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Simultaneous equations

Substitution method

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

  1. Using either of the equations, express one variable in terms of the other.
  2. This expression is then substituted into the other equation to form an equation in one variable only.
  3. Solve this equation to find the value of one of the variables.
  4. Substitute the value of this variable into the equation formed in the first step to find the value of the other variable.
  5. 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.