Model and solve problems that involve matrices of up to dimension 2 × 2, including the solution of systems of linear equations, with and without technology.

A transport company operates two depots. The weekly vehicle transfers satisfy the system: \[\begin{align} 2x + y &= 18 \\ x + 3y &= 24 \end{align}\] where \(x\) is the number of vehicles transferred to Depot A and \(y\) is the number transferred to Depot B. What are the values of \(x\) and \(y\)?

A small business produces two products: deluxe gift boxes and standard gift boxes. Each deluxe box requires 3 hours of assembly time and 2 hours of packaging time. Each standard box requires 1 hour of assembly time and 3 hours of packaging time. In a particular week, the business has 240 hours of assembly time and 300 hours of packaging time available. Use a matrix method to determine how many of each type of box can be produced if all available time is used.