UNLV Algebra Exam Practice

10/4/21, 1:16 PMQuiz: Section 5.2
Section 5.2
Started: Sep 30 at 2:12am
Quiz Instructions
Question 1
5 pts
Consider the following intermediate tableau (not the Initial Simplex tableau):
⎡
P
x1
x2
s1
s2
s3
RH S
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
0
1
0
1
0
0
4
0
0
2
0
1
0
12
0
0
2
−3
0
1
6
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
1
0
−5
3
0
0
12
⎣
⎦
a) Determine the pivot column and the pivot element, and perform all the row
operations for the entire pivot column to obtain the next new tableau. Once you have
the new tableau, look at the numbers you have in the objective row and enter each
one as requested in each box below. Note: Where applicable, fractions must be
entered as 2/5, -1/3, and so on.
Under column
x1
in the objective row, you have:
Under column
x2
in the objective row, you have:
Under column
s1
in the objective row, you have:
Under column
s2
Under column
s3
Under column
RH S
in the objective row, you have:
in the objective row, you have:
in the objective row, you have:
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Quiz: Section 5.2
b) Given the new tableau that you obtained above, three interpretations are possible.
In the box below, type or copy and paste whichever answer shown in boldface letters
below that you think is the correct choice.
1. Enter solved if you think you have obtained the final tableau.
2. Enter ready for another set of pivot operations if you think more pivot
operations are possible.
3. Enter no solution if you think the tableau has no solution.
Question 2
5 pts
Consider the following tableau:
⎡
P
x1
x2
s1
s2
RH S
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
0
8
8
1
0
160
0
4
12
0
1
180
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎣
1
−5
−10
0
0
0
⎦
a) Determine the pivot column and the pivot element, and perform all the row
operations for the entire pivot column to obtain the next new tableau. Once you have
the new tableau, look at the numbers you have in the objective row and enter each
one as requested in each box below. Note: Where applicable, fractions must be
entered as 2/5, -1/3, and so on.
Under column
x1
in the objective row, you have:
Under column
x2
in the objective row, you have:
Under column
s1
in the objective row, you have:
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10/4/21, 1:16 PM
Quiz: Section 5.2
Under column
s2
in the objective row, you have:
Under column
RH S
in the objective row, you have:
b) Given the new tableau that you obtained above, three interpretations are possible.
In the box below, type or copy and paste whichever answer shown in boldface letters
below that you think is the correct choice.
1. Enter solved if you think you have obtained the final tableau.
2. Enter ready for another set of pivot operations if you think more pivot
operations are possible.
3. Enter no solution if you think the tableau has no solution.
Question 3
5 pts
Consider the following intermediate tableau (not the Initial Simplex tableau):
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P
x1
x2
0
0
1
0
1
0
1
0
0
s1
1
2
s2
−
RH S
⎤
1
3
2
2
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
1
1
5
2
2
2
2
−1
7
⎦
a) Determine the pivot column and the pivot element, and perform all the row
operations for the entire pivot column to obtain the next new tableau. Once you have
the new tableau, look at the numbers you have in the objective row and enter each
one as requested in each box below. Note: Where applicable, fractions must be
entered as 2/5, -1/3, and so on.
Under column
x1
in the objective row, you have:
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10/4/21, 1:16 PM
Quiz: Section 5.2
Under column
x2
in the objective row, you have:
Under column
s1
in the objective row, you have:
Under column
s2
Under column
RH S
in the objective row, you have:
in the objective row, you have:
b) Given the new tableau that you obtained above, three interpretations are possible.
In the box below, type or copy and paste whichever answer shown in boldface letters
below that you think is the correct choice.
1. Enter solved if you think you have obtained the final tableau.
2. Enter ready for another set of pivot operations if you think more pivot
operations are possible.
3. Enter no solution if you think the tableau has no solution.
Question 4
5 pts
Consider the following tableau:
⎡
P
x1
x2
s1
s2
RH S
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
0
1
0
1
0
7
0
1
−1
0
1
8
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎣
1
−5
−4
0
0
0
⎦
a) Determine the pivot column and the pivot element, and perform all the row
operations for the entire pivot column to obtain the next new tableau. Once you have
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Quiz: Section 5.2
the new tableau, look at the numbers you have in the objective row and enter each
one as requested in each box below. Note: Where applicable, fractions must be
entered as 2/5, -1/3, and so on.
Under column
x1
in the objective row, you have:
Under column
x2
in the objective row, you have:
Under column
s1
in the objective row, you have:
Under column
s2
Under column
RH S
in the objective row, you have:
in the objective row, you have:
b) Given the new tableau that you obtained above, three interpretations are possible.
In the box below, type or copy and paste whichever answer shown in boldface letters
below that you think is the correct choice.
1. Enter solved if you think you have obtained the final tableau.
2. Enter ready for another set of pivot operations if you think more pivot
operations are possible.
3. Enter no solution if you think the tableau has no solution.
Question 5
8 pts
Use the simplex method to solve the following maximum problem:
Maximize:
P = 7x 1 + 5x 2
Subject to the constraints:
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10/4/21, 1:16 PM
Quiz: Section 5.2
⎧ 2x 1 + x 2 ≤ 100
⎪
⎪
4x 1 + 3x 2 ≤ 240
⎨
x1 ≥ 0
⎪
⎩
⎪
x2 ≥ 0
and using your final tableau answer the questions below by entering the correct
answer in each blank box. Please enter fractions as 3/5, -4/7, and so on.
x1 =
x2 =
P =
Question 6
8 pts
Use the simplex method to solve the following maximum problem:
Maximize:
P = 7x 1 + 12x 2
Subject to the constraints:
⎧ 2x 1 + 3x 2 ≤ 6
⎪
⎪
3x 1 + 7x 2 ≤ 12
⎨
x1 ≥ 0
⎪
⎩
⎪
x2 ≥ 0
and using your final tableau answer the questions below by entering the correct
answer in each blank box. Please enter fractions as 3/5, -4/7, and so on.
x1 =
x2 =
P =
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Quiz: Section 5.2
Question 7
8 pts
Use the simplex method to solve the following maximum problem:
Maximize:
P = 3x 1 + 2x 2
Subject to the constraints:
⎧ 2x 1 + x 2 ≤ 6
⎪
⎪
x 1 + 2x 2 ≤ 6
⎨
x1 ≥ 0
⎪
⎩
⎪
x2 ≥ 0
and using your final tableau answer the questions below by entering the correct
answer in each blank box. Please enter fractions as 3/5, -4/7, and so on.
x1 =
x2 =
P =
Question 8
8 pts
Use the simplex method to solve the following maximum problem:
Maximize:
P = 6x 1 + 3x 2 + 2x 3
Subject to the constraints:
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Quiz: Section 5.2
2x 1 + 2x 2 + 3x 3 ≤ 30
⎧
⎪
⎪
⎪
⎪ 2x + 2x + x ≤ 12
⎪
1
2
3
⎨x ≥ 0
1
⎪
⎪
⎪ x2 ≥ 0
⎪
⎩
⎪
x3 ≥ 0
and using your final tableau answer the questions below by entering the correct
answer in each blank box.
x1 =
x2 =
x3 =
P =
Question 9
8 pts
Use the simplex method to solve the following maximum problem:
Maximize:
P = 2x 1 + 3x 2
Subject to the constraints:
⎧ x1
⎪
⎪
x1
⎨
x1
⎪
⎩
⎪
x2
+ 2x 2 ≤ 30
+ x 2 ≤ 20
≥ 0
≥ 0
and using your final tableau answer the questions below by entering the correct
answer in each blank box. Please enter fractions as 3/5, -4/7, and so on.
x1 =
x2 =
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Quiz: Section 5.2
P =
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Quiz: Section 5.3
Section 5.3
Started: Sep 30 at 2:14am
Quiz Instructions
Question 1
5 pts
Consider the following minimum problem:
Minimize:
C = 6x 1 + 3x 2
Subject to the constraints:
⎧ 4x 1 + x 2 ≥ 4
⎪
⎪
x2 ≥ 2
⎨
x1 ≥ 0
⎪
⎩
⎪
x2 ≥ 0
Write the dual problem for the above minimum problem by selecting the appropriate
number for each blank box shown below (Do not solve the dual problem).
P =
[ Select ]
y1
[ Select ]
y1 ≤
[ Select ]
y1 ≥ 0
y1
;
+
+
[ Select ]
y2
[ Select ]
[ Select ]
y2 ≤ 3
y2 ≥ 0
Question 2
5 pts
Consider the following minimum problem:
Minimize:
C = x 1 + 6x 2
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Quiz: Section 5.3
Subject to the constraints:
⎧ 2x 1 + 3x 2 ≥ 15
⎪
⎪
−x 1 + 2x 2 ≥ 3
⎨
x1 ≥ 0
⎪
⎩
⎪
x2 ≥ 0
Write the dual problem for the above minimum problem by selecting the appropriate
number for each blank box shown below (Do not solve the dual problem).
P =
[ Select ]
y1
[ Select ]
[ Select ]
y1 ≥ 0
;
+
[ Select ]
y1
+
[ Select ]
y1
+
[ Select ]
y2
y2 ≤ 1
y2 ≤ 6
y2 ≥ 0
Question 3
5 pts
Consider the following minimum problem:
Minimize:
C = 2x 1 + x 2
Subject to the constraints:
⎧ 5x 1 + x 2 ≥ 9
⎪
⎪
2x 1 + 2x 2 ≥ 10
⎨
x1 ≥ 0
⎪
⎩
⎪
x2 ≥ 0
Write the dual problem for the above minimum problem by selecting the appropriate
number for each blank box shown below (Do not solve the dual problem).
P =
[ Select ]
[ Select ]
y1
y1
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+
+
[ Select ]
[ Select ]
y2
y2 ≤ 2
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10/4/21, 1:17 PM
Quiz: Section 5.3
[ Select ]
y1 ≥ 0
y1
;
+
[ Select ]
y2 ≤ 1
y2 ≥ 0
Question 4
5 pts
Consider the following minimum problem:
Minimize:
C = 14x 1 + 20x 2
Subject to the constraints:
⎧ x 1 + 2x 2 ≥ 4
⎪
⎪
7x 1 + 6x 2 ≥ 20
⎨
x1 ≥ 0
⎪
⎩
⎪
x2 ≥ 0
Write the dual problem for the above minimum problem by selecting the appropriate
number for each blank box shown below (Do not solve the dual problem).
P =
[ Select ]
[ Select ]
[ Select ]
y1 ≥ 0
;
y1
+
[ Select ]
y1
+
[ Select ]
y1
+
[ Select ]
y2
y2 ≤ 14
y2 ≤ 20
y2 ≥ 0
Question 5
8 pts
Use the simplex method and the Duality Principle to solve the following minimum
problem:
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Quiz: Section 5.3
Minimize:
C = 14x 1 + 20x 2 + 24x 3
Subject to the constraints:
⎧ x 1 + x 2 + 2x 3 ≥ 7
⎪
⎪
⎪
⎪ x + 2x + x ≥ 4
⎪
1
2
3
⎨x ≥ 0
1
⎪
⎪
⎪ x2 ≥ 0
⎪
⎩
⎪
x3 ≥ 0
and using your final tableau answer the questions below by entering the correct
answer in each blank box. Please enter fractions as 3/5, -4/7, and so on.
x1 =
x2 =
x3 =
C =
Question 6
8 pts
Use the simplex method and the Duality Principle to solve the following minimum
problem:
Minimize:
C = 9x 1 + 4x 2 + 10x 3
Subject to the constraints:
2x 1 + x 2 + 3x 3 ≥ 6
⎧
⎪
⎪
⎪
⎪ 6x + x + x ≥ 9
⎪
1
2
3
⎨ x1 ≥ 0
⎪
⎪
⎪ x2 ≥ 0
⎪
⎩
⎪
x3 ≥ 0
and using your final tableau answer the questions below by entering the correct
answer in each blank box. Please enter fractions as 3/5, -4/7, and so on.
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Quiz: Section 5.3
x1 =
x2 =
x3 =
C =
Question 7
8 pts
Use the simplex method to solve the following minimum problem on your own paper.
Then, using your final tableau, enter the answer in each relevant box provided below.
M inimize
:
C = 3x 1 + 4x 2
Subject to the following constraints:
2x 1 + x 2 ≥ 2
2x 1 + x 2 ≥ 6
x1 ≥ 0
;
x2 ≥ 0
Minimum value of
Value of
x1 =
Value of
x2 =
C =
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Quiz: Section 5.3
Question 8
8 pts
Use the simplex method and the Duality Principle to solve the following minimum
problem:
Minimize:
C = 3x 1 + 3x 2
Subject to the constraints:
⎧ 2x 1 + x 2 ≥ 4
⎪
⎪
x 1 + 2x 2 ≥ 4
⎨
x1 ≥ 0
⎪
⎩
⎪
x2 ≥ 0
and using your final tableau answer the questions below by entering the correct
answer in each blank box. Please enter fractions as 3/5, -4/7, and so on.
x1 =
x2 =
C =
Question 9
8 pts
Use the simplex method and the Duality Principle to solve the following minimum
problem:
Minimize:
C = 2x 1 + 2x 2
Subject to the constraints:
⎧ x 1 + 2x 2 ≥ 3
⎪
⎪
3x 1 + 2x 2 ≥ 5
⎨
x1 ≥ 0
⎪
⎩
⎪
x2 ≥ 0
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Quiz: Section 5.3
and using your final tableau answer the questions below by entering the correct
answer in each blank box. Please enter fractions as 3/5, -4/7, and so on.
x1 =
x2 =
C =
Question 10
5 pts
The following intermediate tableau for a dual maximum problem was obtained using
the simplex method for optimizing a minimum problem.
⎡
P
y1
y2
s1
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
0
0
2
1
0
1
1
0
⎣
1
0
−2
0
s2
−
RH S
⎤
1
1
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
3
⎦
1
2
1
2
3
2
Find the pivot element(s) in the above tableau, and perform all the necessary pivot
operations to obtain the final tableau. Then, using the final tableau, answer the
following questions:
The minimum function value is:
The value of
x1
in the minimum problem is :
The value of
x2
in the minimum problem is:
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Quiz: Section 5.3
Question 11
5 pts
The following intermediate tableau for a dual maximum problem was obtained using
the simplex method for optimizing a minimum problem.
⎡
P
y1
y2
s1
s2
RH S
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
0
1
−1
1
0
1
0
0
1
⎥
⎥
⎥
⎥
⎥
⎥
⎥
1
0
−5
⎣
−
1
1
3
3
3
0
1
3
⎦
Find the pivot element(s) in the above tableau, and perform all the necessary pivot
operations to obtain the final tableau. Then, using the final tableau, answer the
following questions:
Note: Where applicable, enter fractions in simplified form as 3/5 , 1/2, and so on.
The minimum function value is:
The value of
x1
in the minimum problem is :
The value of
x2
in the minimum problem is:
Question 12
5 pts
The following intermediate tableau for a dual maximum problem was obtained using
the simplex method for optimizing a minimum problem.
⎡
P
y1
y2
s1
s2
RH S
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
0
−2
1
1
0
2
0
9
0
−3
1
0
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎣
1
−18
0
9
0
18
⎦
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Quiz: Section 5.3
Find the pivot element(s) in the above tableau, and perform all the necessary pivot
operations to obtain the final tableau. Then, using the final tableau, answer the
following questions:
The minimum function value is:
The value of
x1
in the minimum problem is :
The value of
x2
in the minimum problem is:
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