let x = the 10's digit

let y = the units

then

10x+y = "the number"

and

10y+x = "the number reversed

write an equation for each statement

"nine times a two digit number is the same as twice the number obtained on  

interchanging the digits of the number."

9(10x+y) = 2(10y+x)

90x + 9y = 20y + 2x

90x - 2x = 20y - 9y

88x = 11y

simplify, divide by 11

8x = y

"If one digit of the number exceeds the other by 7,"

y = x + 7 (y has to be the larger number)

replace y with 8x

8x = x + 7

8x - x = 7

7x = 7

x = 1

then

y = 8  

18 is the original number

Check this in the 1st statement

9(18) = 2(81)