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To graph the feasible region for the given system of inequalities, follow these steps methodically:
1. Convert the inequalities to their respective equations:
[tex]\( 3y + 4x = 24 \)[/tex]
[tex]\( 3y + 2x = 16 \)[/tex]
[tex]\( y + x = 7 \)[/tex]
[tex]\( y + x = 8 \)[/tex]
[tex]\( x = 0 \)[/tex]
[tex]\( y = 0 \)[/tex]
2. Find the intersection points of these lines to determine the corners of the feasible region:
- Solve for intersection points between pairs of lines.
3. Check each intersection point to see if it satisfies all the inequalities.
- If it does, it's a corner of the feasible region.
### Finding Intersection Points
1. Intersection of [tex]\( 3y + 4x = 24 \)[/tex] and [tex]\( 3y + 2x = 16 \)[/tex]:
[tex]\[ \begin{cases} 3y + 4x = 24 \\ 3y + 2x = 16 \end{cases} \][/tex]
Subtract the second equation from the first:
[tex]\[ (3y + 4x) - (3y + 2x) = 24 - 16 \][/tex]
[tex]\[ 2x = 8 \][/tex]
[tex]\[ x = 4 \][/tex]
Substituting [tex]\( x = 4 \)[/tex] into [tex]\( 3y + 2x = 16 \)[/tex]:
[tex]\[ 3y + 2(4) = 16 \][/tex]
[tex]\[ 3y + 8 = 16 \][/tex]
[tex]\[ 3y = 8 \][/tex]
[tex]\[ y = \frac{8}{3} \][/tex]
Intersection Point: [tex]\((4, \frac{8}{3})\)[/tex]
2. Intersection of [tex]\( y + x = 7 \)[/tex] and [tex]\( y + x = 8 \)[/tex]:
These lines are parallel and will never intersect. So, there is no intersection point.
3. Intersection of [tex]\( y + x = 7 \)[/tex] and [tex]\( 3y + 4x = 24 \)[/tex]:
Substitute [tex]\( y = 7 - x \)[/tex] into [tex]\( 3y + 4x = 24 \)[/tex]:
[tex]\[ 3(7 - x) + 4x = 24 \][/tex]
[tex]\[ 21 - 3x + 4x = 24 \][/tex]
[tex]\[ 21 + x = 24 \][/tex]
[tex]\[ x = 3 \][/tex]
Substitute [tex]\( x = 3 \)[/tex] into [tex]\( y + x = 7 \)[/tex]:
[tex]\[ y + 3 = 7 \][/tex]
[tex]\[ y = 4 \][/tex]
Intersection Point: [tex]\((3, 4)\)[/tex]
4. Intersection of [tex]\( y + x = 8 \)[/tex] and [tex]\( 3y + 4x = 24 \)[/tex]:
Substitute [tex]\( y = 8 - x \)[/tex] into [tex]\( 3y + 4x = 24 \)[/tex]:
[tex]\[ 3(8 - x) + 4x = 24 \][/tex]
[tex]\[ 24 - 3x + 4x = 24 \][/tex]
[tex]\[ x = 0 \][/tex]
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( y + x = 8 \)[/tex]:
[tex]\[ y = 8 \][/tex]
Intersection Point: [tex]\((0, 8)\)[/tex]
5. Intersection of [tex]\( y + x = 7 \)[/tex] and [tex]\( y + x = 8 \)[/tex]:
As noted earlier, these lines are parallel.
6. Intersection of [tex]\( y + x = 7 \)[/tex] and [tex]\( x = 0 \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( y + x = 7 \)[/tex]:
[tex]\[ y = 7 \][/tex]
Intersection Point: [tex]\((0, 7)\)[/tex]
7. Intersection of [tex]\( y + x = 8 \)[/tex] and [tex]\( x = 0 \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( y + x = 8 \)[/tex]:
[tex]\[ y = 8 \][/tex]
Intersection Point: [tex]\((0, 8)\)[/tex]
8. Intersection of [tex]\( y + x = 7 \)[/tex] and [tex]\( y = 0 \)[/tex]:
Substitute [tex]\( y = 0 \)[/tex] into [tex]\( y + x = 7 \)[/tex]:
[tex]\[ x = 7 \][/tex]
Intersection Point: [tex]\((7, 0)\)[/tex]
9. Intersection of [tex]\( y + x = 8 \)[/tex] and [tex]\( y = 0 \)[/tex]:
Substitute [tex]\( y = 0 \)[/tex] into [tex]\( y + x = 8 \)[/tex]:
[tex]\[ x = 8 \][/tex]
Intersection Point: [tex]\((8, 0)\)[/tex]
The key points to consider are:
- [tex]\((4, \frac{8}{3})\)[/tex]
- [tex]\((3, 4)\)[/tex]
- [tex]\((0, 8)\)[/tex]
- [tex]\((0, 7)\)[/tex]
- [tex]\((7, 0)\)[/tex]
- [tex]\((8, 0)\)[/tex]
### Feasibility Check for each Intersection:
- Point [tex]\((4, \frac{8}{3}) \Rightarrow x = 4, y = \frac{8}{3}\)[/tex]:
[tex]\[ y = \frac{8}{3}, x = 4 \][/tex]
All inequalities checked:
[tex]\[ \begin{cases} 3y + 4x = 24 \geq 24 & True \\ 3y + 2x = 16 \geq 16 & True \\ x + y = 7 & y = \frac{8}{3}, x = 4 = \frac{20}{3}, \rightarrow False \end{cases} \][/tex]
- Therefore, this point is not feasible.
### Feasible Vertices:
[tex]\[ (3, 4), (0, 8), (0, 7), (7,0) \][/tex]
Thus, the feasible region is a polygon with vertices at (3,4), (0,8), (0,7), (7,0).
To manually graph this region:
1. Draw the axes and plot the points.
2. Connect points to outline the polygon.
To determine the region partition:
1. Shade above lines [tex]\(3y + 4x = 24\)[/tex] and [tex]\(3y + 2x = 16\)[/tex]
2. Shade the intersection partition below [tex]\(ry= x-y =7,x+y \leq 8\)[/tex].
These vertex points outline where feasible region lies. {Manually graph this region from shaded regions and intersection polygons}.
1. Convert the inequalities to their respective equations:
[tex]\( 3y + 4x = 24 \)[/tex]
[tex]\( 3y + 2x = 16 \)[/tex]
[tex]\( y + x = 7 \)[/tex]
[tex]\( y + x = 8 \)[/tex]
[tex]\( x = 0 \)[/tex]
[tex]\( y = 0 \)[/tex]
2. Find the intersection points of these lines to determine the corners of the feasible region:
- Solve for intersection points between pairs of lines.
3. Check each intersection point to see if it satisfies all the inequalities.
- If it does, it's a corner of the feasible region.
### Finding Intersection Points
1. Intersection of [tex]\( 3y + 4x = 24 \)[/tex] and [tex]\( 3y + 2x = 16 \)[/tex]:
[tex]\[ \begin{cases} 3y + 4x = 24 \\ 3y + 2x = 16 \end{cases} \][/tex]
Subtract the second equation from the first:
[tex]\[ (3y + 4x) - (3y + 2x) = 24 - 16 \][/tex]
[tex]\[ 2x = 8 \][/tex]
[tex]\[ x = 4 \][/tex]
Substituting [tex]\( x = 4 \)[/tex] into [tex]\( 3y + 2x = 16 \)[/tex]:
[tex]\[ 3y + 2(4) = 16 \][/tex]
[tex]\[ 3y + 8 = 16 \][/tex]
[tex]\[ 3y = 8 \][/tex]
[tex]\[ y = \frac{8}{3} \][/tex]
Intersection Point: [tex]\((4, \frac{8}{3})\)[/tex]
2. Intersection of [tex]\( y + x = 7 \)[/tex] and [tex]\( y + x = 8 \)[/tex]:
These lines are parallel and will never intersect. So, there is no intersection point.
3. Intersection of [tex]\( y + x = 7 \)[/tex] and [tex]\( 3y + 4x = 24 \)[/tex]:
Substitute [tex]\( y = 7 - x \)[/tex] into [tex]\( 3y + 4x = 24 \)[/tex]:
[tex]\[ 3(7 - x) + 4x = 24 \][/tex]
[tex]\[ 21 - 3x + 4x = 24 \][/tex]
[tex]\[ 21 + x = 24 \][/tex]
[tex]\[ x = 3 \][/tex]
Substitute [tex]\( x = 3 \)[/tex] into [tex]\( y + x = 7 \)[/tex]:
[tex]\[ y + 3 = 7 \][/tex]
[tex]\[ y = 4 \][/tex]
Intersection Point: [tex]\((3, 4)\)[/tex]
4. Intersection of [tex]\( y + x = 8 \)[/tex] and [tex]\( 3y + 4x = 24 \)[/tex]:
Substitute [tex]\( y = 8 - x \)[/tex] into [tex]\( 3y + 4x = 24 \)[/tex]:
[tex]\[ 3(8 - x) + 4x = 24 \][/tex]
[tex]\[ 24 - 3x + 4x = 24 \][/tex]
[tex]\[ x = 0 \][/tex]
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( y + x = 8 \)[/tex]:
[tex]\[ y = 8 \][/tex]
Intersection Point: [tex]\((0, 8)\)[/tex]
5. Intersection of [tex]\( y + x = 7 \)[/tex] and [tex]\( y + x = 8 \)[/tex]:
As noted earlier, these lines are parallel.
6. Intersection of [tex]\( y + x = 7 \)[/tex] and [tex]\( x = 0 \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( y + x = 7 \)[/tex]:
[tex]\[ y = 7 \][/tex]
Intersection Point: [tex]\((0, 7)\)[/tex]
7. Intersection of [tex]\( y + x = 8 \)[/tex] and [tex]\( x = 0 \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( y + x = 8 \)[/tex]:
[tex]\[ y = 8 \][/tex]
Intersection Point: [tex]\((0, 8)\)[/tex]
8. Intersection of [tex]\( y + x = 7 \)[/tex] and [tex]\( y = 0 \)[/tex]:
Substitute [tex]\( y = 0 \)[/tex] into [tex]\( y + x = 7 \)[/tex]:
[tex]\[ x = 7 \][/tex]
Intersection Point: [tex]\((7, 0)\)[/tex]
9. Intersection of [tex]\( y + x = 8 \)[/tex] and [tex]\( y = 0 \)[/tex]:
Substitute [tex]\( y = 0 \)[/tex] into [tex]\( y + x = 8 \)[/tex]:
[tex]\[ x = 8 \][/tex]
Intersection Point: [tex]\((8, 0)\)[/tex]
The key points to consider are:
- [tex]\((4, \frac{8}{3})\)[/tex]
- [tex]\((3, 4)\)[/tex]
- [tex]\((0, 8)\)[/tex]
- [tex]\((0, 7)\)[/tex]
- [tex]\((7, 0)\)[/tex]
- [tex]\((8, 0)\)[/tex]
### Feasibility Check for each Intersection:
- Point [tex]\((4, \frac{8}{3}) \Rightarrow x = 4, y = \frac{8}{3}\)[/tex]:
[tex]\[ y = \frac{8}{3}, x = 4 \][/tex]
All inequalities checked:
[tex]\[ \begin{cases} 3y + 4x = 24 \geq 24 & True \\ 3y + 2x = 16 \geq 16 & True \\ x + y = 7 & y = \frac{8}{3}, x = 4 = \frac{20}{3}, \rightarrow False \end{cases} \][/tex]
- Therefore, this point is not feasible.
### Feasible Vertices:
[tex]\[ (3, 4), (0, 8), (0, 7), (7,0) \][/tex]
Thus, the feasible region is a polygon with vertices at (3,4), (0,8), (0,7), (7,0).
To manually graph this region:
1. Draw the axes and plot the points.
2. Connect points to outline the polygon.
To determine the region partition:
1. Shade above lines [tex]\(3y + 4x = 24\)[/tex] and [tex]\(3y + 2x = 16\)[/tex]
2. Shade the intersection partition below [tex]\(ry= x-y =7,x+y \leq 8\)[/tex].
These vertex points outline where feasible region lies. {Manually graph this region from shaded regions and intersection polygons}.
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