1. **State the problem:** Dean processed two catering orders. The first order: 6 trays of club sandwiches and 4 trays of vegetarian sandwiches cost 56. The second order: 5 trays of club sandwiches and 4 trays of vegetarian sandwiches cost 52. We need to find the cost of one tray of club sandwiches and one tray of vegetarian sandwiches. 2. **Define variables:** Let $x$ = cost of one tray of club sandwiches. Let $y$ = cost of one tray of vegetarian sandwiches. 3. **Write the system of equations:** From the first order: $$6x + 4y = 56$$ From the second order: $$5x + 4y = 52$$ 4. **Use elimination to solve:** Subtract the second equation from the first to eliminate $y$: $$ (6x + 4y) - (5x + 4y) = 56 - 52 $$ $$ 6x + 4y - 5x - 4y = 4 $$ $$ (6x - 5x) + (4y - 4y) = 4 $$ $$ x + \cancel{0} = 4 $$ $$ x = 4 $$ 5. **Substitute $x=4$ into one of the original equations to find $y$:** Using the second equation: $$ 5(4) + 4y = 52 $$ $$ 20 + 4y = 52 $$ Subtract 20 from both sides: $$ \cancel{20} + 4y - \cancel{20} = 52 - 20 $$ $$ 4y = 32 $$ Divide both sides by 4: $$ \frac{4y}{\cancel{4}} = \frac{32}{\cancel{4}} $$ $$ y = 8 $$ 6. **Answer:** A tray of club sandwiches costs $4$, and a tray of vegetarian sandwiches costs $8$.