Answer to Question #139468 in Linear Algebra for Nkosinathi Fakude

Question

Find the general solution of the system
2x + 4y + 6z = 0
4x + 5y + 6z = 3
7x + 8y + 9z = 6:

Answer 1

$\displaystyle 2x + 4y + 6z = 0 \hspace{1cm} (1)\\ 4x + 5y + 6z = 3\hspace{1cm} (2)\\ 7x + 8y + 9z = 6\hspace{1cm} (3)\\ (2) - (1) \\ 2x + y = 3\\ (2) \times 3\\ 4x + 5y + 6z = 3\\ 12x + 15y + 18 = 9\hspace{1cm} (4)\\ (3) \times 2\\ 14x + 16y + 18z = 12\hspace{1cm} (5)\\ (5) - (4)\\ 2x + y = 3\\ \textsf{Let}\, x = n\, \textsf{where}\, n \in \mathbb{R}\\ y = 3 - 2x\\ y = 3 - 2n\\ \textsf{Substituting the value of}\, x \, \textsf{and}\, y \, \textsf{in}\, (1)\\ \begin{aligned} 2n + 4(3 -2n) + 6z &= 0\\ 2n + 12 - 8n + 6z &= 0\\ 6z &= 6n - 12\\ z &= n - 2 \end{aligned}\\ \therefore\textsf{The general solution of the simultaneous}\\\textsf{linear equation is}\\ x = n, y = 3 - 2n, z = n - 2,\, \textsf{for all}\, n \in \mathbb{R}\\$