Solve over reals the system of equations:
$$\lfloor x\rfloor+\{y\}=1.2$$
$$\{x\}+\lfloor y\rfloor = 3.3$$
My idea: Because $0\le \{y\} < 1$ and $\lfloor x\rfloor$ is an integer, the only possibility is $\lfloor x\rfloor = 1$ and $\{y\}=0.2$.
Because $0\le \{x\} < 1$ and $\lfloor y\rfloor$ is an integer, the only possibility is $\lfloor y\rfloor = 3$ and $\{x\}=0.3$.
The solution is $x=\lfloor x\rfloor+\{x\}=1.3$ and $y=\lfloor y\rfloor+\{y\}=3.2$. Is it correct? Can I write this idea this more cleanly?
That's right. You can make it cleaner by taking the integer and fractional parts, and using the fact these functions are idempotent (in fact, they're linear projection operators summing to the identity).