Solve the LP problems. If no optimal solution exists,...

Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded.
Maximize $\begin{aligned} & p=2 x+3 y \\ \text { subject to } &-x+y \geq 10 \\ & x+2 y \leq 12 \\ & x \geq 0, y \geq 0 . \end{aligned}$

Key Concepts

Linear Programming

Linear programming is an optimization technique where a linear objective function is maximized or minimized subject to a set of linear constraints. It forms the basis for many operations research problems and decision-making models across various industries.

Objective Function

The objective function in linear programming represents the quantity to be optimized, either maximized or minimized. It is a linear function of the decision variables and is used to measure the performance of different feasible solutions.

Constraints

Constraints in linear programming are linear equalities or inequalities that define the limitations or requirements of the problem. They restrict the values that the decision variables can take, thereby shaping the set of feasible solutions.

Feasible Region

The feasible region is the set of all points that satisfy all of the problem's constraints simultaneously. It represents the domain of potential solutions from which an optimal solution is selected.

Boundedness and Unboundedness

Boundedness refers to whether the feasible region is confined to a finite area. If the region is unbounded, it means that the objective function may take on infinitely large or small values, potentially leading to an unbounded optimal solution.

Optimal Solution

The optimal solution is the point within the feasible region that yields the best value for the objective function. It is the solution that maximizes or minimizes the objective based on the problem's requirements, provided that such a solution exists.

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