1. **State the problem:** We want to maximize the objective function $$P = 35x_1 + x_2$$ subject to the constraints: $$2x_1 + x_2 \leq 8$$ $$x_1 + 5x_2 \leq 8$$ with $$x_1, x_2 \geq 0$$. 2. **Introduce slack variables:** Add slack variables $$s_1$$ and $$s_2$$ to convert inequalities into equalities: $$2x_1 + x_2 + s_1 = 8$$ $$x_1 + 5x_2 + s_2 = 8$$ where $$s_1, s_2 \geq 0$$. 3. **Rewrite the objective function for the simplex method:** Express as: $$P - 35x_1 - x_2 = 0$$ 4. **Initial simplex tableau:** \[ \begin{array}{c|cccc|c} & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\ \hline s_1 & 2 & 1 & 1 & 0 & 8 \\ s_2 & 1 & 5 & 0 & 1 & 8 \\ -P & -35 & -1 & 0 & 0 & 0 \end{array} \] This tableau summarizes the system with variables $$x_1, x_2, s_1, s_2$$ and the objective function $$P$$. **Final answer:** The simplex tableau is: \[ \begin{array}{c|cccc|c} & x_1 & x_2 & s_1 & s_2 & \text{RHS} \\ \hline s_1 & 2 & 1 & 1 & 0 & 8 \\ s_2 & 1 & 5 & 0 & 1 & 8 \\ -P & -35 & -1 & 0 & 0 & 0 \end{array} \]