Step 1 :Let's denote the number of DVD rentals as x and the number of popcorn boxes as y.
Step 2 :We know that the total number of new members is 39, so we have the equation \(x + y = 39\).
Step 3 :We also know that the total cost for the incentives was $70, and since each DVD rental costs $1 and each popcorn box costs $2, we have the equation \(x + 2y = 70\).
Step 4 :We can solve this system of equations to find the values of x and y.
Step 5 :The solution to the system of equations is \(x = 8\) and \(y = 31\).
Step 6 :This means that Fran's Flix gave away 8 DVD rentals and 31 boxes of popcorn to new members during the promotional campaign.
Step 7 :Final Answer: Fran's Flix gave away \(\boxed{8}\) DVD rentals and \(\boxed{31}\) boxes of popcorn.