Math 141 Exam 1 Review
1. a) Find the slope-y-intercept form of the line that passes through the point (-2,1) and
has x-intercept 3.
b) For the line above, what is the change in y whenever x decreases by 4?
c) What is the change in x whenever y increases by 3?
2. Find the slope-int. form of the line through the point (0,4) given that each increase of 2
in x produces a decrease of 5 in y.
3. Find the general form of the line parallel to 5 x  4 y  8 and passing through (1, 4).
4. In the production of a certain product, a company has fixed costs of $375 and each unit
costs an additional $30 to produce. No one will buy the product until the price is below
$70. For each decrease of $5 in the price 20 more units can be sold. Find the break even
quantities.
5. In a specific week, receipts at a theatre totaled $2882. Adult tickets were sold for $7
and child tickets were $5. The theatre sold three times as many child tickets as adult
tickets. How many adult and how many child tickets were sold that week?
6. Joe wants to invest in a stock fund earning 12%, a mutual fund earning 7% and a bond
fund earning 3%. Because the stock is riskier, he will put only 20% of the money into the
stock fund. His total earning should come to $3900. Give the complete solution set to
how much he should invest in each fund and also give two practical particular solutions.
7. Towit rental car company charges $25 per day plus gas. Their cars get 30 miles per
gallon of gas. Jumpstart rental charges $0.50 per mile plus gas and their cars get 25 miles
per gallon of gas. Gas is currently $2.70 per gallon. How many miles would you have to
drive in a day for these charges to be equal? Which company is cheaper if you drive 70
miles?
8. A manufacturer has fixed costs of $6000 per month and each unit costs him an
additional $6 to produce. If he sells each unit for $7.50, what are the cost, revenue and
profit functions? What is the break even point?
9. A heifer was purchased for $1050 in the year 2000. It depreciated linearly to a value of
0 in 2007. What was the value in the year 2004?
10. Demand for a certain commodity is 500 units when the price is $325 and is 1500 units
when the price is $275. Suppliers will provide 2000 units at a price of $400 per unit. For
each $10 increase in price, they will provide 100 more units.
a) Find the demand equation. What is the lowest price for which none are demanded?
b) Find the supply equation. What is the highest price at which suppliers will provide
none?
c) Find the equilibrium point.
11.. A pollution control office is monitoring two waste treatment plants. They measure
the amount of nitrates, lead and oxygen in the plant discharge in parts per million. The
measurements at the beginning of the monitoring period are given by matrix E. The
measurements at the end of the period are given by matrix F.
E=
 13

 20
plantA
plantB
17
22
30 

50 
F=
plantA
plantB
10

15
15
16
30 

45 
What matrix represents the
change in the amounts of pollutants in the discharge?
12. . The Blacks are each investing in two stocks. Matrix A shows the amounts each of
them had in stocks 1 and 2 at the beginning of the year. Matrix B shows the amounts each
had at the end of the year. If they owe 25% in taxes on the gains, what matrix shows their
after tax gains in each stock for each person?
A=
13. .
his 10 , 000

hers 15 , 000
2
A=  a

 0
1
5
6
 3

7

 5 
20 , 000 

12 , 000 
2
B=  4

 b
B=
 1

3

7 
his  11 , 000

hers 16 , 500
21 , 000 

12 , 600 
Find the entry in row 2 column 1 of AB.
14. For A and B as in problem 13, find
B
T
A
.
15. A police department employs two grades of personnel: rookies and sergeants. A
rookie spends 20 hours training and 20 hours on patrol each week. A sergeant spends 5
hours training and 30 hours on patrol each week. The training center can handle 240
person hours each week while the department needs 440 person hours each week for
patrol duty. How many persons at each grade should they have?
16. A chemistry department wants to make 2 liters of an 18% acid solution by mixing a
21% solution with a 14% solution. How many liters of each type of acid should they use?
17. Each augmented matrix represents a system of equations in x, y, and z. Find each
solution or state none exists. If the solution is infinite, give the parametric solution and
two particular solutions.
1

0
a) 
0

0
1

d) 0

 0
0
0
1
0
3 

 2

5 

0 
0
1
0
0
0
 1

3

0 
1
0
1
b) 
0
0
1
1
2
 4

 3
1

0
c) 
0

0
0
0
1
1
0
0
0
0
 3

 4

1 

0 
18. .Use rref in your calculator to solve each system of equations. If infinite, give the
parametric solution and two particular solutions.
a)
x  2 y  z  2
x  2 y  z  2
 2x  3y  z  1
b)
 2x  3y  z  1
2 x  3 y  4 z  12
c)
x  3y  2z  0
3x  2 y  z  5
7 x  4 y  7 z  29
x  y  z  6
d)
2x  3y  z  5
4 x  5 y  z  17
 2 x  y  3z  5
19. Each of the matrices is not fully row reduced. Perform the next pivot of the row
reduction. Show all work and row operations.
a)
1

0

 0
5
2
 3
 3
4
6
7 

8

 9 
b)
1

0

0

0
0
1
2
1
3
4
0
0
5
0
0
7
 1

2

 5

3 
c)
1

0

 0
0
2
1
 3
0
 4
5 

1

 8 
20. Perform the indicated operations for A, B, C.
a
1
2
0
3
 a
A= 
a) A + B b)
A
0

1
T
 2B
b
2
1
1
b
0
B= 
T
1

2
 1

2
C= 
 1

 0
2
1
3
1
0 

1

2 

 1
c) BC
21. A kitchen appliance manufacturer makes can openers and dough cutters using two
machines, a press and a riveter. Each can opener requires 0.2 minutes in the press and 0.4
minutes in the riveter. A dough cutter requires 0.5 minutes in the press and 0.3 minutes in
the riveter. The press can be operated only 3 hours per day and the riveter 2.5 hours per
day. If these two machines are fully used, how many can openers and dough cutters can
be produced each day?
22. Write a product of 3 matrices to answer the following: A company produces two nut
mixtures, regular and premium. The regular mix contains 60% peanuts, 25% cashews and
15% pecans. The premium mix contains 30% peanuts, 40% cashews and 30% pecans.
Peanuts cost them $.80 per pound, cashews cost $1.15 per pound and pecans cost $1.50
per pound. How much will it cost them to fill an order for 4 pounds of regular mix and 12
pounds of premium mix?
23. The table shows the grades of 4 students, A, B, C and D on each of 3 quizzes.
Q1 Q 2 Q 3



A
5
7
8


B
9
10
9 


C
8
9
7


 D
6
10
9 
Treat this as a matrix. what matrix would you multiply it by, and on which side to find:
a) the class average for each quiz.
b) the total quiz points for each student.
c) the quiz average for each student.
24. Which matrix operations can be performed for
 0
C=   1

 1
1

2

0 
1
A= 
4
3
0
 2

6 
1
B=  0

 3
2
1
1
 1

 2

0 
If not possible, why not, and if possible, what is the size of the resulting
matrix?
a) A+B b) BC c) AC
d) CA
e) AB+C T
25. True or False questions:
a) If a system of equations has more variables than equations then the solution is not
unique.
b) If a system of equations has more variables than equations then there are infinitely
many solutions.
c) If an augmented matrix in rref has a row of all 0’s then the system of equations has
infinitely many solutions.
d) If any row of an augmented has only 0’s on the left of the bar and a nonzero on the
right there is no solution.