$-\frac{1355}{2}$}\text{ and }z=\frac{
\begin{array}{|c|c|c}
 -23 & 15 & -7 \\
 -6 & 24 & -22 \\
 -10 & 8 & 21 \\
\end{array}
}{14} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Evaluate }\text{the }\text{determinant }
\begin{array}{|c|c|c}
 -23 & 15 & -7 \\
 -6 & 24 & -22 \\
 -10 & 8 & 21 \\
\end{array}
=-11794: \\
 x=\frac{-12335}{14}\text{ and }y=\frac{-1355}{2}\text{ and }z=\frac{\fbox{$-11794$}}{14} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{gcd }\text{of }-11794 \text{and }\text{14 }\text{is }2, \text{so }\frac{-11794}{14}=\frac{-5897\times 2}{7\times 2}=-\frac{5897}{7}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & x=\frac{-12335}{14}\text{ and }y=\frac{-1355}{2}\text{ and }z=\fbox{$-\frac{5897}{7}$} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/algebra/system_of_equations_w_steps/4774.txt'}