4.1 Simultaneous Equations


4.1 Simultaneous Equations

(A) Steps in solving simultaneous equations:
  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 \(a{x^2} + bx + c = 0\)
  4. Solve the quadratic equation. 
  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.
\[\begin{array}{l} xy = 20\\ x(9 - x) = 20\\ 9x - {x^2} = 20 \end{array}\] 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. 
x 2 9x+20=0 (x4)(x5)=0 x=4 or x=5 Find the value of the second unknown by substituting the value obtained into the linear equation.
When x=4, y=9x=94=5 When x=5, y=9x=95=4