Let x be the speed of Santa's sleigh in still air and y be the speed of the wind. We can use the formula d = rt, where d is the distance traveled, r is the speed at which the sleigh is traveling, and t is the time elapsed, to find x and y.
For the first case, where Santa is flying with the wind, the distance traveled is 35 miles, the time elapsed is 5 minutes, and the speed of the sleigh is x + y. Since there are 60 minutes in an hour, the time elapsed in hours is 5 minutes / 60 minutes/hour = 1/12 hours. So, we can write the equation 35 miles = (x + y) * (1/12 hours). Solving for x + y, we get x + y = 35 miles / (1/12 hours) = 420 miles/hour.
For the second case, where Santa is flying against the wind, the distance traveled is 35 miles, the time elapsed is 7 minutes, and the speed of the sleigh is x - y. Since there are 60 minutes in an hour, the time elapsed in hours is 7 minutes / 60 minutes/hour = 7/60 hours. So, we can write the equation 35 miles = (x - y) * (7/60 hours). Solving for x - y, we get x - y = 35 miles / (7/60 hours) = 300 miles/hour.
We can solve for x and y by equating the expressions for x + y and x - y and then solving for x and y. Since x + y = 420 miles/hour and x - y = 300 miles/hour, we can add these equations together to get 2x = 720 miles/hour. Dividing both sides of the equation by 2, we get x = 360 miles/hour. We can then substitute this value for x into the equation x + y = 420 miles/hour to solve for y, giving us y = 420 miles/hour