Showing equivalence of group members

Question

Let $G$ be a group, $x,y \in G, \text{ord } x = 5,\ x^3y = xy^3$. Show that $xy = yx$.

I tried to reform $x^3y$ and $xy^3$, but all I got was $yx^3x^3x = yxy$.

Can you please help me to solve this?

Answer 1

$$x^3y=xy^3\implies x^2=y^2\implies \begin{cases}xy=x^6y=(x^2)^3y=(y^2)^3y=y^7\\{}\\yx=yx^6=y(x^2)^3=y(y^2)^3=y^7\end{cases}$$