Let x = the 10's digit

Let y = the units

then

10x + y = "the number"

:

Write an equation for each statement:

"the digits of a two-digit number are reversed, the new number is 9 more than the original number,"

10y + x = 10x + y + 9

Combine like terms

10y - y = 10x - x + 9

9y = 9x + 9

Simplify, divide by 9

y = (x + 1)

"the sum of the digits of the original number is 13."

x + y = 13

Substitute (x+1) for y

x + (x+1) = 13

2x = 13 - 1

2x = 12

x = 6 is the 10's digit

then since the sum is 13, obviously

y = 7 is the units

and

67 is the original two digit number

:

Check this in the statement:

"when the digits of a two-digit number are reversed, the new number is 9 more than the original number,

76 = 67 + 9

:

When the number 67 if reversed it is 9 more than the original