To maximize the function [tex]\( F = 6x + 5y \)[/tex] subject to the constraints:
1. [tex]\( x + y \leq 4 \)[/tex]
2. [tex]\( 2x + y \leq 7 \)[/tex]
3. [tex]\( x \geq 0 \)[/tex]
4. [tex]\( y \geq 0 \)[/tex]
we proceed with the following steps:
### Step 1: Identify the Feasible Region
Firstly, let's identify the feasible region determined by the constraints:
- The line [tex]\( x + y = 4 \)[/tex] can be rewritten as [tex]\( y = 4 - x \)[/tex]. This represents a line that passes through the points [tex]\( (0,4) \)[/tex] and [tex]\( (4,0) \)[/tex].
- The line [tex]\( 2x + y = 7 \)[/tex] can be rewritten as [tex]\( y = 7 - 2x \)[/tex]. This represents a line that passes through the points [tex]\( (0,7) \)[/tex] and [tex]\( (3.5,0) \)[/tex].
Next, determine the intersection points and which portion of the plane satisfies all the constraints [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex]. The region of interest is bounded by these lines and the coordinate axes.
### Step 2: Find the Intersection Points
Calculate the intersection points of the lines:
1. Intersection of [tex]\( x + y = 4 \)[/tex] and [tex]\( 2x + y = 7 \)[/tex]:
[tex]\[
\begin{cases}
x + y = 4 \\
2x + y = 7
\end{cases}
\][/tex]
Subtract the first equation from the second:
[tex]\[
(2x + y) - (x + y) = 7 - 4 \\
x = 3
\][/tex]
Substitute [tex]\( x = 3 \)[/tex] back into [tex]\( x + y = 4 \)[/tex]:
[tex]\[
3 + y = 4 \\
y = 1
\][/tex]
So, the intersection point is [tex]\( (3, 1) \)[/tex].
### Step 3: Evaluate the Objective Function at Corner Points
Evaluate [tex]\( F = 6x + 5y \)[/tex] at each vertex of the feasible region. The vertices are the intersection points and the points where the lines intersect the axes:
1. At [tex]\( (0,0) \)[/tex]:
[tex]\[
F = 6(0) + 5(0) = 0
\][/tex]
2. At [tex]\( (4,0) \)[/tex]:
[tex]\[
F = 6(4) + 5(0) = 24
\][/tex]
3. At [tex]\( (0,7) \)[/tex]:
[tex]\[
F = 6(0) + 5(7) = 35 \quad (\text{Note: This point does not satisfy } x + y \leq 4)
\][/tex]
4. At [tex]\( (3,1) \)[/tex]:
[tex]\[
F = 6(3) + 5(1) = 18 + 5 = 23
\][/tex]
### Step 4: Determine the Maximum Value
Compare the values obtained at each feasible vertex:
- [tex]\( F(0, 0) = 0 \)[/tex]
- [tex]\( F(4, 0) = 24 \)[/tex]
- [tex]\( F(3, 1) = 23 \)[/tex]
The maximum value is [tex]\( F = 24 \)[/tex] achieved at the point [tex]\( (3, 1) \)[/tex].
### Conclusion
[a] The maximum value of [tex]\( F \)[/tex] is [tex]\(\boxed{23.0}\)[/tex].
[b] The point where this maximum value occurs is [tex]\(\boxed{(3.0, 1.0)}\)[/tex].
Therefore, the final answer is:
(a) The maximum value [tex]\(F\)[/tex] is [tex]\(23.0\)[/tex].
(b) The point where the maximum value occurs is [tex]\((3.0, 1.0)\)[/tex].
These are the optimal solutions within the feasible region for the given linear programming problem.
