Problem:
Solve the following system of three equations: 
$4 \sqrt{2} x-15 \sqrt{2} y-14 \sqrt{2} z+2 \sqrt{2}=0$, $-2 \sqrt{2} x+8 \sqrt{2} y-3 \sqrt{2} z-11 \sqrt{2}=0$, $-5 \sqrt{2} x-16 \sqrt{2} y-\sqrt{2} z-5 \sqrt{2}=0$
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Solve the following system}: \\
 \{
\begin{array}{l}
 2 \sqrt{2}+4 \sqrt{2} x-15 \sqrt{2} y-14 \sqrt{2} z=0 \\
 -11 \sqrt{2}-2 \sqrt{2} x+8 \sqrt{2} y-3 \sqrt{2} z=0 \\
 -5 \sqrt{2}-5 \sqrt{2} x-16 \sqrt{2} y-\sqrt{2} 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}
 4 \sqrt{2} x-15 \sqrt{2} y-14 \sqrt{2} z=-2 \sqrt{2} \\
 -2 \sqrt{2} x+8 \sqrt{2} y-3 \sqrt{2} z=11 \sqrt{2} \\
 -5 \sqrt{2} x-16 \sqrt{2} y-\sqrt{2} z=5 \sqrt{2} \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Express }\text{the }\text{system }\text{in }\text{matrix }\text{form}: \\
 \left(
\begin{array}{ccc}
 4 \sqrt{2} & -15 \sqrt{2} & -14 \sqrt{2} \\
 -2 \sqrt{2} & 8 \sqrt{2} & -3 \sqrt{2} \\
 -5 \sqrt{2} & -16 \sqrt{2} & -\sqrt{2} \\
\end{array}
\right)\left(
\begin{array}{c}
 x \\
 y \\
 z \\
\end{array}
\right) \text{= }\left(
\begin{array}{c}
 -2 \sqrt{2} \\
 11 \sqrt{2} \\
 5 \sqrt{2} \\
\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}
 4 \sqrt{2} & -15 \sqrt{2} & -14 \sqrt{2} & -2 \sqrt{2} \\
 -2 \sqrt{2} & 8 \sqrt{2} & -3 \sqrt{2} & 11 \sqrt{2} \\
 -5 \sqrt{2} & -16 \sqrt{2} & -\sqrt{2} & 5 \sqrt{2} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Swap}\text{ row }1 \text{with}\text{ row }3: \\
 \left(
\begin{array}{cccc}
 -5 \sqrt{2} & -16 \sqrt{2} & -\sqrt{2} & 5 \sqrt{2} \\
 -2 \sqrt{2} & 8 \sqrt{2} & -3 \sqrt{2} & 11 \sqrt{2} \\
 4 \sqrt{2} & -15 \sqrt{2} & -14 \sqrt{2} & -2 \sqrt{2} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Subtract }\frac{2}{5}\, \times \, (\text{row }1) \text{from}\text{ row }2: \\
 \left(
\begin{array}{cccc}
 -5 \sqrt{2} & -16 \sqrt{2} & -\sqrt{2} & 5 \sqrt{2} \\
 0 & \frac{72 \sqrt{2}}{5} & -\frac{13 \sqrt{2}}{5} & 9 \sqrt{2} \\
 4 \sqrt{2} & -15 \sqrt{2} & -14 \sqrt{2} & -2 \sqrt{2} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide}\text{ row }1 \text{by }\sqrt{2}: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & \frac{72 \sqrt{2}}{5} & -\frac{13 \sqrt{2}}{5} & 9 \sqrt{2} \\
 4 \sqrt{2} & -15 \sqrt{2} & -14 \sqrt{2} & -2 \sqrt{2} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Multiply}\text{ row }2 \text{by }\frac{5}{\sqrt{2}}: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & 72 & -13 & 45 \\
 4 \sqrt{2} & -15 \sqrt{2} & -14 \sqrt{2} & -2 \sqrt{2} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide}\text{ row }3 \text{by }\sqrt{2}: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & 72 & -13 & 45 \\
 4 & -15 & -14 & -2 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{4}{5}\, \times \, (\text{row }1) \text{to}\text{ row }3: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & 72 & -13 & 45 \\
 0 & -\frac{139}{5} & -\frac{74}{5} & 2 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Multiply}\text{ row }3 \text{by }5: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & 72 & -13 & 45 \\
 0 & -139 & -74 & 10 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Swap}\text{ row }2 \text{with}\text{ row }3: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & -139 & -74 & 10 \\
 0 & 72 & -13 & 45 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }\frac{72}{139}\, \times \, (\text{row }2) \text{to}\text{ row }3: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & -139 & -74 & 10 \\
 0 & 0 & -\frac{7135}{139} & \frac{6975}{139} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Multiply}\text{ row }3 \text{by }\frac{139}{5}: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & -139 & -74 & 10 \\
 0 & 0 & -1427 & 1395 \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide}\text{ row }3 \text{by }-1427: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & -139 & -74 & 10 \\
 0 & 0 & 1 & -\frac{1395}{1427} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }74\, \times \, (\text{row }3) \text{to}\text{ row }2: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & -139 & 0 & -\frac{88960}{1427} \\
 0 & 0 & 1 & -\frac{1395}{1427} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide}\text{ row }2 \text{by }-139: \\
 \left(
\begin{array}{cccc}
 -5 & -16 & -1 & 5 \\
 0 & 1 & 0 & \frac{640}{1427} \\
 0 & 0 & 1 & -\frac{1395}{1427} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add }16\, \times \, (\text{row }2) \text{to}\text{ row }1: \\
 \left(
\begin{array}{cccc}
 -5 & 0 & -1 & \frac{17375}{1427} \\
 0 & 1 & 0 & \frac{640}{1427} \\
 0 & 0 & 1 & -\frac{1395}{1427} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Add}\text{ row }3 \text{to}\text{ row }1: \\
 \left(
\begin{array}{cccc}
 -5 & 0 & 0 & \frac{15980}{1427} \\
 0 & 1 & 0 & \frac{640}{1427} \\
 0 & 0 & 1 & -\frac{1395}{1427} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Divide}\text{ row }1 \text{by }-5: \\
 \left(
\begin{array}{cccc}
 1 & 0 & 0 & -\frac{3196}{1427} \\
 0 & 1 & 0 & \frac{640}{1427} \\
 0 & 0 & 1 & -\frac{1395}{1427} \\
\end{array}
\right) \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Collect }\text{results}: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & \{
\begin{array}{l}
 x=-\frac{3196}{1427} \\
 y=\frac{640}{1427} \\
 z=-\frac{1395}{1427} \\
\end{array}
 \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/algebra/system_of_equations_w_steps/4336.txt'}