Final answer:
The appropriate mathematical technique to maximize the given linear function under constraints is linear algebra, specifically linear programming.
Step-by-step explanation:
To maximize z = 10x₁ + 15x₂ + 20x₃ + 25x₄ under certain constraints, the appropriate mathematical technique is linear algebra. This problem is an example of linear programming, which is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships.
Linear algebra is used in linear programming to find the maximum or minimum value of a linear function subjected to various constraints. The constraints also need to be linear, which means they can be written in the form ax + by + ... ≤ c, where a, b, c, etc., are constants. The typical method to solve such a problem would be through the simplex algorithm or graphical methods, depending on the number of variables and constraints.
This type of problem is typical in economics, business, engineering, and military applications, where decisions have to be made regarding the allocation of resources such as materials, time, and labor, in the most beneficial manner.