__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.

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