Suppose that $f(x,z)$ and $g(x,y)$ are homogeneous rational functions of the 1st degree, that is, $f(tx,tz)=tf(x,z)$ and $g(tx,ty)=tf(x,y)$ and consider the function $$ F(x,y,z)=\big[f(x,z)+ay\big]\big[g(x,y)+bz\big], $$ where $a,b\neq0$ are constants. I want to prove that $F$ must depend upon $x$, that is, that $\frac{\partial F}{\partial x}\not\equiv0$.
Comment: By $f(x,z)$ we mean that $f$ is depend upon both $x$ and $z$. The same for $g(x,y)$.
My attempt: Since $f$ and $g$ are homogeneous of the 1st degree, we can write $$ f(x,z)=zu\left(\frac{x}{z}\right)\qquad g(x,y)=yv\left(\frac{x}{y}\right) $$ where $u(t)$ and $v(t)$ are rational functions. Hence $$ F(x,y,z)=\big[zu\left(\frac{x}{z}\right)+ay\big]\big[yv\left(\frac{x}{y}\right)+bz\big]. $$ I would appreciate help on how to proceed from here.