Solving two greatest integer function equations

Question

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.

Answer 1

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.

Answer 2

$$ x |x| = 39 $$

Means $x$ is at the very least positive, since otherwise 39 could not be positive:

$$ -x |-x| = -xx = -39 $$

So this is pretty much equivalent, but not equal to:

$$x^2 = 39$$

Likewise for the other expression. So the result should be something like:

$$\sqrt{39} + \sqrt{68}$$