Answer:
Ivy is 6 years old and Rose is 12 years old.
Explanation:
To solve this problem, two variables are necessary since we have two unknown variables: Ivy's age and Rose's age.
Ivy's age will be

and Rose's age will be

"Ivy is half as old as her sister":

this will be our fist equation.
"In six years, Ivy will be two-thirds as old as Rose"
in six years Ivy will have
years, and Rose will have
years, so for this sentence we have the equation:

this will be our second equation.
Substituting the value of x from the first equation into the second, and solving for y(Rose):

Rose is 12 years old.
and Ivy (using the fist equation):

Ivy is 6 years old.