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# Simultaneous equations

### Elimination method

In this method we eliminate one variable and form one equation. The steps involved are:

1. Multiply one or both equations by constants so that one of the variables has the same coefficient.
2. Subtract one equation from the other so that the variable with the same coefficient is eliminated.
3. Solve this equation to find the value of the variable.
4. Substitute the value of this variable into one of the equations to find the value of the other variable.
For example: Consider:
\begin{eqnarray}
2x+y &=& 21 \label{eqn:elim1} \\
3x+4y &=& 44 \label{eqn:elim2}
\end{eqnarray}
Multiply $$(\ref{eqn:elim1})\times 3$$:
\begin{eqnarray}
6x+3y &=& 63 \label{eqn:elim3}
\end{eqnarray}
Multiply $$(\ref{eqn:elim2})\times 2$$:
\begin{eqnarray}
6x+8y &=& 88 \label{eqn:elim4}
\end{eqnarray}
Subract equation $$(\ref{eqn:elim4})$$ from $$(\ref{eqn:elim3})$$:\begin{eqnarray}
-5y&=&-25 \nonumber \\
y &=& 5 \label{eqnelim3}
\end{eqnarray}
Substitute $$(\ref{eqnelim3})$$ into $$(\ref{eqn:elim1})$$:
\begin{eqnarray*}
2x+5 &=& 21 \\
2x &=& 16 \\
x &=& 8
\end{eqnarray*}
Therefore, the solution is $$(8,5)$$.

Note: Remember to check the solution, by substituting into the original equations.