System of linear equations solution?

Question

Let suppose we have for all $x,y,z$ an identity given by: $$xf(x,y,z)+yg(x,y,z)+zh(x,y,z)=0$$ If I have an equation $$ax+by+cz=0$$ Does this imply that the coefficients of $x,y,z$ in both equations are proportional? I mean assuming $abc\neq 0$ $$\dfrac{f(x,y,z)}{a}=\dfrac{g(x,y,z)}{b}=\dfrac{h(x,y,z)}{c}=\dfrac{p}{q}$$ where $\gcd(p,q)=1$