1. **State the problem:** Convert the inequality system to an equation system using slack variables and find all basic solutions, indicating feasibility. 2. **Convert inequality to equation:** Given inequality: $$3x_1 + 7x_2 \leq 21$$ Add slack variable $$s_1 \geq 0$$ to convert to equation: $$3x_1 + 7x_2 + s_1 = 21$$ 3. **Basic solutions:** Basic solutions occur when two variables are set to zero and the third is solved from the equation. - Set $$x_1 = 0$$ and $$x_2 = 0$$: $$s_1 = 21$$ Since $$s_1 = 21 \geq 0$$, this solution is feasible. - Set $$x_1 = 0$$ and $$s_1 = 0$$: $$7x_2 = 21 \implies x_2 = 3$$ Since $$x_2 = 3 \geq 0$$, this solution is feasible. - Set $$x_2 = 0$$ and $$s_1 = 0$$: $$3x_1 = 21 \implies x_1 = 7$$ Since $$x_1 = 7 \geq 0$$, this solution is also feasible. 4. **Summary table of basic solutions:** | $x_1$ | $x_2$ | $s_1$ | Feasible | |-------|-------|-------|----------| | 0 | 0 | 21 | Yes | | 0 | 3 | 0 | Yes | | 7 | 0 | 0 | Yes | All basic solutions are feasible because all variables are non-negative. **Final answer:** The e-system is $$3x_1 + 7x_2 + s_1 = 21$$ with slack variable $$s_1 \geq 0$$. All basic solutions are feasible: $(0,0,21)$, $(0,3,0)$, and $(7,0,0)$.