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Math 141 Review for Exam 1
Sections 1.1-1.5, 2.1-2.6, 3.1-3.2
9/19/05
1. Jim’s Toy Trucks sells trucks for $7 each. The cost to manufacture a truck is $5.20 after
the initial setup of $2000. Write the Cost, Revenue, and Profit functions. How many
trucks must be sold to break even? Give the cost, revenue, and profit at the break-even
point.
C(x) = 2000 + 5.2x
R(x) = 7x
P(x) = 7x – (2000 + 5.2x) = 1.8x - 2000
Break-even: 1.8x – 2000 = 0  x  1111 (round to nearest unit)
C(1111) = 7777.20
R(1111) = 7777
P(1111) = 0
2. The price for a certain product is $5 per unit when 14,000 units are produced. At a price
of $4, an additional 3,000 units will be sold. The manufacturer will not make the product
to sell for less than $3.50 per unit. For each $1.25 more, he will make 1000 more units.
Write the supply and demand equations. How many units are manufactured at the
equilibrium point?
Supply: (14000, 5) and (17000, 4)
y – 5 = (-1/3000)(x – 14000)
Demand: (0, 3.5) and (1000, 4.75)
y – 3.5 = (1/800)(x – 0) 
Solving simultaneously:

m = -1/3000
y = (-1/3000)x + 29/3
m = 125/100000 = 1/800
y = (1/800)x + 3.5
(-1/3000)x + 29/3 = (1/800)x + 3.5
37/6 = (19/12000)x
x  3895
3. The Lemon Growers Cooperative records the price per bin of lemons and how many
hundreds of bins were shipped each week. Write the equation of the Least Squares Line.
Predict the price of a bin of lemons when 70 bins are shipped.
Bins
Shipped
(hundreds)
Price
($ per bin)
47
25
61
32
38
52
40
209
165
237
179
190
219
196
y = 2.00x + 114.91
When x = 70, y = 2.00(70) + 114.91 = $255.06
4. Set up the following problem using a system of linear equations:
One group of customers bought 8 deluxe hamburgers, 6 orders of large fries, and 6 large
colas for $26.10. A second group ordered 10 deluxe hamburgers, 6 large fries, and 8
large colas and paid $31.60. A third group purchased 3 deluxe hamburgers, 2 large fries,
and 4 colas for $10.95. Determine the price of each food item.
X = price of hamburger
Y = price of fries
Z = price of cola
8x + 6y + 6z = 26.10
10x + 6y + 8z = 31.60
3x + 2y + 4z = 10.95
5. Solve the system in #4 using augmented matrices and Gauss-Jordan Elimination. Show
the row operations.
 8 6 6 26.1 
4 3 3 13.05
10 6 8 31.6  1
/ 2 R1
1/ 2 R2
1 R 3
  5 3 4 15.8  R



 3 2 4 10.95
3 2 4 10.95
1 1  1 2.1 
1
5 3 4 15.8  R
2 5 R1
R 33 R1
  0


3 2 4 10.95
0
1
2.1 
1 1
0 1  4.5  2.65 R
1 R 2
3 R 2
 R



0  1
7
4.65 
1  1 2.1 
1/ 2 R2
 2 9 5.3  

 1 7 4.65
4.75 
1 0 3.5
0 1  4.5  2.65 2
/ 5 R3



0 0 2.5
2 
4.75 
1 0 3.5
1 0 0 1.95
0 1  4.5  2.65 R
17 / 2 R 3
R 29 / 2 R 3
   0 1 0 .95 


0 0
0 0 1 .8 
1
.8 
6. Write a matrix equation for the system in #4. Solve using inverse matrices.
 8 6 6  X   26.1 
10 6 8  Y    31.6 

  

 3 2 4  Z  10.95
1

 X   8 6 6  26.1 
 Y   10 6 8   31.6 
  
 

 Z   3 2 4 10.95
Use the given matrices for problems #7-10.
 2  2 4
A  1 3 1
1 0 2
 2  4 5 
B
1  2
3
 2  3 5
C    1 1 2
 3
0 4
 X  1.95
  Y    .95 
  

 Z   .8 
3
0 
E 
2
 
1 
5
4
1
2

D   6  2 3  3
 5 0  1  4
7. Find
a) BA
b) BC
c) AC
d) DC
e) CD
 10  8 22
AC =  2 0 15 


 4  3 13 
DC cannot be done because the # of columns of D (4) must equal the # of rows of C (3)
  3  8  2
BA = 

 5 3 9 
2 2
 23
BC = 

 13  8 9
 47  4  22  13
CD =   6  7  3  12


  14 15
8
 13
8. Find the inverse of each matrix above that has an inverse. How do you know it is the
inverse?
2  7
 3
1 0 0


1
1
A    .5 0
1
A  A  I  0 1 0
 1.5  1 4 
0 0 1
B, D, and E don’t have inverses because they are not square matrices.
9. Find 2A – 3CD
4  4 8 
2A = 2 6 2


2 0 4
 141  12  66  39
3CD =   18  21  9  36


  42 45
24  39
2A–3CD does not exist because matrices must be of the same size to add or subtract
them.
10. Find the transpose of A, B, and E.
 2 1 1
 2 3 


T
T
A   2 3 0
B   4 1 
 4 1 2
 5  2
E T  3 0 2 1
Graph the solution to the following system:
x≥0
3x + 4y  6
2x – y  0
11. Write a system of linear inequalities for the following problem.
Pete’s Nuts has 75 pounds of cashews and 120 pounds of peanuts. These are to be mixed
in 1 pound packages as follows: a low-grade mixture that contained 4 ounces of cashews
and 12 ounces of peanuts and a high-grade mixture that contains 8 ounces of cashews and
8 ounces of peanuts. How many packages of each mixture should be made?
X = pkg of low-grade
Y = pkg of high-grade
4x + 8y  75(16) 
12x + 8y 120(16) 
4x + 8y  1200
12x + 8y 1920
x  0, y  0
Be sure to review the following:
y
x
2. Writing the equation of a line using the point-slope formula: y  y1  m( x  x1 )
3. Relationship between slopes of parallel and perpendicular lines.
4. Writing Cost, Revenue, and Profit equations and using them to find values and breakeven points: Profit = Revenue – Cost.
5. Writing Supply and Demand equations and using them to find values and equilibrium
points.
6. Solving problems involving linear depreciation.
7. Finding the Least Squares Line (linear regression on the calculator) and using it to make
predictions.
8. Setting up a system of linear equation from a word problem.
9. Finding solutions for systems of equations using substitution, elimination, Gauss-Jordan
(including GJ program and rref on calculator).
10. Adding, Subtracting, and Multiplying Matrices by hand and on the calculator.
11. Identity matrix and its properties, finding the transpose and inverse of matrices.
12. Determine whether a matrix is in row reduced form and how to get to row reduced form.
13. Graphing systems of linear inequalities.
14. Setting up a linear programming problem.
1. Meaning of slope (rate of change) and how to find the slope: m 