What does homogeneous mean?

Question

I saw this in an example problem:

Question: IV By substituting $y=tx$, when the equations are homogeneous in terms which contain $x$ and $y$

(1)\begin{align*} & 52x^2+7xy=5y^2\\ & 5x-3y=17\end{align*} \begin{align*} & \hspace{5mm}52x^2+7tx^2=5t^2x^2\\ & \hspace{5mm}5x-3tx=17\\ & \hspace{15mm}\vdots\end{align*}

I have two main questions:

1. How did the author know to substitute $y=tx$ here, and not try some other method
2. What in the world does "homogeneous" mean?!?

Google search says "homogeneous" in math means something with the same degree.

Obviously, $x^2$ and $x$ are not the same degree. So at this point, I am extra confused.

Answer 1

Let us take the term 7xy of your first equation as an example.

Degree (7xy) = degree (xy) = degree(x) + degree(y) = 1 + 1 = 2.

The degrees of all other terms are calculated in such manner. Since all of them are of degree 2, we can say that the equation is homogeneous in degree 2.