# Simultaneous equations

### Substitution method

In this method, we substitute one variable from one equation into the other. The steps involved are:

- Using either of the equations, express one variable in terms of the other.
- This expression is then substituted into the other equation to form an equation in one variable only.
- Solve this equation to find the value of one of the variables.
- Substitute the value of this variable into the equation formed in the first step to find the value of the other variable.
- Check your answer in both of the original equations.

For example, solve for \(x\) and \(y\):\[ \begin{eqnarray} 2x+y &=& 21 \label{eqn:sub1} \\ 3x+4y &=& 44\ \label{eqn:sub2} \end{eqnarray}\] Rearranging \((\ref{eqn:sub1})\) to give \begin{eqnarray} y&=& 21-2x \label{eqn:sub3} \end{eqnarray} Substituting \((\ref{eqn:sub3})\) into \((\ref{eqn:sub2})\) and solving for \(x\) gives: \begin{eqnarray} 3x+4(21-2x) &=& 44 \nonumber \\ 3x+84-8x &=& 44 \nonumber \\ -5x + 84 &=& 44 \nonumber \\ -5x &=& -40 \nonumber \\ x&=& 8 \label{eqn:sub4} \end{eqnarray} Finally, substituting \((\ref{eqn:sub4})\) into \((\ref{eqn:sub3})\) to solve for \(y\) gives \(y = 21-2\times 8 = 5\). Therefore the solution is \((8,5)\).

Need to check by substituting the solution into both original equations.