1. **State the problem:** We have the system of equations: $$2x_1 + 3x_2 + s_1 = 162$$ $$4x_1 + 3x_2 + s_2 = 216$$ and six basic solutions (A) to (F) with values for $x_1$, $x_2$, $s_1$, and $s_2$. We need to determine which of these solutions are feasible. 2. **Feasibility criteria:** A solution is feasible if all variables $x_1$, $x_2$, $s_1$, and $s_2$ are **non-negative** (i.e., $\geq 0$) because these represent quantities or slack variables in linear programming. 3. **Check each solution:** - (A) $x_1=0$, $x_2=0$, $s_1=162$, $s_2=216$; all values $\geq 0$ so feasible. - (B) $x_1=0$, $x_2=54$, $s_1=0$, $s_2=54$; all values $\geq 0$ so feasible. - (C) $x_1=0$, $x_2=72$, $s_1=-54$, $s_2=0$; $s_1$ is negative, so not feasible. - (D) $x_1=81$, $x_2=0$, $s_1=0$, $s_2=-108$; $s_2$ is negative, so not feasible. - (E) $x_1=54$, $x_2=0$, $s_1=54$, $s_2=0$; all values $\geq 0$ so feasible. - (F) $x_1=27$, $x_2=36$, $s_1=0$, $s_2=0$; all values $\geq 0$ so feasible. 4. **Conclusion:** The feasible basic solutions are (A), (B), (E), and (F). Solutions (C) and (D) are not feasible because they have negative slack variables. **Final answer:** Feasible solutions are (A), (B), (E), and (F).