4.1 Simultaneous Equations

Example:
Solve the following simultaneous equations.
$\begin{array}{l}y+x=9\\ xy=20\end{array}$

Solution:
For the linear equation, arrange so that one of the unknown becomes the subject of the equation.
$\begin{array}{l}y+x=9\\ y=9-<!--\; -\; -->x\end{array}$

Substitute the linear equation into the non-linear equation.
$\begin{array}{l}xy=20\\ x(9-<!--\; -\; -->x)=20\\ 9x-<!--\; -\; -->x^2=20\end{array}$

Simplify and expressed the equation in the general form of quadratic equation $a{x}^2+bx+c=0$
$\begin{array}{l}9x-<!--\; -\; -->x^2=20\\ x^2-<!--\; -\; -->9x+20=0\end{array}$

Solve the quadratic equation. 
$\begin{array}{l}{x}^2-9x+20=0\\ (x-4)(x-5)=0\\ x=4\text{ or }x=5\end{array}$

Find the value of the second unknown by substituting the value obtained into the linear equation.
$\begin{array}{l}\text{When }x=4\text{,}\\ y=9-x=9-4=5\\ \\ \text{When }x=5\text{,}\\ y=9-x=9-5=4\end{array}$