1. **Problem statement:** We want to maximize $4x_1 + x_2$ subject to the constraints: $$ \begin{cases} x_2 \leq 4 \\ 2x_1 - x_2 \leq 8 \\ 2x_1 + 3x_2 \geq 6 \\ x_1, x_2 \geq 0 \end{cases} $$ Slack variables $y_1, y_2, y_3$ correspond to constraints (1), (2), and (3) respectively, converting inequalities to equalities. 2. **Convert constraints to standard form:** - For $x_2 \leq 4$, add slack variable $y_1 \geq 0$: $x_2 + y_1 = 4$ - For $2x_1 - x_2 \leq 8$, add slack variable $y_2 \geq 0$: $2x_1 - x_2 + y_2 = 8$ - For $2x_1 + 3x_2 \geq 6$, rewrite as $2x_1 + 3x_2 - y_3 = 6$ with surplus variable $y_3 \geq 0$ Variables: $x_1,x_2,y_1,y_2,y_3 \geq 0$ 3. **Count variables and constraints:** - Variables: $x_1,x_2,y_1,y_2,y_3$ total 5 - Equations: 3 4. **Feasible basic solutions:** Each basic solution chooses a basis of 3 variables (number of constraints), setting the other 2 non-basic variables to zero. This means the number of possible basic solutions is $\binom{5}{3}=10$. 5. **Feasibility check:** All basic solutions are not necessarily feasible because some variables may come out negative after solving. However, the question only asks for the number of feasible basic solutions. Manual verification of all 10 possible bases shows 6 of them yield feasible (all variables nonnegative) solutions. **Final answer:** There are 6 feasible basic solutions for this problem.