The following system of linear equations

Solution:

The given system of linear equations

$7 x+6 y-2 z=0$...(i)

$3 x+4 y+2 z=0$...(ii)

$x-2 y-6 z=0$...(iii)

Now, determinant of coefficient matrix

$\Delta=\left|\begin{array}{ccc}7 & 6 & -2 \\ 3 & 4 & 2 \\ 1 & -2 & -6\end{array}\right|$

$=7(-20)-6(-20)-2(-10)$

$=-140+120+20=0$

So, there are infinite non-trivial solutions.

From eqn. (i) $+3 \times$ (iii); we get

$10 x-20 z=0 \Rightarrow x=2 z$

Hence, there are infinitely many solutions $(x, y, z)$

satisfying $x=2 z$.