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Sagot :
Certainly! Let's solve the system of inequalities step by step, determining the feasible region where all conditions are satisfied.
Given inequalities:
1. [tex]\( x + y \leq 9 \)[/tex]
2. [tex]\( -x + y \geq 4 \)[/tex]
3. [tex]\( x \geq 0 \)[/tex]
4. [tex]\( y \geq 0 \)[/tex]
### Step-by-Step Solution:
#### Step 1: Find the Intersection Points of the Lines
1. For [tex]\( x + y = 9 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 9 \)[/tex]
- When [tex]\( y = 0 \)[/tex]: [tex]\( x = 9 \)[/tex]
So the points are (0, 9) and (9, 0).
2. For [tex]\( -x + y = 4 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 4 \)[/tex]
- When [tex]\( y = 0 \)[/tex]: [tex]\( x = -4 \)[/tex] (but for [tex]\( x \geq 0 \)[/tex], we don't need this point)
So the points are (0, 4).
3. Intersection of [tex]\( x + y = 9 \)[/tex] and [tex]\( -x + y = 4 \)[/tex]:
- Solve the system:
[tex]\[ \begin{cases} x + y = 9 \\ -x + y = 4 \end{cases} \][/tex]
Add the two equations:
[tex]\[ (x + y) + (-x + y) = 9 + 4 \Rightarrow 2y = 13 \Rightarrow y = 6.5 \][/tex]
Substitute [tex]\( y = 6.5 \)[/tex] into [tex]\( x + y = 9 \)[/tex]:
[tex]\[ x + 6.5 = 9 \Rightarrow x = 2.5 \][/tex]
So the intersection point is [tex]\((2.5, 6.5)\)[/tex].
#### Step 2: Determine the Feasible Region
All constraints must be satisfied together:
- [tex]\( x \geq 0 \)[/tex]: This implies we are considering points in the right half-plane including the y-axis.
- [tex]\( y \geq 0 \)[/tex]: This implies we are considering points in the upper half-plane including the x-axis.
The corner points of the feasible region (polygon) are:
- (0, 4)
- (0, 9)
- (2.5, 6.5)
- (9, 0)
These are determined by the intersections found and the constraints [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex].
#### Step 3: Draw the Feasible Region
To graph the feasible region:
1. Plot the points (0, 4), (0, 9), (2.5, 6.5), and (9, 0).
2. Connect these points to form a polygon.
When you draw the graph, it should look something like this:
```plaintext
10 | (0,9)
9 |
8 | / \
7 | / (2.5,6.5)
6 | / \
5 | / \
4 |(0,4) \
3 | / \
2 | / \
1 | / \
0 --------------------*(9,0)
0 1 2 3 4 5 6 7 8 9
```
### Conclusion
Every point inside and on the boundary of this polygon is a solution to the system of inequalities. This is your feasible region.
Given inequalities:
1. [tex]\( x + y \leq 9 \)[/tex]
2. [tex]\( -x + y \geq 4 \)[/tex]
3. [tex]\( x \geq 0 \)[/tex]
4. [tex]\( y \geq 0 \)[/tex]
### Step-by-Step Solution:
#### Step 1: Find the Intersection Points of the Lines
1. For [tex]\( x + y = 9 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 9 \)[/tex]
- When [tex]\( y = 0 \)[/tex]: [tex]\( x = 9 \)[/tex]
So the points are (0, 9) and (9, 0).
2. For [tex]\( -x + y = 4 \)[/tex]:
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 4 \)[/tex]
- When [tex]\( y = 0 \)[/tex]: [tex]\( x = -4 \)[/tex] (but for [tex]\( x \geq 0 \)[/tex], we don't need this point)
So the points are (0, 4).
3. Intersection of [tex]\( x + y = 9 \)[/tex] and [tex]\( -x + y = 4 \)[/tex]:
- Solve the system:
[tex]\[ \begin{cases} x + y = 9 \\ -x + y = 4 \end{cases} \][/tex]
Add the two equations:
[tex]\[ (x + y) + (-x + y) = 9 + 4 \Rightarrow 2y = 13 \Rightarrow y = 6.5 \][/tex]
Substitute [tex]\( y = 6.5 \)[/tex] into [tex]\( x + y = 9 \)[/tex]:
[tex]\[ x + 6.5 = 9 \Rightarrow x = 2.5 \][/tex]
So the intersection point is [tex]\((2.5, 6.5)\)[/tex].
#### Step 2: Determine the Feasible Region
All constraints must be satisfied together:
- [tex]\( x \geq 0 \)[/tex]: This implies we are considering points in the right half-plane including the y-axis.
- [tex]\( y \geq 0 \)[/tex]: This implies we are considering points in the upper half-plane including the x-axis.
The corner points of the feasible region (polygon) are:
- (0, 4)
- (0, 9)
- (2.5, 6.5)
- (9, 0)
These are determined by the intersections found and the constraints [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex].
#### Step 3: Draw the Feasible Region
To graph the feasible region:
1. Plot the points (0, 4), (0, 9), (2.5, 6.5), and (9, 0).
2. Connect these points to form a polygon.
When you draw the graph, it should look something like this:
```plaintext
10 | (0,9)
9 |
8 | / \
7 | / (2.5,6.5)
6 | / \
5 | / \
4 |(0,4) \
3 | / \
2 | / \
1 | / \
0 --------------------*(9,0)
0 1 2 3 4 5 6 7 8 9
```
### Conclusion
Every point inside and on the boundary of this polygon is a solution to the system of inequalities. This is your feasible region.
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