Solving two greatest integer function equations If $$x\lfloor x\rfloor =39 \quad \text{and}\quad  y\lfloor y \rfloor=68.$$
What is the value of:
$$\lfloor x\rfloor+\lfloor y \rfloor $$
I don't know how to solve such problems.
I would appreciate an insight regarding the general approach to such problems.

Notice that $\lfloor x\rfloor$ is not very different from $x$, so $x\lfloor x\rfloor $ is not much different from $ x^2$. If you want $x\lfloor x\rfloor  = 39$, you need $x^2$ to be about $ 39$ also, which means $x$ is going to be around 6 or so, and $\lfloor x\rfloor$ will be exactly 6.  Then $x=6\frac12$ does the trick.
Do the same for $y$, and then add the results.
