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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 $$z = -1$$.