What is optimal solution?

Answer

### Step-by-Step Solution

1. **Understanding the Concept of Optimal Solution**:
- An optimal solution in linear programming refers to the best possible outcome (maximum or minimum value) of an objective function, given a set of constraints.

**Hint**: Remember that the optimal solution is about finding the best value for the objective function while adhering to the constraints.

2. **Identifying the Objective Function**:
- The objective function is a mathematical expression that we want to optimize. For example, if we have \( z = 2x + 3y \), this is our objective function.

**Hint**: Look for expressions that represent what you want to maximize or minimize in the problem.

3. **Defining Constraints**:
- Constraints are the restrictions or limitations on the variables in the objective function. For instance, if we have constraints like \( x + y \geq 10 \) and \( x + y \leq 5 \), these define the feasible region.

**Hint**: Constraints limit the values that the variables can take, so identify them clearly.

4. **Graphing the Constraints**:
- To visualize the feasible region, graph the constraints on a coordinate plane. The area where all constraints overlap is known as the feasible region.

**Hint**: Use graphing techniques to find where the lines representing constraints intersect and form a bounded area.

5. **Determining the Feasible Region**:
- The feasible region is the common area that satisfies all constraints, including non-negativity constraints (e.g., \( x \geq 0 \) and \( y \geq 0 \)).

**Hint**: Ensure that the feasible region is clearly defined by checking all constraints.

6. **Finding the Optimal Solution**:
- Evaluate the objective function at the vertices (corner points) of the feasible region. The optimal solution will be the vertex that gives the highest or lowest value of the objective function, depending on whether you are maximizing or minimizing.

**Hint**: Check each vertex of the feasible region to see which one provides the optimal value for the objective function.

7. **Conclusion**:
- The optimal solution is thus the point in the feasible region that yields the best value of the objective function, confirming the definition that any point in the feasible region which gives the optimal value of the objective function is considered an optimal solution.

**Hint**: Always summarize your findings to confirm that you have identified the optimal solution correctly.