Let the number be "xy", that is to say 10*x + y.

Then we know that

x + y = 10

and

"yx = 2*xy - 1" or 10*y + x = 2*(10*x + y) - 1

From the first condition

y = 10 - x

Inserting this into the second condition:

10*[10 - x] + x = 2*(10*x + [10 - x]) - 1

100 - 10x + x = 2*(10x + 10 - x) - 1

100 - 9x = 2*(9x + 10) - 1

100 - 9x = 18x + 20 - 1

100 - 9x = 18x + 19

-27x = -81

x = -81/(-27) = 3

Thus

y = 10 - (3) = 7

and the number is 37, as advertised.

-Dan