Answer:
Ivy is 15 years old and Audrey is 12 years old.
Explanation:
Let Ivy's age be
and Audrey's age be
.
Since the sum of their ages is 27, we can write the equation
.
Next, we'll write a second equation from the fact that 9 years ago Ivy was twice as old as Audrey. Nine years ago, Ivy and Audrey's ages were
and
, respectively. Therefore, we have

Let's isolate
by adding 9 to both sides:

Distribute:

Now substitute
into our first equation:

Therefore, Ivy's age must be:

Thus, Ivy must be 15 years old and Audrey must be 12 years old.