The sum of the digits of a two-digit number is 12. the number formed by interchanging the digits is 54 more than the original number. what is the original number?

The number is xy

x+y=12

10y+x=54+10x+y

so

x+y=12

minus x both sides

y=12-x

sub that for y

10 (12-x) + x=54+10x + (12-x)

120-10x+x=54+10x+12-x

120-9x=66+9x

add 9x both sides

120=66+18x

minus 66 both sides

54=18x

divide both sides by 18

3=x

sub back

y=12-x

y=12-3

y=9

the number is 39