Solution: let this two-digit number be x, then the ten-digit number is 8-X.

So this two-digit number is expressed as10 (8-x)+x = 80-9x; The number after digit position exchange is expressed as 10x+(8-x)=9x+8.

According to the equation, it is: (80-9x)-(9x+8)= 18.

Solution: x=3

So, 8-x=8-3=5.

So the original two-digit number is 53.