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Sagot :
To determine the vertices of the feasible region defined by the given constraints, follow these steps:
1. Define the Constraints:
[tex]\[ \begin{align*} 1. & \quad x + y \leq 7 \\ 2. & \quad x - 2y \leq -2 \\ 3. & \quad x \geq 0 \\ 4. & \quad y \geq 0 \\ \end{align*} \][/tex]
2. Find Intersection Points:
Solve for the pairs of equations to find where the lines intersect:
- Intersection of [tex]\( x + y = 7 \)[/tex] and [tex]\( x - 2y = -2 \)[/tex]:
[tex]\[ \begin{cases} x + y = 7 \\ x - 2y = -2 \end{cases} \][/tex]
Solving these equations simultaneously:
[tex]\[ x + y = 7 \\ x - 2y = -2 \][/tex]
Subtract the second equation from the first:
[tex]\[ (x + y) - (x - 2y) = 7 - (-2) \\ 3y = 9 \\ y = 3 \][/tex]
Substitute [tex]\( y = 3 \)[/tex] into [tex]\( x + y = 7 \)[/tex]:
[tex]\[ x + 3 = 7 \\ x = 4 \][/tex]
Hence, the intersection is [tex]\( (4, 3) \)[/tex].
- Intersection of [tex]\( x + y = 7 \)[/tex] and [tex]\( x = 0 \)[/tex]:
[tex]\[ x + y = 7 \quad \text{and} \quad x = 0 \][/tex]
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( x + y = 7 \)[/tex]:
[tex]\[ 0 + y = 7 \\ y = 7 \][/tex]
Hence, the intersection is [tex]\( (0, 7) \)[/tex].
- Intersection of [tex]\( x - 2y = -2 \)[/tex] and [tex]\( y = 0 \)[/tex]:
[tex]\[ x - 2y = -2 \quad \text{and} \quad y = 0 \][/tex]
Substitute [tex]\( y = 0 \)[/tex] into [tex]\( x - 2y = -2 \)[/tex]:
[tex]\[ x - 2(0) = -2 \\ x = -2 \][/tex]
But [tex]\( x \geq 0 \)[/tex], so this intersection is not feasible and shouldn't be considered.
3. Check Boundaries:
- Intersection at [tex]\((x, y)\)[/tex] that meet non-negativity conditions:
[tex]\[ x, y \geq 0 \][/tex]
4. Vertices of the Feasible Region:
Combine the intersection points [tex]\( (4, 3) \)[/tex] and [tex]\( (0, 7) \)[/tex] with boundary conditions points.
Considering the options provided:
a) [tex]\( (0, 0), (0, 1), (4, 3), (7, 0) \)[/tex]
b) [tex]\( (0, 1), (4, 3), (7, 0) \)[/tex]
c) [tex]\( (0, 1), (0, 7), (2, 5) \)[/tex]
d) [tex]\( (0, 1), (0, 7), (4, 3) \)[/tex]
Among these options, the vertices we found [tex]\( (4, 3) \)[/tex] and [tex]\( (0, 7) \)[/tex] coincide with option (d). Therefore, the vertices of the feasible region are:
[tex]\[ \boxed{(0,1),(0,7),(4,3)} \][/tex]
1. Define the Constraints:
[tex]\[ \begin{align*} 1. & \quad x + y \leq 7 \\ 2. & \quad x - 2y \leq -2 \\ 3. & \quad x \geq 0 \\ 4. & \quad y \geq 0 \\ \end{align*} \][/tex]
2. Find Intersection Points:
Solve for the pairs of equations to find where the lines intersect:
- Intersection of [tex]\( x + y = 7 \)[/tex] and [tex]\( x - 2y = -2 \)[/tex]:
[tex]\[ \begin{cases} x + y = 7 \\ x - 2y = -2 \end{cases} \][/tex]
Solving these equations simultaneously:
[tex]\[ x + y = 7 \\ x - 2y = -2 \][/tex]
Subtract the second equation from the first:
[tex]\[ (x + y) - (x - 2y) = 7 - (-2) \\ 3y = 9 \\ y = 3 \][/tex]
Substitute [tex]\( y = 3 \)[/tex] into [tex]\( x + y = 7 \)[/tex]:
[tex]\[ x + 3 = 7 \\ x = 4 \][/tex]
Hence, the intersection is [tex]\( (4, 3) \)[/tex].
- Intersection of [tex]\( x + y = 7 \)[/tex] and [tex]\( x = 0 \)[/tex]:
[tex]\[ x + y = 7 \quad \text{and} \quad x = 0 \][/tex]
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( x + y = 7 \)[/tex]:
[tex]\[ 0 + y = 7 \\ y = 7 \][/tex]
Hence, the intersection is [tex]\( (0, 7) \)[/tex].
- Intersection of [tex]\( x - 2y = -2 \)[/tex] and [tex]\( y = 0 \)[/tex]:
[tex]\[ x - 2y = -2 \quad \text{and} \quad y = 0 \][/tex]
Substitute [tex]\( y = 0 \)[/tex] into [tex]\( x - 2y = -2 \)[/tex]:
[tex]\[ x - 2(0) = -2 \\ x = -2 \][/tex]
But [tex]\( x \geq 0 \)[/tex], so this intersection is not feasible and shouldn't be considered.
3. Check Boundaries:
- Intersection at [tex]\((x, y)\)[/tex] that meet non-negativity conditions:
[tex]\[ x, y \geq 0 \][/tex]
4. Vertices of the Feasible Region:
Combine the intersection points [tex]\( (4, 3) \)[/tex] and [tex]\( (0, 7) \)[/tex] with boundary conditions points.
Considering the options provided:
a) [tex]\( (0, 0), (0, 1), (4, 3), (7, 0) \)[/tex]
b) [tex]\( (0, 1), (4, 3), (7, 0) \)[/tex]
c) [tex]\( (0, 1), (0, 7), (2, 5) \)[/tex]
d) [tex]\( (0, 1), (0, 7), (4, 3) \)[/tex]
Among these options, the vertices we found [tex]\( (4, 3) \)[/tex] and [tex]\( (0, 7) \)[/tex] coincide with option (d). Therefore, the vertices of the feasible region are:
[tex]\[ \boxed{(0,1),(0,7),(4,3)} \][/tex]
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