📘 linear programming

1. **State the problem:** Minimize the cost function $$C = 15x + 25y$$ subject to the constraints:

1. **State the problem:** Maximize the objective function $$P = 30x + 12y$$ subject to the constraints $$3x + y \leq 18$$ and $$x, y \geq 0$$. 2. **Convert inequalities to equation

1. **State the problem:** We want to maximize the objective function $$P = 30x + 12y$$ subject to the constraints $$3x + y \leq 18$$ and $$x, y \geq 0$$. 2. **Understand the constr

1. **State the problem:** We want to maximize the objective function $$P = 20x + 70y$$ subject to the constraints: $$x + 4y \leq 12$$

1. **State the problem:** We want to maximize the objective function $$P = 20x + 80y$$ subject to the constraints $$x + 3y \leq 12$$ and $$x, y \geq 0$$. 2. **Identify the feasible

1. **Problem statement:** We have 120 mangoes and 180 papayas. We want to make two types of packages. Type A contains 3 mangoes and 3 papayas and sells for 6 each. Type B contains

1. **State the problem:** Visualize the feasible region defined by the constraint $-2x_1 + x_2 \leq 8$ and $x_1, x_2 \geq 0$. 2. **Rewrite the constraint:**

1. **State the problem:** Maximize the objective function $$P = 2x_1 + 5x_2$$ subject to the constraint $$-2x_1 + x_2 \leq 8$$ and $$x_1, x_2 \geq 0$$. 2. **Set up the problem for

1. **State the problem:** We want to maximize the objective function $$P = 35x_1 + 140x_2$$

1. **State the problem:** We need to find the pivot element in the simplex tableau after identifying the pivot column.

1. **State the problem:** We need to find the pivot column in the simplex tableau for maximizing $P = 35x_1 + x_2$.

1. **State the problem:** We want to maximize the objective function $$P = 35x_1 + x_2$$ subject to the constraints:

1. **State the problem:** We want to maximize the objective function $$P = 35x_1 + x_2$$ subject to the constraints:

1. **State the problem:** We want to maximize the objective function $$P = 35x_1 + x_2$$ subject to the constraints:

1. **State the problem:** Graph the system of inequalities:

1. **State the problem:** We want to maximize the objective function $$P = 22x_1 + 11x_2$$

1. **State the problem:** Graph the system of inequalities:

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

1. **State the problem:** Convert the inequality system to an equation system using slack variables and write the associated linear equation. 2. **Given inequality:**

1. **State the problem:** We are given six basic solutions to the system: $$\begin{cases} 2x_1 + 3x_2 + s_1 = 9 \\ 4x_1 + 3x_2 + s_2 = 12 \end{cases}$$

1. **State the problem:** We have the system of equations: $$2x_1 + 3x_2 + s_1 = 162$$