Answer
$f(x,y)$ is homogeneous of degree zero.
$$f(x,y)=\frac{3 +10 v}{6v}$$
Work Step by Step
Given
$$f(x, y)=\frac{x-2}{2 y}+\frac{5 y+3}{3 y}$$
So, we get
$$f(x, y)=\frac{3x-6+10y+6}{6 y} =\frac{3x+10y}{6 y}$$
Let
\begin{aligned}
f (t x, ty)&=\frac{3tx+10ty}{6t y}\\
&=\frac{t(3x+10y)}{6 ty}\\
&=\frac{3x+10y}{6 y}\\
&=f(x,t)
\end{aligned}
Thus, $f(x,y)$ is homogeneous of degree zero.
Transforming the given function
\begin{aligned}f(x, y)&=\frac{3 \frac{x}{x}+10 \frac{y}{x}}{6 \frac{y}{x}}\\
&=\frac{3 +10 \frac{y}{x}}{6 \frac{y}{x}}
\\
\text{Put} \ \ \ v=\frac{y}{x} \ \ \ \text{so, we get}\\
f(x,y)&=\frac{3 +10 v}{6v}
\end{aligned}
