1. **State the problem:** We want to minimize the objective function $$f = x_1 + x_2 + 3x_3 + 2x_5$$ subject to the constraints $$\begin{cases} x_1 + x_2 + x_3 = 4, \\ x_1 - 3x_2 + x_4 = 2, \\ 4x_1 + 2x_2 + x_5 = 7, \end{cases}$$ with non-negativity conditions $$x_j \geq 0, \quad j = 1, \ldots, 5.$$ 2. **Convert to standard form for the simplex method:** The system is already equalities with slack variables $x_4$ and $x_5$ included. 3. **Set up the initial simplex tableau:** Variables: $x_1, x_2, x_3, x_4, x_5$ Objective: minimize $f = x_1 + x_2 + 3x_3 + 2x_5$ Constraints: $$\begin{cases} x_1 + x_2 + x_3 = 4, \\ x_1 - 3x_2 + x_4 = 2, \\ 4x_1 + 2x_2 + x_5 = 7. \end{cases}$$ 4. **Express the objective function in terms of variables:** We want to minimize $f$, so in the simplex tableau, we use $-f$ for maximization: $$-f = -x_1 - x_2 - 3x_3 - 2x_5.$$ 5. **Initial basic variables:** From the constraints, the basic variables can be $x_3, x_4, x_5$ with values: $$x_3 = 4 - x_1 - x_2,$$ $$x_4 = 2 - x_1 + 3x_2,$$ $$x_5 = 7 - 4x_1 - 2x_2.$$ 6. **Substitute into objective function:** $$f = x_1 + x_2 + 3(4 - x_1 - x_2) + 2(7 - 4x_1 - 2x_2)$$ $$= x_1 + x_2 + 12 - 3x_1 - 3x_2 + 14 - 8x_1 - 4x_2$$ $$= (x_1 - 3x_1 - 8x_1) + (x_2 - 3x_2 - 4x_2) + (12 + 14)$$ $$= -10x_1 - 6x_2 + 26.$$ 7. **Minimize $f$ by choosing $x_1, x_2 \\geq 0$ subject to non-negativity of $x_3, x_4, x_5$:** From constraints: $$x_3 = 4 - x_1 - x_2 \geq 0 \Rightarrow x_1 + x_2 \leq 4,$$ $$x_4 = 2 - x_1 + 3x_2 \geq 0 \Rightarrow x_1 \leq 2 + 3x_2,$$ $$x_5 = 7 - 4x_1 - 2x_2 \geq 0 \Rightarrow 4x_1 + 2x_2 \leq 7.$$ 8. **Find feasible region for $(x_1, x_2)$:** - $x_1 + x_2 \leq 4$ - $x_1 \leq 2 + 3x_2$ - $4x_1 + 2x_2 \leq 7$ - $x_1, x_2 \geq 0$ 9. **Check vertices of feasible region:** - At $(0,0)$: $f = 26$ - At $(0,4)$: check constraints: - $x_3 = 4 - 0 - 4 = 0 \geq 0$ - $x_4 = 2 - 0 + 12 = 14 \geq 0$ - $x_5 = 7 - 0 - 8 = -1 < 0$ (not feasible) - At $(1,3)$: - $x_3 = 4 - 1 - 3 = 0$ - $x_4 = 2 - 1 + 9 = 10$ - $x_5 = 7 - 4 - 6 = -3 < 0$ (not feasible) - At $(1.25, 2.75)$ (intersection of $x_1 + x_2 = 4$ and $4x_1 + 2x_2 = 7$): - $x_3 = 4 - 1.25 - 2.75 = 0$ - $x_4 = 2 - 1.25 + 8.25 = 9$ - $x_5 = 7 - 5 - 5.5 = -3.5 < 0$ (not feasible) - At $(1,1)$: - $x_3 = 4 - 1 - 1 = 2$ - $x_4 = 2 - 1 + 3 = 4$ - $x_5 = 7 - 4 - 2 = 1$ - $f = -10(1) - 6(1) + 26 = 10$ 10. **Check if $(1,1)$ is feasible and gives minimum:** All variables non-negative, $f=10$ is less than $f(0,0)=26$. 11. **Conclusion:** The optimal plan is $$x_1 = 1, x_2 = 1, x_3 = 2, x_4 = 4, x_5 = 1$$ with minimum value $$f_{min} = 10.$$