How to solve these $3$ equations for three unknowns $x$,$y$,$z$?
Hint: Put $$X=x+1$$ $$Y=y+1$$ $$Z=z+1$$
Then we have
$$XY=24$$ $$YZ=32$$ $$ZX=48$$
Can you take it from there?
We can use Simon's favorite factoring trick.
$$xy+x+y+1=(x+1)(y+1)$$
This tells us
$$(x+1)(y+1)=24$$$$(y+1)(z+1)=32$$$$(x+1)(z+1)=48$$So, we know that $x+1 = \pm\frac{\sqrt{24\cdot32\cdot48}}{32}\to x=5,-7$. Likewise, you can find the other variables.