Amovie theater earns $3946 by selling 428 tickets. if a full price costs $10.25 and a discount costs $8, how many full price tickets were sold?

We need a system of equations here, one equation based on the NUMBER of tickets sold and another based on the MONEY earned by the sales.  We have 2 different types of tickets:  full price (f) and discount (d).  The total number of tickets sold is 428; therefore, the first equation is f + d = 428.  That accounts for the number of tickets sold.  Each full price is 10.25 which can be represented as 10.25f, and each discount ticket costs 8 which can be represented as 8d.  The money earned by selling these tickets at those prices was 3946.  That means that the second equation is 10.25f + 8d = 3946.  We will solve the first bolded equation for f to get f = 428 - d.  Sub that value in for f in the second bolded equation:  10.25(428-d) + 8d = 3946.  Distribute to get 4387 - 10.25d + 8d = 3946.  Combine like terms to get -2.25d = -441.  Solving for d we get 196.  That means that there were 196 discounted tickets sold.  Put that in for d in the first bolded equation to find the number of full price tickets.  f + 196 = 428,  and f = 232.  There were 232 full price tickets sold.  There you go!