To minimize the objective function [tex]\( z = 4x + 2y \)[/tex] subject to the given constraints, we can use the following steps:
### Step 1: Convert the Inequalities to Standard Form
The inequalities given are:
1. [tex]\( 4y + 5x \geq 32 \)[/tex]
2. [tex]\( 3y + 2x \geq 16 \)[/tex]
3. [tex]\( y + x \geq 7 \)[/tex]
We need to convert these inequalities into a standard form for linear programming, which involves ensuring all the inequalities are less than or equal to (≤) type. For this, we will multiply each inequality by -1 to reverse the inequality signs:
1. [tex]\( 4y + 5x \geq 32 \)[/tex] becomes [tex]\( -5x - 4y \leq -32 \)[/tex]
2. [tex]\( 3y + 2x \geq 16 \)[/tex] becomes [tex]\( -2x - 3y \leq -16 \)[/tex]
3. [tex]\( y + x \geq 7 \)[/tex] becomes [tex]\( -x - y \leq -7 \)[/tex]
Additionally, the constraints [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex] ensure that we are working in the first quadrant.
### Step 2: Identify the Objective Function and Constraints
The objective function to minimize is:
[tex]\[ z = 4x + 2y \][/tex]
With the inequalities:
1. [tex]\( -5x - 4y \leq -32 \)[/tex]
2. [tex]\( -2x - 3y \leq -16 \)[/tex]
3. [tex]\( -x - y \leq -7 \)[/tex]
4. [tex]\( x \geq 0 \)[/tex]
5. [tex]\( y \geq 0 \)[/tex]
### Step 3: Determine the Feasible Region
The feasible region is the set of all points that satisfy the system of inequalities. Graphically, each inequality represents a half-plane, and the feasible region is the intersection of all these half-planes.
### Step 4: Identify the Corner Points
In linear programming, the minimum or maximum value of the objective function occurs at one of the vertices (corner points) of the feasible region. By solving the system of equations formed by the boundaries of these inequalities, we find the corner points.
### Step 5: Evaluate the Objective Function at Each Corner Point
To find the minimum value of [tex]\( z \)[/tex], we evaluate [tex]\( z = 4x + 2y \)[/tex] at each feasible corner point.
### Step 6: Conclusion and Result
Upon solving the system and evaluating the objective function, we find:
[tex]\[ z = 4x + 2y \text{ achieves its minimum value when} x = 0 \text{ and} y = 8. \][/tex]
Thus, the minimum value is:
[tex]\[ z = 4(0) + 2(8) = 16. \][/tex]
### Final Answer
The minimum value of [tex]\( z \)[/tex] is [tex]\( \boxed{16} \)[/tex] when [tex]\( x = 0 \)[/tex] and [tex]\( y = 8 \)[/tex].
