Finite Math 
Test #1

Oct. 10, 2000

F Track 
R. Hammack


Name: ________________________ 
Score: _________

(1) Perform the indicated matrix operation, or explain why it cannot be done.
(a)  [ 
5

1

][ 
1

4

] 
=


2

3

2

0

[ 
5(1) + 1(2)

5(4) + 1(0)

] 
=

[ 
7

20

] 
2(1) + 3(2)

2(4) + 3(0)

8

8



(2) Sketch the solutions of the following system of inequalities.
2x_{1}

+

x_{2}

≤

8 
x_{1}

+

x_{2}

≤

5 
x_{1}

+

2x_{2}

≤

8 
x_{1}

≥

0 
x_{2}

≥

0 
The solution is sketched on the right. The corner points are found to be (0, 0), (0, 4), (0,4), (2, 3) and (3,2). 
(3)

Maximize subject to ...

P = 10x_{1} + x_{2}

You may use any method. (However, notice that you sketched the feasible region in the previous problem. Feel free to use that information here.)
Using the corner points from the previous problem, we create a table:

From looking at the table, it becomes clear that the maximum value of P is 40, and this occurs when x_{1} = 40 and x_{2} = 0. 
(4) Use GaussJordan elimination to solve the following system of equations:
2x_{1}

+

4x_{2}

+

2x_{4}

=

6  
x_{1}

+

2x_{2}

+

x_{3}

+

2x_{4}

=

4 

R_{1} <> R_{2} 

2R_{1} + R_{2} > R_{2}  

1/2R_{2} > R_{2} 

R_{2} + R_{1} > R_{1} 

(Reduced) 
This corresponds to the system 

...or ...


Thus, the solution are
x_{1}

=

3  2m n 
x_{2}

=

m 
x_{3}

=

1  n 
x_{4}

=

n 
(5) Use the simplex method to solve the following problem.
A small publishing company is considering whether to publish 3 books. Let's call them book A, book B and book C. Two machines are needed to print the books, a printer and a binder. First a book is printed on the printer, then it is fed into the binder. Each copy of book A takes 6 minutes on the printer, then 4 minutes on the binder. Each copy of book B takes 4 minutes on the printer, then 2 minutes on the binder. Each copy of book C takes 2 minutes on the printer, then 1 minute on the binder. The printer is available for 1000 minutes per week, and the binder is available for 800 minutes per week. Each copy of book A will bring a profit of $2, each copy of book B will bring a profit of $1, and each copy of book C will bring a profit of $3. How many copies of each book shuld be made to realize a maximum profit?
First, let's organize all the data into a table.
A

B

C

Maximum


Printer 
6

4

2

1000 minutes

Binder 
4

2

1

800 minutes

Profit 
$2

$1

$3

Let x = copies of A. Let y = copies of B. Let z = copies of C. 

There are two problem constrains, so there will be two slack variables. The problem is transformed into the following system of equations.
6x

+ 4y

+ 2z

+ s_{1}

=

1000


4x

+ 2y

+ z

+ s_{2}

=

800










2x

y

3z

+ P

=

0

x

y

z

s_{1}

s_{2}

P


s_{1} 
6

4

2

1

0

0



1000

1000/2 = 500  
s_{2} 
4

2

1

0

1

0



800

800/1 = 800  









P 
2

1

3

0

0

1



0

Now it's time to do a pivot operation. We select the pivot column (3rd column) and the pivot row (1st row). Thus, z will enter and s_{1} will exit. The pivot element is 2, so first we need to change it to a 1 by doing 1/2R_{1} > R_{1}
3

2

1

1/2

0

0



500


4

2

1

0

1

0



800











2

1

3

0

0

1



0

Now the pivot entry is 1. To finish the pivot operation, we need to get 0's
elsewhere in the pivot coulum.
Thus we do R_{1} + R_{2} > R_{2} and
3R_{1} + R_{3 }> R_{3.}
x

y

z

s_{1}

s_{2}

P


z 
3

2

1

1/2

0

0



500


s_{2} 
1

0

0

1/2

1

0



800











P 
7

5

0

3/2

0

1



1500

There are no more negative indicators, so we can just read off the solution. A maximum profit of $1500 will be made by making z=500 copies of book C, 0 copies of book A and 0 copies of book B. That's the best that can be done given the constraints.