__4.1 Simultaneous Equations__

**(A) Steps in solving simultaneous equations:**

- For the linear equation, arrange so that one of the unknown becomes the subject of the equation.
- Substitute the linear equation into the non-linear equation.
- Simplify and expressed the equation in the general form of quadratic equation $a{x}^{2}+bx+c=0$
- Solve the quadratic equation.
- 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.

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

$$\begin{array}{l}\text{When}x=4\text{,}\\ y=9-x=9-4=5\\ \\ \text{When}x=5\text{,}\\ y=9-x=9-5=4\end{array}$$