# 3.2 Simultaneous Equations Involving One linear Equation and One Non-linear Equation

(A) Steps in solving simultaneous equations involving one linear equation and one non-linear equation:

1. For the linear equation, arrange so that one of the unknown becomes the subject of the equation.
2. Substitute the linear equation into the non-linear equation.
3. Simplify and expressed the equation in the general form of quadratic equation $a{x}^{2}+bx+c=0$
5. Find the value of the second unknown by substituting the value obtained into the linear equation.

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.
$\begin{array}{l}xy=20\\ x\left(9-x\right)=20\\ 9x-{x}^{2}=20\end{array}$

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