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Sagot :
To solve the given linear programming problem, follow these step-by-step instructions:
### Step 1: Convert the Problem to Standard Form
The original problem is:
[tex]\[ \begin{array}{ll} \text{Maximize:} & z = 6x + 2y \\ \text{subject to:} & 6x - y \leq 15 \\ & 2x + y \geq 10 \\ & x \geq 2 \\ & y \leq 9 \end{array} \][/tex]
First, convert all constraints to the form [tex]\(Ax \leq b\)[/tex].
The second constraint [tex]\(2x + y \geq 10\)[/tex] can be rewritten as:
[tex]\[ -2x - y \leq -10 \][/tex]
Thus, our inequalities become:
[tex]\[ \begin{array}{ll} 6x - y \leq 15 \\ -2x - y \leq -10 \\ x \geq 2 \\ y \leq 9 \end{array} \][/tex]
### Step 2: Identify Variable Bounds
From the inequalities:
- [tex]\( x \geq 2 \)[/tex] implies a lower bound for [tex]\( x \)[/tex] of 2.
- [tex]\( y \leq 9 \)[/tex] implies an upper bound for [tex]\( y \)[/tex] of 9.
### Step 3: Solve the System of Linear Inequalities
Use linear programming techniques to find the optimal values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
First, consider the intersection of the constraints by graphing or mathematical methods.
To find the feasible region, solve the system of equations formed by the intersecting constraints:
1. Solve [tex]\( 6x - y = 15 \)[/tex] and [tex]\( -2x - y = -10 \)[/tex]:
[tex]\[ \begin{align*} 6x - y &= 15 \quad \text{(i)} \\ -2x - y &= -10 \quad \text{(ii)} \end{align*} \][/tex]
Add equation (i) and (ii):
[tex]\[ 6x - y - 2x - y = 15 - 10 \][/tex]
[tex]\[ 4x - 2y = 5 \][/tex]
[tex]\[ 4x = 20 \Rightarrow x = 4 \][/tex]
Plug [tex]\( x = 4 \)[/tex] back into one of the original equations:
[tex]\[ 6(4) - y = 15 \][/tex]
[tex]\[ 24 - y = 15 \][/tex]
[tex]\[ y = 9 \][/tex]
The intersection point is [tex]\( (4, 9) \)[/tex].
### Step 4: Evaluate the Objective Function at Feasible Corners
Considering all constraints and variable bounds, the key feasible points you need to evaluate are located at:
- [tex]\( (2, 9) \)[/tex] (from [tex]\( x \geq 2 \)[/tex] and [tex]\( y \leq 9 \)[/tex])
- [tex]\( (4, 9) \)[/tex] (from the intersection above)
Evaluate [tex]\( z = 6x + 2y \)[/tex] at these points:
- For [tex]\( (2, 9) \)[/tex]:
[tex]\[ z = 6(2) + 2(9) = 12 + 18 = 30 \][/tex]
- For [tex]\( (4, 9) \)[/tex]:
[tex]\[ z = 6(4) + 2(9) = 24 + 18 = 42 \][/tex]
### Step 5: Determine the Maximum Value
The maximum value of [tex]\( z \)[/tex] among the feasible points is [tex]\( 42 \)[/tex], achieved at [tex]\( (4, 9) \)[/tex].
Hence, the maximum value is:
[tex]\[ \boxed{42} \][/tex]
### Step 1: Convert the Problem to Standard Form
The original problem is:
[tex]\[ \begin{array}{ll} \text{Maximize:} & z = 6x + 2y \\ \text{subject to:} & 6x - y \leq 15 \\ & 2x + y \geq 10 \\ & x \geq 2 \\ & y \leq 9 \end{array} \][/tex]
First, convert all constraints to the form [tex]\(Ax \leq b\)[/tex].
The second constraint [tex]\(2x + y \geq 10\)[/tex] can be rewritten as:
[tex]\[ -2x - y \leq -10 \][/tex]
Thus, our inequalities become:
[tex]\[ \begin{array}{ll} 6x - y \leq 15 \\ -2x - y \leq -10 \\ x \geq 2 \\ y \leq 9 \end{array} \][/tex]
### Step 2: Identify Variable Bounds
From the inequalities:
- [tex]\( x \geq 2 \)[/tex] implies a lower bound for [tex]\( x \)[/tex] of 2.
- [tex]\( y \leq 9 \)[/tex] implies an upper bound for [tex]\( y \)[/tex] of 9.
### Step 3: Solve the System of Linear Inequalities
Use linear programming techniques to find the optimal values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
First, consider the intersection of the constraints by graphing or mathematical methods.
To find the feasible region, solve the system of equations formed by the intersecting constraints:
1. Solve [tex]\( 6x - y = 15 \)[/tex] and [tex]\( -2x - y = -10 \)[/tex]:
[tex]\[ \begin{align*} 6x - y &= 15 \quad \text{(i)} \\ -2x - y &= -10 \quad \text{(ii)} \end{align*} \][/tex]
Add equation (i) and (ii):
[tex]\[ 6x - y - 2x - y = 15 - 10 \][/tex]
[tex]\[ 4x - 2y = 5 \][/tex]
[tex]\[ 4x = 20 \Rightarrow x = 4 \][/tex]
Plug [tex]\( x = 4 \)[/tex] back into one of the original equations:
[tex]\[ 6(4) - y = 15 \][/tex]
[tex]\[ 24 - y = 15 \][/tex]
[tex]\[ y = 9 \][/tex]
The intersection point is [tex]\( (4, 9) \)[/tex].
### Step 4: Evaluate the Objective Function at Feasible Corners
Considering all constraints and variable bounds, the key feasible points you need to evaluate are located at:
- [tex]\( (2, 9) \)[/tex] (from [tex]\( x \geq 2 \)[/tex] and [tex]\( y \leq 9 \)[/tex])
- [tex]\( (4, 9) \)[/tex] (from the intersection above)
Evaluate [tex]\( z = 6x + 2y \)[/tex] at these points:
- For [tex]\( (2, 9) \)[/tex]:
[tex]\[ z = 6(2) + 2(9) = 12 + 18 = 30 \][/tex]
- For [tex]\( (4, 9) \)[/tex]:
[tex]\[ z = 6(4) + 2(9) = 24 + 18 = 42 \][/tex]
### Step 5: Determine the Maximum Value
The maximum value of [tex]\( z \)[/tex] among the feasible points is [tex]\( 42 \)[/tex], achieved at [tex]\( (4, 9) \)[/tex].
Hence, the maximum value is:
[tex]\[ \boxed{42} \][/tex]
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