Elimination method for three equations in three unknowns

The use of the elimination method can be extended to larger systems of equations, for example, for three unknowns in three equations.

• The elimination process is the same as for two unknowns.
• Use one equation to eliminate one of the variables from the other two equations.

For Example:
\begin{eqnarray}
2x - y + z &=& 3 \label{eqn:elim3:1} \\
x + 3 y -2z &=& 11 \label{eqn:elim3:2} \\
3x-2y+4z&=& 1 \label{eqn:elim3:3}
\end{eqnarray}
Use \( (\ref{eqn:elim3:1}) \) to eliminate \( y \) from  \( (\ref{eqn:elim3:2}) \) and \( (\ref{eqn:elim3:3}) \).

Multiply \( (\ref{eqn:elim3:1}) \times 3\)  
\begin{eqnarray}
6x - 3y + 3 z&=& 9 \label{eqn:elim3:4}
\end{eqnarray} 
\( (\ref{eqn:elim3:2}) + (\ref{eqn:elim3:4})\) gives
\begin{eqnarray}
7x + z &=& 20 \label{eqn:elim3:5}
\end{eqnarray}
Multiply \( (\ref{eqn:elim3:1}) \times 2\) gives:
\begin{eqnarray}
4x-2y+2z &=& 6 \label{eqn:elim3:6}
\end{eqnarray}
\( (\ref{eqn:elim3:3}) - (\ref{eqn:elim3:6})\) gives
\begin{eqnarray}
-x + 2z &=& -5 \label{eqn:elim3:7}
\end{eqnarray}
Now we have two equations \( (\ref{eqn:elim3:5}) \) and \( (\ref{eqn:elim3:7}) \) which can be solved using either elimination or substitution to give the solution \( x = 3\), \( y = 2 \) and \( z = -1 \).