Let the age of Suzy be S. It is given that she is 10 years older than Billy. Let Billy be B years old. So S = B+10

Next year Suzy will be S+1 years old and Billy will be B+1 years. Also, Suzy will be twice as old as Billy. Therefore S+1 = 2(B+1)

=> S+1 = 2B + 2

=> B+ 10 +1 = 2B +2

=> B+11 = 2B +2

=> B = 9.

So Billy is 9 years old and Suzy is 9 +10 = 19 years old.

There ages of Billy and Suzy now are 9 and 19 respectively.

Let Suzy's age be S

Let Billy's age be B

==> given that Suzy's 10 years older that Billy:

==> S = B + 10 ..............(1)

Next year :

Suzy's age will be S + 1

Billy Will be B+1

But given that Suzy will be twice Billy's age:

==> S+1 = 2 ( B+1)

==> S + 1=  2B + 2

==> S = 2B + 1

But from (1) we know that S = 10 + B

==> 10 + B = 2B + 1

==> B = 9

==> S = 19

Then Billy's age= 9

And Suzy's age = 19