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Simultaneous equations

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 \( z = -1 \).