This morning, Ellie processed two catering orders at the sandwich shop where she works. The first order was for 6 trays of club sandwiches and 2 trays of vegetarian sandwiches, at a cost of $72. The second order, which cost $108, was for 6 trays of club sandwiches and 6 trays of vegetarian sandwiches. How much do the trays cost?
Let's assume the cost of one tray of club sandwiches is $x and the cost of one tray of vegetarian sandwiches is $y.
From the first order, we can write the equation 6x + 2y = 72.
From the second order, we can write the equation 6x + 6y = 108.
To solve this system of equations, we can multiply the first equation by 3 to make the coefficients of x in both equations equal. We get 18x + 6y = 216.
By subtracting the second equation from this equation, we get 12x = 108.
Dividing both sides of the equation by 12, we get x = 9.
Substituting the value of x in the first equation, we get 6*9 + 2y = 72.
Simplifying, we get 54 + 2y = 72.
Subtracting 54 from both sides of the equation, we get 2y = 18.
Dividing both sides of the equation by 2, we get y = 9.
Therefore, the cost of one tray of club sandwiches is $9 and the cost of one tray of vegetarian sandwiches is $9. Answer: \boxed{9}.