Simultaneous equations
Elimination method
In this method we eliminate one variable and form one equation. The steps involved are:
- Multiply one or both equations by constants so that one of the variables has the same coefficient.
- Subtract one equation from the other so that the variable with the same coefficient is eliminated.
- Solve this equation to find the value of the variable.
- Substitute the value of this variable into one of the equations to find the value of the other variable.
- Check your answer in both of the original equations.
\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.