The sum of the digits of a three-place number is 19. if the tens and units digits are interchange the number is diminished by 27, and if the hundreds and tens digits are interchange the number is increased by 180. what is the number? set up sum of three didgits with only 2 variables.

I cant really help you out with the last part since setting up the sum of three digits with two variable complicates things but here is the calculation and answer as follows:

let the number be a be represented as xyz (where x is the hundredth digit, y is the tenth digit and z is the one digit)

which means a = 100x+10y+z (equation 1)
if you add the three digits x+y+z = 19 (equation 2)

if you switch positions of the tenth digit and one digit then
a-27 = 100x + 10z +y (equation 3) (number diminish by 27 hence why a -27)

if you switch the position of the hundredth and tenth digit then
a+180 = 100y + 10x + z (equation 4) (number increases by 180)

if you use equation 1, 2, 3, and 4 and solve for x, y and z which gives you a then the number is 685.

Sorry for the long explanation but this was the first approach that came to mind might be more simpler ones and also one that helps you set up the sum of three digits with only 2 variables.