Non zero solution of homogenous system equations

Question

If the system of linear equations $$x+2ay+az=0$$ $$x+3by+bz=0$$ $$x+4cy+cz=0$$ has a non-zero solution, then find a relation between $a, b, c$ .

My attempt

I tried to find the discriminant to be zero assuming this to be a non trivial solution but it came out wrong.

Answer 1

Hint: Eliminating $x$ we get $$y(3b-2a)+z(b-a)=0$$ $$y(4c-2a)+z(c-a)=0$$ Now you have distinguish several cases.

Answer 2

As the determinant is $ab+bc-2ac$ we have a non-trivial kernel iff $b(a+c)=2ac$. Now if $a=-c$ it follows that $a=c=0$ and $b$ may be any real number. Otherwise $b=\frac{2ac}{a+c}$.