x, y - the numbers
[tex]x-y=13 \\
x^2+y^2=449 \\ \\
\hbox{solve the first equation for x:} \\
x-y=13 \\
x=13+y \\ \\
\hbox{substitute the result for x in the second equation:} \\
(13+y)^2+y^2=449 \\
169+26y+y^2+y^2=449 \\
2y^2+26y+169-449=0 \\
2y^2+26y-280=0 \ \ \ \ \ \ \ \ \ \ \ |\div 2 \\
y^2+13y-140=0 \\
y^2+20y-7y-140=0 \\
y(y+20)-7(y+20)=0 \\
(y-7)(y+20)=0 \\
y-7=0 \ \lor \ y+20=0 \\
y=7 \ \lor \ y=-20[/tex]
[tex]x=13+y \\ \Downarrow \\ \hbox{for } y=7: \\
x=13+7 \\
x=20 \\ \\
\hbox{for } y=-20: \\
x=13-20 \\
x=-7 \\ \\
(x,y)=(20,7) \hbox{ or } (x,y)=(-7,-20)[/tex]
The numbers are 20 and 7 or -7 and -20.
