Answer:
Ivy is 15 years old and Audrey is 12 years old.
Step-by-step explanation:
Let Ivy's age be [tex]i[/tex] and Audrey's age be [tex]a[/tex].
Since the sum of their ages is 27, we can write the equation [tex]i+a=27[/tex].
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 [tex]i-9[/tex] and [tex]a-9[/tex], respectively. Therefore, we have [tex]i-9=2(a-9)[/tex]
Let's isolate [tex]i[/tex] by adding 9 to both sides:
[tex]i=2(a-9)+9[/tex]
Distribute:
[tex]i=2a-18+9,\\i=2a-9[/tex]
Now substitute [tex]i=2a-9[/tex] into our first equation:
[tex]2a-9+a=27,\\3a-9=27, \\3a=36, \\a=\boxed{12}[/tex]
Therefore, Ivy's age must be:
[tex]i+12=27,\\i=27-12=\boxed{15}[/tex]
Thus, Ivy must be 15 years old and Audrey must be 12 years old.
