Finite Math Practice Exam
Aubuchon
1. Solve the system of equations:
x  2y  1
3 x  2 y  11
Solution: (3, 1)
2. Solve the system of equations:
x  3y  0
4x  12 y  4
Solution: There is no solution to this system.
3. Determine the value of k for which the system of the linear equations
3x  4 y  12
has infinitely many solutions. Then find all solutions

 x  ky  4
corresponding to this value of k.
Solution: For there to be infinitely many solutions, both lines represented by the
equations must be the same. This can only happen if k  4/ 3 . In this case the
solution can be represented as
3
y   x  3,
4
x is free
4. Four large cheeseburgers and two chocolate shakes cost a total of $7.90. Two
shakes cost $0.15 more than one cheeseburger. What is the cost of a
cheeseburger? A shake?
Solution: Let x = the cost of a cheeseburger, and y = the cost of a shake, both
measured in $. Then we have the following system to solve:
4x  2 y 
7.90
x  2 y   .15
The solution is: A cheeseburger costs $1.55 and a shake costs $.85.
5. Solve the following system of equations using the Gauss-Jordan elimination
method.
x  2y  z  7
 x  y  2 z  8
2 x  5 y  11
Solution: The solution is (3,1, 2) .
6. Tracy has $20,000 to invest. As her financial planner, you recommend that she
diversify into three investments: Treasury bills that yield 5% simple interest,
Treasury bonds that yield 7% simple interest, and corporate bonds that yield 10%
simple interest. Tracy wishes to earn $1390 per year in income. She also wants
her investment in Treasury bills to be $3000 more than her investment in
corporate bonds. How much money should she place in each investment?
Solution: Let x = the amount invested in Treasury bills, y = the amount invested
in Treasury bonds, and z = the amount invested in corporate bonds. Then we have
the following system to solve:
x 
y 
z  20000
.05 x  .07 y  .10 z  1390
x

z  3000
The solution is: (8000, 7000,5000)
So $8000 should be invested in Treasury bills, $7000 invested in Treasury bonds,
and $5000 in corporate bonds.
7. Solve the following system of equations using the Gauss-Jordan elimination
method.
x  2y  z  7
2x  5 y  z  3
Solution: The original augmented matrix reduces to
 1 0 3 29 
0 1 1 11


This corresponds to the following solution:
x  29  3z
y  11  z
z is free
8. Given the following augmented matrix for a system of linear equations, find the
solution to the system if it exists.
1 0 0

0 1 3
0 0
1
1

4 
2 
Solution: Finishing up the row-reduction yields the matrix
 1 0 0 1
0 1 0 2 


0 0 1 2 
So the solution is (1, 2, 2) .
9. The management of Hartman Rent-A-Car has allocated $1080,000 to purchase 60
new automobiles to add to their existing fleet of rental cars. The company will
choose from compact (x), mid-sized (y), and full-sized (z) cars costing $10,000,
$16,000, and $22,000 each, respectively. Find formulas giving the options
available to the company. Then find 3 specific options for the company.
Solution: Let x = the number of compact cars purchased, y = the number of midsized cars purchased, and z = the number of full-sized cars purchased. Then we
have the following system to solve:
x 
y 
z 
60
10000 x  16000 y  22000 z  1080000
Row-reducing the augmented matrix from this system yields
 1 0 1 20 
0 1 2 80 


The corresponding system can be written as the desired formulas:
x  20 
z
y 
80  2 z
Here are three specific options: (0, 40, 20),
z is free (sort of)
(10, 20,30),
 4 3  4 w
10. Perform the indicated operation(s):  6 4    x y 

 

 10 0   z 0 
w  3
 8

Solution:  x  6 y  4 
 z  10
0 
(20, 0, 40)
0
x
x  2 0 3  5



11. Given 3 2 y  3 1  x 12 , find ( x, y, z ) .

 

 0
8 z   0
8
4 
Solution: x = 3, y = 12, z = 4.
 1 5
 4 5 1
 4 4  , C   3 4 4 , and D   2 3 .
12. Let A  
,
B


0 3 2
 4 3


 4 1 5




 5 2 
Find 2A  C , D 2 , and AB .
 11 6 2 2  8 15
 11 42
Solution: 2 A  C  
, D 
, AB  



 8 5 8
 20 3
 25 14
13. Tom and Kathy’s stock holdings are given by the matrix
BAC GM IBM TRW
A   200
 100

300 100 200  T . At the close of trading on a certain day, the
200 400 0  K
BAC 50 
GM  41
prices (in dollars per share) of the stocks are given by the matrix B 
.
IBM 90 
 
TRW 82 
Find the matrix AB, and state what it represents.
 47700
Solution: The matrix AB  
 , and it represents the total value of Tom and
 49200
Kathy’s stock holdings.
14. Solve the system of linear equations by using the inverse of the coefficient matrix.
 2z  2
x

3

15.  x  2 y  3z  
2

 2
 x  y
 3 2
Solution: The inverse of the coefficient matrix is A   3 2

 1 1
2


Multiplying this inverse by the right hand side matrix B  3 / 2 



2 
1
 1
solution X  A B   1 . Hence the solution is (1, 1, 0.5) .
 
1/ 2 
1
4 
5 .
2 
yields the