Pentru orice a numar real are loc (identitatea lui Hermite):
[tex][a]+[a+\frac{1}{2}]=[2a]\\
[/tex]
Aplicand partea intreaga celor doua ecuatii ale sistemului obtinem:
[tex][x]+[y]=13\\
\ [x]+[2y]=26
[/tex]
Scadem relatiile si avem:[tex][2y]-[y]=13[/tex]
Dar
[tex][2y]=[y]+[y+\frac{1}{2}][/tex]
De unde rezulta
[tex][y+\frac{1}{2}]=13 \Rightarrow y+\frac{1}{2}\in[13,14)\Rightarrow y\in[12.5,13.5)\\
\\Cazul ~I\\
~ [y]=12\Rightarrow x=13.9-12=1.9\\
1+2y=26.3\Rightarrow y=\frac{25.3}{2}=12.65\\
\\Cazul ~II\\
~ [y]=13\Rightarrow x=13.9-13=0.9\\
0+2y=26.3\Rightarrow y=\frac{26.3}{2}=13.15\\
\\
S=\{(1.9;12.65),(0.9;13.15) \}[/tex]
