Step 1 :Let's denote the number of sandwiches Pete's Deli sold as x. According to the problem, Bill's Roast Beef sells 6 times as many sandwiches as Pete's Deli, so we can denote the number of sandwiches Bill's Roast Beef sold as 6x.
Step 2 :The problem also states that the difference between their sales is 350 sandwiches. This gives us the equation \(6x - x = 350\).
Step 3 :We can solve this equation to find the value of x, which represents the number of sandwiches sold by Pete's Deli. The solution to the equation is \(x = 70\). This means Pete's Deli sold 70 sandwiches.
Step 4 :To find the number of sandwiches sold by Bill's Roast Beef, we multiply x by 6, which gives us 420 sandwiches.
Step 5 :Final Answer: Pete's Deli sold \(\boxed{70}\) sandwiches and Bill's Roast Beef sold \(\boxed{420}\) sandwiches.