Question

What is the most efficient way to solve a system of 3 equations in three variables such as:

ax1+bx2+cx3=0
dx1+ex2+fx3=0
gx1+hx2+ix3=0

so as to write x1, x2 and x3 in terms of an arbitrary constant, for example:
x1=t, x2=3t, x3=-3t/2

Answer 1

There is interaction between three planes which they are crossing the origin.

Answer 2

if the interaction is a line so you can suppose x1=t then find x2 and x3 as a functions of t.