dependence on a certain variable in a system

Question

I am considering the following system:

$$ a_{1}x+a_{2}\gamma y=0 \\ \frac{a_{3}}{\gamma}x + a_{4}y=0 $$ and want to solve for x and y (theoretically).

My question: can I conclude that x and y depend on $\gamma$?

Intuitively I think not because of the following: there is relationship between x and y as

$$ x=-\frac{a_{2}\gamma y}{a_{1}} \text{ and } y=-\frac{a_{3}x}{a_{4}\gamma} $$ so x is dependent on $\gamma$ in a "proportional" way and y is dependent on $\gamma$ in a "inverse-proportional" way. So when I consider $$x=-\frac{a_{2}\gamma y}{a_{1}}$$ for example, I would think that $\gamma$ would cancel out, and hence x is independent of $\gamma$. Similarly for y. But this contradicts the initial assumption. Any ideas?

Answer 1

Define $z=\gamma y$, and rewrite the system of equations in terms of $(x, z) $. This can be done in such a way that $\gamma$ does not appear, thus the solution in terms of $(x, z)$ is independent of $\gamma$, and $y$ is inversely proportional to $\gamma$.

Answer 2

From the first equation we get $$y=-\frac{a_1x}{a_2\gamma}$$ plugging this in the second equation we get $$\left(\frac{a_3}{\gamma}-\frac{a_1a_4}{a_2\gamma}\right)=0$$ Now we have $$x=0$$ and $$\frac{a_3}{\gamma}-\frac{a_1a_4}{a_2\gamma}\neq 0$$ or $$\frac{a_3}{\gamma}-\frac{a_1a_4}{a_2\gamma}=0$$ and $x$ is an arbitrary real number.