jmc

Consider the first equation, $$\lfloor x\rfloor +\{y\}=\mathrm{2.4.}$$Because $\lfloor x\rfloor$ is an integer, while $0\le \{y\}<1,$ the only possibility is that $\lfloor x\rfloor =2$ and $\{y\}=\mathrm{0.4.}$ Similarly, from the second equation, we get $\{x\}=0.1$ and $\lfloor y\rfloor =5.$ Then $$x=\lfloor x\rfloor +\{x\}=2.1$$and $$y=\lfloor y\rfloor +\{y\}=5.4,$$so $|x-y|=|2.1-5.4|=\boxed{3.3}.$