True or False. Determine whether the following statements are true or false.
1. The solution set of a system of linear inequalities is unbounded if it can be enclosed by a circle.
2. A linear programming problem consists of a linear objective function to be maximized or minimized subject to certain constraints in the form of linear equations or inequalities.
3. If a linear programming problem has a solution, then it must occur at an interior point of the feasible set \( S \) .
4. If a feasible set \( S \) of a linear programming problem is bounded, then the objective function has a minimum and a maximum value on the feasible set \( S \) .
5. Non-negative variables known as slack variables are used to replace the system of equality constraints with a system of inequality constraints.
6. A solution obtained by setting the basic variables equal to zero is known as the optimal solution of the system of linear inequalities.
7. When selecting the pivot column for solving standard maximization problems using the sim- plex method, you must first locate the leatt negative entry to the left of the vertical line in the last row of the initial simplex tableau.
8. The pivot element is the element common to both the pivot column and the pivot row.
9. A dual problem has a solution if and only if the corresponding primal problem has a solution.
10. The objective functions of both the primal and dual problem attain the same optimal value.