Are these solutions to a LP problem feasible? basic?

Consider the following LP: \begin{align*} \max 8x_1 + 14x_2 + 12x_3 + 50x_4\\ \text{s. t. } x_1 + 2x_2 + 2x_3 + 16x_4 &\le 8\\ 2x_1 + 3x_2 + 4x_3 + 5x_4 &\le 15\\ 5x_1 + 6x_2 + 8x_3 + 10x_4 &\le 40\\ x_1, x_2, x_3, x_4 &\ge 0 \end{align*}

a) Is the solution $(x_1, x_2, x_3, x_4) = (1,3,0,0)$ feasible? basic?

b) Is the solution $(x_1, x_2, x_3, x_4) = (8,0,0,0)$ feasible? basic? degenerate?

So far, I have: (a) The solution is feasible because it satisfies all the three constraints. (b) The solution is not feasible because it does not satisfy the second constraint.

How do you know if the solutions are basic and degenerate?

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I will name the objective function eq-1.

I will name the constraints with ($\leq$) eq-2. I will name the constraints $x_{i} >= 0$ eq-3.

(a) Is the solution $(1,3,0,0)$ feasible? basic?

Feasibility is granted when eq-2 is satisfied.

A basic feasible solution must satisfy eq-2 and eq-3. Since both are satisfied in your case, (1,3,0,0) is a basic feasible solution.

(b) Hint: use the same definitions as in (a) above.

For all the other definitions in your question clearly stated, see this nice book (page 20 - Section 1.6): Operations Research for Management.

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