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