1. **State the problem:** We want to maximize the objective function $$P = 6x_1 + 2x_2$$ subject to the constraint $$7x_1 + 9x_2 \leq 35$$ and non-negativity conditions $$x_1, x_2 \geq 0$$. 2. **Introduce slack variable:** To convert the inequality constraint into an equation, we add a slack variable $$s_1 \geq 0$$ which represents the unused portion of the resource. 3. **Form the equation:** The inequality $$7x_1 + 9x_2 \leq 35$$ becomes the equation $$7x_1 + 9x_2 + s_1 = 35$$. 4. **Interpretation:** The slack variable $$s_1$$ measures how much less than 35 the left side is, ensuring the equation holds exactly. **Final answer:** $$7x_1 + 9x_2 + s_1 = 35$$ with $$x_1, x_2, s_1 \geq 0$$.