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:

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 $a x^{2} + b x + c = 0$
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 \\ x y = 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} x y = 20 \\ x (9 - x) = 20 \\ 9 x - x^{2} = 20 \end{array}

Simplify and expressed the equation in the general form of quadratic equation

a x^{2} + b x + c = 0

\begin{array}{l} 9 x - x^{2} = 20 \\ x^{2} - 9 x + 20 = 0 \end{array}

Solve the quadratic equation.

\begin{array}{l} x^{2} - 9 x + 20 = 0 \\ (x - 4) (x - 5) = 0 \\ x = 4 or x = 5 \end{array}

Find the value of the second unknown by substituting the value obtained into the linear equation.

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