During a one-month promotional campaign, Fran's Flix gave either a free DVD rental or a 12-serving box of microwave popcorn to new members. It cost the store $\$ 1$ for each free rental and $\$ 2$ for each box of popcorn. In all, 39 new members were signed up and the store's cost for the incentives was $\$ 70$. How many of each incentive were given away?

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.