If the numbers are a and b, then a+b=63 and a-b=9. From the first equation a=63-b, so we can substitute this value in the other: 63-b-b=9, 63-2b=9, so 2b=63-9=54 and b=27. From this value of b we can find a: a=27+9=36.
Another way to eliminate a or b is to add the two equations: 2a=72, so a=36; and subtract the second from the first: 2b=54, so b=27.