Problem:
Solve the following system of three equations: 
$11 x-15 y-12 z-6=0$, $-9 x+6 y-8 z+21=0$, $4 x-3 y+23 z+15=0$
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Solve the following system}: \\
 \{
\begin{array}{l}
 -6+11 x-15 y-12 z=0 \\
 21-9 x+6 y-8 z=0 \\
 15+4 x-3 y+23 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}
 11 x-15 y-12 z=6 \\
 -9 x+6 y-8 z=-21 \\
 4 x-3 y+23 z=-15 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Express }\text{the }\text{system }\text{in }\text{matrix }\text{form}: \\
 \left(
\begin{array}{ccc}
 11 & -15 & -12 \\
 -9 & 6 & -8 \\
 4 & -3 & 23 \\
\end{array}
\right)\left(
\begin{array}{c}
 x \\
 y \\
 z \\
\end{array}
\right) \text{= }\left(
\begin{array}{c}
 6 \\
 -21 \\
 -15 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Write }\text{the }\text{system }\text{in }\text{augmented }\text{matrix }\text{form }\text{and }\text{use }\text{Gaussian }\text{elimination}: \\
 \left(
\begin{array}{cccc}
 11 & -15 & -12 & 6 \\
 -9 & 6 & -8 & -21 \\
 4 & -3 & 23 & -15 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{9}{11}\, \times \, (\text{row }1) \text{to}\text{ row }2: \\
 \left(
\begin{array}{cccc}
 11 & -15 & -12 & 6 \\
 0 & -\frac{69}{11} & -\frac{196}{11} & -\frac{177}{11} \\
 4 & -3 & 23 & -15 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Multiply}\text{ row }2 \text{by }-11: \\
 \left(
\begin{array}{cccc}
 11 & -15 & -12 & 6 \\
 0 & 69 & 196 & 177 \\
 4 & -3 & 23 & -15 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }\frac{4}{11}\, \times \, (\text{row }1) \text{from}\text{ row }3: \\
 \left(
\begin{array}{cccc}
 11 & -15 & -12 & 6 \\
 0 & 69 & 196 & 177 \\
 0 & \frac{27}{11} & \frac{301}{11} & -\frac{189}{11} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Multiply}\text{ row }3 \text{by }11: \\
 \left(
\begin{array}{cccc}
 11 & -15 & -12 & 6 \\
 0 & 69 & 196 & 177 \\
 0 & 27 & 301 & -189 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }\frac{9}{23}\, \times \, (\text{row }2) \text{from}\text{ row }3: \\
 \left(
\begin{array}{cccc}
 11 & -15 & -12 & 6 \\
 0 & 69 & 196 & 177 \\
 0 & 0 & \frac{5159}{23} & -\frac{5940}{23} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Multiply}\text{ row }3 \text{by }\frac{23}{11}: \\
 \left(
\begin{array}{cccc}
 11 & -15 & -12 & 6 \\
 0 & 69 & 196 & 177 \\
 0 & 0 & 469 & -540 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide}\text{ row }3 \text{by }469: \\
 \left(
\begin{array}{cccc}
 11 & -15 & -12 & 6 \\
 0 & 69 & 196 & 177 \\
 0 & 0 & 1 & -\frac{540}{469} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }196\, \times \, (\text{row }3) \text{from}\text{ row }2: \\
 \left(
\begin{array}{cccc}
 11 & -15 & -12 & 6 \\
 0 & 69 & 0 & \frac{26979}{67} \\
 0 & 0 & 1 & -\frac{540}{469} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide}\text{ row }2 \text{by }69: \\
 \left(
\begin{array}{cccc}
 11 & -15 & -12 & 6 \\
 0 & 1 & 0 & \frac{391}{67} \\
 0 & 0 & 1 & -\frac{540}{469} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }15\, \times \, (\text{row }2) \text{to}\text{ row }1: \\
 \left(
\begin{array}{cccc}
 11 & 0 & -12 & \frac{6267}{67} \\
 0 & 1 & 0 & \frac{391}{67} \\
 0 & 0 & 1 & -\frac{540}{469} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }12\, \times \, (\text{row }3) \text{to}\text{ row }1: \\
 \left(
\begin{array}{cccc}
 11 & 0 & 0 & \frac{37389}{469} \\
 0 & 1 & 0 & \frac{391}{67} \\
 0 & 0 & 1 & -\frac{540}{469} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide}\text{ row }1 \text{by }11: \\
 \left(
\begin{array}{cccc}
 1 & 0 & 0 & \frac{3399}{469} \\
 0 & 1 & 0 & \frac{391}{67} \\
 0 & 0 & 1 & -\frac{540}{469} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Collect }\text{results}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & \{
\begin{array}{l}
 x=\frac{3399}{469} \\
 y=\frac{391}{67} \\
 z=-\frac{540}{469} \\
\end{array}
 \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/algebra/system_of_equations_w_steps/3099.txt'}