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Chapter 5: Systems of Equations and Inequalities
1. Determine which ordered pair is a solution of the system.
 x + 5y = 5

 –8 x – 5 y = 30
A) (2, –5)
B) (5, 2)
C) (–3, 2)
D) (–5, 2)
E) (–2, –3)
Ans: D

9x + 5 y = 5

2 x + 6 y = –1
A)  35 1 
 , 
 44 44 
B)  25 1 
 , 
 44 44 
C) (–9, 86)
D)  35 19 
 ,– 
 44 44 
E) (–9, 17)
Ans: D

 –2 x – y 2 = 4

 –3 x + y = 10
A) (–4, –2)
B) (–8, 9)
C) (–8, –1)
D) (–2, 4)
E) (–4, 2)
Ans: A
Larson/Hodgkins, College Algebra with Applications for Business and the Life Sciences, 2e Page 227
4. Solve each system of equations by the substitution method.
4 x – 3 y = 14
x – 5 y = 18
A)  86 124 
 ,– 
 17 17 
B)  58 16 
– , 
 17 17 
C)  16 58 
 ,– 
 17 17 
D)  124 86 
– , 
 23 23 
E) no solution
Ans: C
5. Solve the system by the method of substitution.

 2 1
 – 5 x – 5 y = 1

 2 x – 9 y =1
 5 5
A) (–2, 1)
B) (–1, –2)
C) (1, –2)
D) (–2, –1)
E) (2, 1)
Ans: D

 x − y = –1
 2
 x − y =
1
A) (2, 1), (–1, –2)
B) (2, 3), (–1, 0)
C) (2, –1), (–1, –4)
D) (2, –1)
E) no real solution
Ans: B
7. Solve the system of equations below.
 y = x2 + x – 5

 y = 2 x +1
A) ( 3, 7 ) and ( 4,9 )
B) ( –2, –3) and ( 3, 7 )
C) ( –2, –3) and ( 2,5)
D) ( 2,5) and ( 3, 7 )
E) ( 2,5) and ( 4,9 )
Ans: B

 x 2 + y 2 = 169

5 x – 12 y = 0
A) (12, 5), (–12, –5)
B) (12, 5)
C) (12, 5), (–12, 5), (–12, –5), (12, –5)
D) (5, 12), (5, –12)
E) no real solution
Ans: A
9. Solve the system below by method of substitution, if possible.
 x2 + y 2 =
16

 x– y=6
A) no solution
B) ( 3, –3)
C) ( –3, –9 )
D) ( 3, –3) , ( –3, –9 )
E) ( –2, –8)
Ans: A

 y = x3 + 8 x 2 + 3
 2
 y = x – 12 x + 3
A) (–3, 48), (4, –29), (1, –8)
B) (–3, 48), (–1, 16)
C) (–4, 67), (1, –8)
D) (–3, 48), (–4, 67), (0, 3)
E) no real solution
Ans: D
11. Solve the system graphically.
A)
B)
C)
D)
E)
Ans: C
12. Determine whether the system of equations below has one solution, two solutions, or no
solution.
 y = x2 + 5x + 5

 y = − x + 3x – 9
2
A) two solutions
B) no solution
C) one solution
Ans: B
13. Solve the system of equations below by the substitution method.
–5 x + 2 y = –3
10 x – 4 y = 13
A)  7 7
– ,– 
 20 8 
B)  7 7 
– ,– 
 8 20 
C)  19 19 
– , 
 8 20 
D)  19 19 
 ,– 
8 20 
E) no solution
Ans: E
14. Use a graphing utility to find the point(s) of intersection of the graphs.
 y = 3e5 x +5

15 x − y +18 = 0
A) ( –3, –1)
B) ( –3,1)
C) ( –1,3)
D) ( –1, –3)
E) ( 3,1)
Ans: C
15. Find the sales necessary to break even (R – C = 0) for the cost C of producing x units
and the revenue R obtained by selling x units. (Round to the nearest whole unit.)
= C 7.9= x + 6000 R 8.7 x
A) 666 units
B) 666 units or 714 units
C) 714 units
D) 698 units
E) no real solution
Ans: C
16. You invest $3700 in a fishing lure business. A lure costs $1.60 to produce and will be
sold for $6.20. How many lures must you sell to break even?
A) 1716 lures
B) 805 lures
C) 2313 lures
D) 475 lures
E) 597 lures
Ans: B
17. The sales of various types of lawn and garden items vary according to the season. At a
certain home improvement store, the monthly sales H of garden hoses (hoses sold per
month) declines from July to October whereas the monthly sales of lawn rakes R
(rakes sold per month) increase during this same interval. The sales of these two items
during the calendar months July-October are modeled by the equations:
H(t) = 64 – 6t
R(t) = 17t – 97,
where t is the month (t = 7 corresponds to July). In which month does the number of
rakes sold equal the number of hoses sold?
A) August
B) September
C) October
D) November
E) July
Ans: E
18. A total of $50,000 is invested in two funds paying 6.5% and 7.5% simple interest. The
total yearly interest is $3600. How much is invested at the 6.5% rate?
A) $28,000
B) $12,000
C) $26,000
D) $15,000
E) $20,000
Ans: D
19. You are offered two different jobs. Company A offers an annual salary of $34,000 plus a
year-end bonus of 3.5% of your total sales. Company B offers a salary of $28,000 plus a
year-end bonus of 7.5% of your total sales. What is the amount you must sell in one
year to earn the same salary working for either company?
A) $170,000
B) $120,000
C) $150,000
D) $90,000
E) $130,000
Ans: C
20. Solve the system by the method of elimination.
 3 x + 8 y = 69

 –2 x − y = –20
A) ( –1, 22 )
B) ( 7, 6 )
C)  229 368 
 , 
 5 5 
D)  53 
 2, 
 3
E) inconsistent
Ans: B
21. Solve the system by elimination.

 x – 2 y = –10

 –9 x + 9 y = –4
A)  7
 –3, 
 2
B)  23 
 3, 
 9 
C)  98 94 
 , 
 9 9 
D)  10 + a 
 a,  (dependent)
 2 
E) inconsistent
Ans: C
22. Solve each system of equations by the elimination method.
3x – 3 y = 6
x – 6 y = –1
A)  3 13 
 , 
5 5 
B)  1 11 
 ,– 
7 7 
C)  3 13 
 , 
7 7 
D)  13 3 
– ,– 
 7 7
E) no solution
Ans: A
23. Solve the system below by elimination if possible. Then state whether the system below
is consistent or inconsistent.
 2b + m = 3

4b + 2m = 2
A) The system is consistent and its solution is ( 3, 6 ) .
B) The system is consistent and its solution is ( –3, 6 ) .
C) The system is consistent and its solution is ( –6, –3) .
D) The system is consistent and its solution is ( 6, –3) .
E) The system is inconsistent and no solution exists.
Ans: E

7 1 2
 x+ y = –
6 6 3
 7 x + y =–4
A) ( 2, –18)
B) ( 4, –32 )
C) ( 6, –46 )
D) ( a, –4 − 7a ) (dependent)
E) inconsistent
Ans: D

0.08 x + 0.07 y = 0.2

0.04 x + 0.05 y = 0.17
A)  12 
 4, – 
 7
B)  13 
1, 
 5
C)  19 14 
– , 
 12 3 
D)  20 – 8a 
 a,  (dependent)
 7 
E) inconsistent
Ans: C
26. Use the statements below to write a system of equations. Solve the system by
elimination.
The sum of twice a number r and a number s is –14. The difference of r and s is
2.
A) = =
r –4, s 6
B) = r 4,=s –6
C) = =
r –6, s –4
D) = =
r –4, s –6
E) = r 4,=s 6
Ans: D
27. An airplane flying into a headwind travels 280 miles in 2 hours and 48 minutes. On the
return flight, the distance is traveled in 2 hours. Find the airspeed of the plane and the
speed of the wind, assuming that both remain constant.
A) plane speed = 135 mph; wind speed = 27 mph
B) plane speed = 135 mph; wind speed = 20 mph
C) plane speed = 97 mph; wind speed = 9 mph
D) plane speed = 97 mph; wind speed = 20 mph
E) plane speed = 120 mph; wind speed = 20 mph
Ans: E
28. One acetic acid solution is 60% water and another is 40% water. How many liters of
each solution should be mixed to produce 20 liters of a solution that is 49% water?
A) 5 liters of the 60% solution and 15 liters of the 40% solution
B) 9 liters of the 60% solution and 11 liters of the 40% solution
C) 15 liters of the 60% solution and 5 liters of the 40% solution
D) 6 liters of the 60% solution and 14 liters of the 40% solution
E) 14 liters of the 60% solution and 6 liters of the 40% solution
Ans: B
29. A total of $28,000 is invested in two corporate bonds that pay 11% and 4% simple
interest. The total annual interest is $2170. How much is invested in the 4% bond?
A) $17,000
B) $15,000
C) $13,000
D) $18,000
E) $16,000
Ans: C
30. Find the equilibrium point of the demand and supply equations. (The equilibrium point
is the price p and number of units x that satisfy both the demand and supply equations.)
Demand Supply
p = 36 – 0.02x p = 0.5x – 380
A) (800, 20 )
B) ( 200,32 )
C) ( 761, 0.5)
D) (100,34 )
E) inconsistent
Ans: A
31. The supply and demand equations for a small LCD television are given by
 p + 0.55 x = 1484 Demand


 p – 0.25 x = 300 Supply
where p is the price (in dollars) and x represents the number of televisions. For how
many units will the quantity demanded equal the quantity supplied? What price
corresponds to this value?
A) 1855 units at $463.75
B) 2230 units at $257.50
C) 1480 units at $257.50
D) 2230 units at $670.00
E) 1480 units at $670.00
Ans: E
32. Find the least squares regression line y = ax + b for the points
( x1, y1 ) , ( x2 , y2 ) , , ( xn , yn )
by solving the system for a and b.
 n   n 
nb +  ∑ xi  a =  ∑ yi 
   
=  i 1=  i 1 
 n   n 2  n 
 ∑ xi  b +  ∑ xi  a =  ∑ xi yi 
     
=  i 1=  i 1 =  i 1 
Points: (1, –1) , ( 3, –7 ) , ( 6, –15 ) , ( 7, –19 )

space
A) y = 1.92x – 2.92
B) y = –0.15x + 1.92
C) y = –2.53x + 1.63
D) y = –2.92x + 1.92
E) y = 1.19x + 1.63
Ans: D
33. The concentration C (in parts per million) of carbon dioxide in the atmosphere is
measured at the Mauna Loa Observatory in Hawaii. The greatest monthly carbon
dioxide concentration for each year from 2002 to 2006 is shown in the table.
Year t Concentration, C
2002 0 375.55
2003 1 378.35
2004 2 380.63
2005 3 382.26
2006 4 384.92
Solve the following system for a and b to find the least squares regression line
y= at + b for the data. Let t represent the year, with t = 0 corresponding to 2002.
 5b + 10a =
1901.71

10b + 30a =
3826.07
Use the least square regression line to predict the largest monthly carbon dioxide
concentration in 2014. Round your answer to the nearest hundredths part per million.
A) 396.20 parts per million
B) 400.73 parts per million
C) 389.40 parts per million
D) 398.46 parts per million
E) 402.99 parts per million
Ans: E
34. Determine which one of the ordered triples below is a solution of the given system of
equations.
8 x – 2 y – 6 z = –56

7 x + 3 y – z = –33
5 x + 6 y + 4 z = –3

A) (7, 5, 17)
B) (–3, –2, 6)
C) (–2, –6, 3)
D) (7, 5, 6)
E) (6, –2, –3)
Ans: B
35. Determine which of the following systems of equations is in row-echelon form.
System I:
 x + 9 y – z = 20

 y + 6 z = 10
 2 z = –3

System II:
 x – 9 y + 8z = 2

 y – 3 z = –15
 z = 20

System III:
 x – 9 y – 3 z = 19

 y + 7z = 8
 z = –13

A) None of the systems are in row-echelon form.
B) Only system II is in row-echelon form.
C) Only systems II and III are in row-echelon form.
D) All systems are in row-echelon form.
E) Only systems I and III are in row-echelon form.
Ans: C
36. Use back-substitution to solve the system of linear equations.

 –4 x + 8 y – 7 z = –126

 –5 y + 3 z = 58
 z=6

A) ( –8,5, 6 )
B) ( 5, –8, 6 )
C) ( 5, 6, –8)
D) ( 6, –8,5)
E) ( 0, –8, 6 )
Ans: B
37. Solve the system of linear equations.
 x – 4 y + z = 18

 2 x – 4 y + 3 z = 20
 –2 x + 4 y – z = –20

A) ( 3, 4,3)
B) ( 2, –4, 0 )
C) ( –4, 0, 2 )
D) ( 4,3,3)
E) inconsistent
Ans: B
38. Solve the system of equations below, if possible.
5 x – 4 y – 4 z = –1

 2 x + 4 y + 3 z = –1
 3x – 8 y – 7 z = 1

A) ( 0,3, 4 )
B) ( 4, 0,3)
C) ( 3, 4, 0 )
D) ( 3, 0, 4 )
E) The system is inconsistent.
Ans: E

 x+ y+z = –15

 x – y – 7 z = 31
 4 y + z = –17

A) ( –8, –4, –3)
B) ( –4, –8, –3)
C) ( –7, –3, –5)
D) ( –3, –7, –5)
E) ( –4, –5, –6 )
Ans: C
– x – 3 y + 4z = 5

 x + 5 z = –5
 5 x – 4 y + z = 11

A) ( 2,1, 0 )
B) ( 0, –3, –1)
C) ( –3, –1, 0 )
D) (1, 0, 2 )
E) ( 0, –3, 4 )
Ans: B
41. Solve the system of equations below:
– x + 8 y – 2 z = –9
x – 5 y + 2z = 6
– x + 7 y – 2 z = –8
A) ( –7, –1, –6 )
B) ( –1, 6, –7 )
C) ( –2c +1, –1, c )
D) ( –1, –2c +1, c )
E) ( –1, –2c –1, c )
Ans: C
–5 x + 8 y – 7 z = 0
2 x – 7 y – 8z = 0
–2 x + 3 y + 2 z = 0
A) ( 5, –3, –5)
B) ( 5c, –5c, –3c )
C) ( 5, –5, –3)
D) ( –5c,5c, –3c )
E) ( 0, 0, 0 )
Ans: E
5 x + 4 y + 7 z = –4
–4 x – 3 y – 6 z = 2
A) ( 2c + 4, –3c – 6, c )
B) ( –3c + 4, 2c – 6, c )
C) ( –3c – 6, 2c + 4, c )
D) ( 2c – 6, –3c + 4, c )
E) ( 4c – 6, –3c + 2, c )
Ans: B

 x+ y+ z+w= 10
 x – 5 y + 4z + w = 1


 2x – 5 z – 5w = –16
 –3 x + 4 y – 2 z + 3w = –3
A) ( 2, 4, –1,5)
B) ( 2,5, 4, –1)
C) ( –1, 4,5, 2 )
D) ( 2, 4,5, –1)
E) inconsistent
Ans: D
45. Which of the following systems of equations has as a solution the ordered triple
( 4, –1, –2 ) .
System I:
 5 x – 6 y – 4 z = 34

 –2 x + 3 y – 6 z = 1
 –5 x + 4 y – 6 z = –12

System II:
 –2 x + 3 y + 3 z = –17

 –4 x + 5 y + z = –23
 –2 x – 3 y + 4 z = –13

System III:
 – x – 4 y – 6z = 7

 –5 x + 4 y + 6 z = –37
 6 x + y – 3 z = 27

A) Only system II has the ordered triple ( 4, –1, –2 ) as a solution.
B) Only systems I and II have the ordered triple ( 4, –1, –2 ) as a solution.
C) Each of the systems I, II, and III has the ordered triple ( 4, –1, –2 ) as a solution.
D) Only system III has the ordered triple ( 4, –1, –2 ) as a solution.
E) None of the systems has the ordered triple ( 4, –1, –2 ) as a solution.
Ans: B
46. Which of the following three ordered triples are of the given form below.
 1 
 a, a + 3, a + 2 
 4 
Triple I : ( 4,3,3)
Triple II : ( –12, –9, –1)
Triple III : ( –5, –5, 0 )
A) Only triple I and II are in the given form.
B) None of the triples are in the given form.
C) Only triple I and III are in the given form.
D) Only triple III is in the given form.
E) Only triple II is in the given form.
Ans: E
47. Find an equation of the form y = ax 2 + bx + c whose graph passes through the points
( –2,11) , ( 0,9 ) , and ( 2, –1) .
A) y = – x 2 – 3x + 9
B) y = – x2 + 9x – 3
C) y = –3 x 2 – x + 9
D) y = 9 x2 – x – 3
E) y = 9 x 2 – 3 x –1
Ans: A
48. Find the equation of the parabola y = ax 2 + bx + c that passes through the points.
( 0, –1) , (1, –1) , ( 2,1)
A) y = x2 + x + 1
B) y = x2 – x + 1
C) y = 2 x2 – x – 1
D) y = 2 x2 – x – 2
E) y = x2 – x – 1
Ans: E
49. Find the equation of the circle

x 2 + y 2 + Dx + Ey + F = 0
that passes through the points ( 5,1) , ( 3, –1) , ( 7, –1) .
A) x 2 + y 2 – 10 x + 2 y + 22 =
0
B) x 2 + y 2 – 5 x + y + 22 =
0
C) x 2 + y 2 – 10 x + 2 y + 30 =
0
D) x 2 + y 2 – 10 x + 2 y – 4 =
0
E) x2 + y 2 – 5x + y – 4 =
0
Ans: A
50. A real estate company borrows $2,000,000. Some of the money is borrowed at 4%,
some at 8%, and some at 11% simple annual interest. How much is borrowed at the 11%
rate when the total annual interest is $130,000 and the amount borrowed at 4% is the
same as the amount borrowed at 8%?
A) $500,000
B) $200,000
C) $300,000
D) $600,000
E) $900,000
Ans: B
51. A mixture of 5 gallons of chemical A, 6 gallons of chemical B, and 22 gallons of
chemical C is required to kill a crop destroying insect. Commercial spray X contains 1,
2, and 3 parts of these chemicals, respectively. Commercial spray Y contains only
chemical C. Commercial spray Z contains chemicals A, B, and C in equal amounts.
How much of commercial spray X is needed to obtain the desired mixture?
A) 2 gallons
B) 3 gallons
C) 1 gallon
D) 6 gallons
E) 15 gallons
Ans: D
52. A chemist needs 20 liters of a 45% acid solution. The solution is to be mixed from three
solutions whose acid concentrations are 10%, 20%, and 50%. How many liters of the
50% solution should the chemist use if trying to use as little as possible of the 50%
solution?
A) 2
5 liters
3
B) 2
7 liters
3
C) 2
16 liters
3
D) 1
3 liters
3
E) 2
11 liters
3
Ans: C
53. A residential building contractor borrowed $31,000 to complete a new home. Some of
the money was borrowed at 5%, some at 7%, and some at 9%. How much was borrowed
at each rate if the annual interest owed was $2050 and the amount borrowed at 7% is
three times more than the amount borrowed at 9%?
A) $9000 at 5%; $17,000 at 7%; $5000 at 9%
B) $11,000 at 5%; $15,000 at 7%; $5000 at 9%
C) $9000 at 5%; $16,000 at 7%; $6000 at 9%
D) $10,000 at 5%; $13,000 at 7%; $7000 at 9%
E) $9000 at 5%; $18,000 at 7%; $6000 at 9%
Ans: B
54. The federal debt of the United States as a percentage of the Gross Domestic Product
(GDP) from 2001 to 2005 is shown in the table. In the table, x represents the year,
with x = 0 corresponding to 2002.
Year, x % of GDP
−1 57.4
0 59.7
1 62.6
2 63.7
3 64.3
Find the least squares regression parabola y = ax 2 + bx + c for the data by solving the
following system.
 5c + 5b + 15a =307.7

 5c + 15b + 35a =
325.5
15c + 35b + 99a =
 953.5
Use the model to predict the federal debt as percents of GDP in 2010. Round to the
nearest tenth percents.
A) 64.2%
B) 52.7%
C) 56.5%
D) 61.9%
E) 48.2%
Ans: C
55. Write an inequality for the shaded region shown in the figure.
A)
B)
C)
D)
E)
Ans: E
56. Sketch the graph of the inequality below.
x < –3
A)
B)
C)
D)
E)
Ans: A
x + y > –3
A)
B)
C)
D)
E)
Ans: D
y < –x +3
A)
B)
C)
D)
E)
Ans: D
59. Use a graphing utility to graph the inequality. Shade the region representing the
solution.
A)
B)
C)
D)
E)
Ans: B
60. Graph the solution set of the system of inequalities below.
– x + y ≤ 2

 x+ y ≤2
 y≥0

A)
B)
C)
D)
E)
Ans: D
61. Sketch the graph of the solution set of each system of inequalities.
2x – 5 y < 6
3x + y < 8
A)
B)
C)
D)
E)
Ans: E
62. Sketch the graph and label the vertices of the solution set of the system of inequalities.
Shade the solution set.
A)
B)
C)
D)
E)
Ans: E
63. Sketch the graph of the solution set of each system of inequalities.
y < 2x −1
y < x2 − 2x + 2
A)
B)
C)
E)
Ans: E
64. Graph the solution set of the system of inequalities below.
y ≤ x + 2

y ≥ x – 2

y ≥ 0
 y ≤ 2
A)
B)
C)
D)
E)
Ans: D
65. Write a system of inequalities whose solution set is graphed below.
A) 0 ≤ x ≤ 3

0 ≤ y ≤ 2
B)  –3 ≤ x ≤ 0

 0≤ y≤2
C)  –3 ≤ x ≤ 0

 –2 ≤ y ≤ 0
D)  0≤ x≤2

 –3 ≤ y ≤ 0
E)  0≤ x≤3

 –2 ≤ y ≤ 0
Ans: A
66. Derive a set of inequalities to describe the region.
Triangle: vertices at (0, 0), (5, 0), (5, 4)
A)
 4
y ≤ 5 x

y ≥ 0
x ≤ 5


B)
 4
y ≤ x
 5
 x ≤ 5
C)
 4
y ≥ 5 x

y ≥ 0
x ≤ 5


D)
 4
y ≤ x
 5
 x ≥ 0
E)
 5
 y ≤ x
4

y ≥ 0
x ≤ 5


Ans: A
67. A furniture company produces tables and chairs. Each table requires 1 hour in the
assembly center and 2.25 hours in the finishing center. Each chair requires 0.75 hour in
the assembly center and 0.5 hour in the finishing center. The company's assembly center
is available 18 hours per day, and its finishing center is available 12 hours per day. Let
x and y be the number of tables and chairs produced per day, respectively. Find a
system of inequalities describing all possible production levels.
A) 0.75 x + 0.5 y ≤ 18
 x + 2.25 y ≤ 12


 x≥0
 y≥0
B) 2.25 x + 0.5 y ≤ 18
 x + 0.75 y ≤ 12


 x≥0
 y≥0
C)  x + 0.5 y ≤ 18
0.75 x + 2.25 y ≤ 12


 x≥0
 y≥0
D)  x + 2.25 y ≤ 18
0.75 x + 0.5 y ≤ 12


 x≥0
 y≥0
E)  x + 0.75 y ≤ 18
2.25 x + 0.5 y ≤ 12


 x≥0
 y≥0
Ans: E
68. For the given supply and demand equations, find the consumer surplus. Round to the
nearest dollar.
Demand Supply
p = 170 – 0.00003x p = 140 + 0.00004x
A) $2,755,102
B) $3,030,612
C) $3,306,122
D) $4,132,653
E) $3,673,469
Ans: A
69. Find the consumer surplus for the pair of demand and supply equations below.
p = 35 – 0.00003 x Demand
p = 10 + 0.00002 x Supply
A) $1,250,000
B) $3,750,000
C) $5,000,000
D) $2,500,000
E) $1,875,000
Ans: B
70. You plan to invest up to $30,000 in two different interest-bearing accounts. Each
account is to contain at least $9000. Moreover, one account should have at least three
times the amount that is in the other account. Find a system of inequalities that describes
the amount that you can invest in each account.
A)  x + y ≤ 30, 000
 x ≥ 9000


 y ≥ 9000
 x − 3 y ≤ 0
B)  x + y ≥ 30, 000
 x ≥ 9000


 y ≥ 9000
 x − 3 y ≤ 0
C)  x + y ≥ 30, 000
 x ≤ 9000


 y ≤ 9000
 x − 3 y ≥ 0
D)  x + y ≤ 30, 000
 x ≥ 9000


 y ≥ 9000
 x − 3 y ≥ 0
E)  x + y ≥ 30, 000
 x ≥ 9000


 y ≥ 9000
 x − 3 y ≥ 0
Ans: D
71. A dietitian designs a special diet supplement using two different foods. Each ounce of
food X contains 10 units of calcium, 15 units of iron, and 20 units of vitamin B. Each
ounce of food Y contains 15 units of calcium, 10 units of iron, and 20 units of vitamin
B. The minimum daily requirements for the diet are 380 units of calcium, 240 units of
iron, and 390 units of vitamin B. Which combinations of foods X and Y below, if any,
can be given to the patient to meet the minimum daily requirements?
Combination I : 21 ounces of food X and 22 ounces of food Y

Combination II : 16 ounces of food X and 19 ounces of food Y
Combination III : 10 ounces of food X and 10 ounces of food Y
A) Only combinations II and III meet the minimum daily requirement.
B) Only combination II meets the minimum daily requirement.
C) None of the combinations meet the minimum daily requirement.
D) Only combinations I and II meet the minimum daily requirement.
E) Only combination III meets the minimum daily requirement.
Ans: D
72. Find which of the following system of inequalities has a right triangle as a graphed
solution set.
System I:
 2
y ≤ 5 x + 2

y ≥ 0
x ≥ 0


System II:
 3
 y ≤ 5 x –1

y ≥ 0
x ≥ 0


System III:
 3
y ≤ 5 x + 4

y ≥ 0
x ≥ 0


A) No systems have a right triangle as a graphed solution set.
B) Only systems II and III have a right triangle as a graphed solution set.
C) Only systems I and III have a right triangle as a graphed solution set.
D) Only system I has a right triangle as a graphed solution set.
E) Only systems I and II have a right triangle as a graphed solution set.
Ans: B
73. Find the minimum and maximum values of the objective function and where they occur,
subject to the indicated constraints.
Objective function:
Constraints:
A)
B)
C)
D)
E)
Ans: E
74. Find the minimum and maximum values of the objective function and where they occur,
subject to the indicated constraints. (You should graph the feasible solutions on the grid
below before you attempt to find the minimum and maximum values.)
Objective function:
Constraints:
A)
B)
C)
D)
E)
Ans: B
75. Which of the following vertices of the constraint region shown is a maximum value of
the objective function below.
z = x+ y+2
A) only vertex C
B) vertices A and D
C) vertices B and C
D) only vertex D
E) only vertex B
Ans: E
76. Which of the following vertices of the constraint region shown is a minimum value of
the objective function below.
z = –5 x + 5 y +1
A) vertices C and D
B) only vertex C
C) only vertex A
D) vertices B and C
E) only vertex D
Ans: B
77. An ice cream supplier has two machines that produce vanilla and chocolate ice cream.
To meet one of its contractual obligations, the company must produce at least 11 gallons
of vanilla ice cream and 90 gallons of chocolate ice cream per day. One machine makes
5 gallons of vanilla and 6 gallons of chocolate ice cream per hour. The second machine
makes 1 gallons of vanilla and 14 gallons of chocolate ice cream per hour. It costs $21
per hour to run machine 1 and $26 per hour to run machine 2. How many hours should
each machine be operated to fulfill the contract at the least expense?
A) Machine 1 for 6 hours and machine 2 for 1 hour.
B) Machine 1 for 1 hour and machine 2 for no hours.
C) Machine 1 for 1 hour and machine 2 for 6 hours.
D) Machine 1 for 6 hours and machine 2 for no hours.
E) Machine 1 for no hours and machine 2 for 1 hour.
Ans: C
78. A company has budgeted a maximum of $1,200,000 for national advertising an allergy
medication. Each minute of television time costs $120,000 and each one-page
newspaper ad costs $60,000. Each television ad is expected to be viewed by 24 million
viewers, and each newspaper ad is expected to be seen by 7 million readers. What is the
optimal amount that should be spent on advertising for each type ad?
A) $1,200,000 on television time and $0 on newspaper ads
B) $1,000,000 on television time and $200,000 on newspaper ads
C) $500,000 on television time and $700,000 on newspaper ads
D) $600,000 on television time and $600,000 on newspaper ads
E) $900,000 on television time and $300,000 on newspaper ads
Ans: A
79. An investor has $300,000 to invest in two types of investments. Type A pays 5%
annually and type B pays 7% annually. To have a well-balanced portfolio, the investor
imposes the following conditions. At least one-third of the total portfolio is to be
allocated to type A investments and at least one-third of the portfolio is to be allocated
to type B investments. What is the optimal amount that should be invested in each
investment?
A) $100,000 in type A (5%), $200,000 in type B (7%)

B) $0 in type A (5%), $300,000 in type B (7%)
C) $200,000 in type A (5%), $100,000 in type B (7%)
D) $300,000 in type A (5%), $0 in type B (7%)
E) $110,000 in type A (5%), $190,000 in type B (7%)
Ans: A
80. A company makes two types of telephone answering machines: the standard model and
the deluxe model. Each machine passes through three processes: P1 , P2 , and P3 . One
standard answering machine requires 2 hours in P1 , 5 hours in P2 , and 3 hours in P3 .
One deluxe answering machine requires 3 hours in P1 , 5 hours in P2 , and 1 hour in
P3 . Because of employee work schedules, P1 is available for 30 hours, P2 is available
for 55 hours, and P3 is available for 27 hours. If the profit is $47 for each standard
model and $28 for each deluxe model, how many units of each type should the company
produce to maximize profit?
A) 8 standard models and 3 deluxe models
B) 9 standard models and 0 deluxe models
C) 10 standard models and 0 deluxe models
D) 0 standard models and 10 deluxe models
E) 3 standard models and 8 deluxe models
Ans: A
81. Find the maximum value of the objective function and where it occurs, if one exists.
Objective function:
z = x + 3y
Constraints:
x≥0
y≥0
−x + y ≤ 1
–4 x + y ≥ 4
A) The solution region is empty. There is no maximum value.
B) The maximum value is 1 and occurs at (1, 0 ) .
C) The maximum value is 3 and occurs at ( 0,1) .
D) The maximum value is 12 and occurs at ( 0, 4 ) .
E) The maximum value is 4 and occurs at ( 4, 0 ) .
Ans: A
82. Find the minimum and maximum values of the objective function = z 2 x + 8 y and
where they occur, subject to the constraints x ≥ 0, y ≥ 0, 2 x + y ≤ 4 .
A) The minimum value is 0 at ( 0, 0 ) and the maximum value is 32 at ( 0, 4 ) .

B) The minimum value is 0 at ( 0, 2 ) and the maximum value is 32 at ( 0, 4 ) .
C) The minimum value is 0 at ( 0, 0 ) and the maximum value is 32 at ( 0, 2 ) .
D) The minimum value is 0 at ( 0, 4 ) and the maximum value is 32 at ( 0, 2 ) .
E) The minimum value is 0 at ( 0, 4 ) and the maximum value is 32 at ( 0, 0 ) .
Ans: A
83. Maximize the object function = z 2 x + y subject to the constraints
3 x + y ≤ 15, 4 x + 3 y ≤ 30, x ≥ 0, and y ≥ 0 .
A) The maximum value is 10 at ( 2, 6 ) .
B) The maximum value is 4 at (1, 2 ) .
C) The maximum value is 10 at ( 3, 4 ) .
D) The maximum value is 12 at ( 3, 6 ) .
E) The maximum value is 5 at ( 2,1) .
Ans: D
84. Maximize the object function = z 4 x + 3 y subject to the constraints

3 x + y ≤ 15, 4 x + 3 y ≤ 30, x ≥ 0, and y ≥ 0 .
A) The maximum value is 24 along the line segment connection ( 0,10 ) and ( 3, 4 ) .
B) The maximum value is 30 along the line segment connection ( 0,15 ) and ( 3, 6 ) .
C) The maximum value is 30 along the line segment connection ( 0,10 ) and ( 3, 6 ) .
D) The maximum value is 24 along the line segment connection ( 0,15 ) and ( 3, 4 ) .
E) The maximum value is 30 along the line segment connection ( 0,30 ) and ( 3, 6 ) .
Ans: C
85. Maximize the object function= z 10 x + 8 y subject to the constraints

x + 5 y ≤ 25, x + y ≤ 9, 4 x + 2 y ≤ 32, x ≥ 0, and y ≥ 0 .
A) The maximum value is 90 at ( 5, 4 ) .
B) The maximum value is 122 at ( 5,9 ) .
C) The maximum value is 130 at ( 9,5 ) .
E) The maximum value is 56 at ( 4, 2 ) .
Ans: D
86. Find an objective function that has a minimum value at the indicated vertex D of the
constraint region shown below.
A) Answer will vary. Sample answer: z= x + 3 y .

B) Answer will vary. Sample answer: =
z 3x + y .
C) z 3( x + y ) .
Answer will vary. Sample answer:=
D) z 3( x + 2 y ) .
Answer will vary. Sample answer:=
E) =
Answer will vary. Sample answer: z 3( 2x + y ) .
Ans: A
87. Find an objective function that has a minimum value at the indicated vertices A and D
of the constraint region shown below.
A) Answer will vary. Sample answer is z=0

B) Answer will vary. Sample answer is =
z 5x + 6 y
C) Answer will vary. Sample answer is =
z 6x + 5 y .
D) Answer will vary. Sample answer is z = 5y
E) Answer will vary. Sample answer is z = 6x
Ans: C
88. The costs to a store two models of Global Positioning System (GPS) receivers are $80
and $100. The $80 model yields a profit of $25 and the $100 model yields a profit of
$30. Market tests and available resources indicate the following constraints.
The merchant estimates that the total monthly demand will not be exceed 200 units.
The merchant does not want to invest more than $18,000 in GPS receiver inventory.
What is the optimal inventory level for each model what is optimal profit?
A) A maximum profit of $4500 occurs when 180 of each model of GPS receiver is
sold.
B) A maximum profit of $5000 occurs when 200 of each model of GPS receiver is
sold.
C) A maximum profit of $5400 occurs when 180 of each model of GPS receiver is
sold.
D) A maximum profit of $6000 occurs when 200 of each model of GPS receiver is
sold.
E) A maximum profit of $5500 occurs when 100 of each model of GPS receiver is
sold.
Ans: E
89. A farming cooperative mixes two brands of cattle feed. Brand X costs $30 per bag, and
brand Y costs $25 per bag. Research and available resources have indicated the
following constraints.
Brand X contains two units nutritional element A, two units of element B, and two units
of element C.
Brand Y contains one unit of nutritional element A, nine units of element B, and three
units of element C.
The minimum requirements for nutrients A, B, and C are 12 units, 36 units, and 24
units, respectively.
What is the optimal number of bags of each brand that should be mixed? What is
optimal cost?
A) To minimize cost, use three bags of brands X and six bags of brand Y for
optimal an cost of $540.
B) To minimize cost, use three bags of brands X and six bags of brand Y for
C) To minimize cost, use three bags of brands X and six bags of brand Y for
D) To minimize cost, use three bags of brands X and six bags of brand Y for
E) To minimize cost, use three bags of brands X and six bags of brand Y for
Ans: C
90. A humanitarian agency can use two models of vehicles for a refugee rescue mission.
Each model A vehicle costs $1000 and each model B vehicle costs $1500. Mission
strategies and objectives indicate the following constraints.
A total of at least 20 vehicles must be used.
A model A vehicle can hold 45 boxes of supplies.A model B vehicle can hold 30 boxes
of supplies. The agency must deliver at least 690 boxes of supplies to the refugee camp.
A model A vehicle can hold 20 refugees. A model B vehicle can hold 32 refugees. The
agency must rescue at least 520 refugees.
What is the optimal number of vehicles of each model that should be used? What is the
optimal cost?
A) A minimum cost of $27,000 occurs when 6 vehicles of each model are used.
B) A minimum cost of $27,000 occurs when 14 vehicles of each model are used.
C) A minimum cost of $34,500 occurs when 23 vehicles of each model are used.
D) A minimum cost of $26,000 occurs when 26 vehicles of each model are used.
E) A minimum cost of $25,000 occurs when 10 vehicles of each model are used.
Ans: E
91. A manufacturer produces two models of bicycles. The times (in hours) required for
assembling, painting, and packaging each model are shown in the table.
Process Model A Model B
Assembling 2 2.5
Painting 4 1
Packaging 1 0.75
The total times available for assembling, painting, and packing are 4000 hours, 4800
hours, and 1500 hours, respectively. The profit per unit are $50 for model A and $75 for
model B. What is the optimal production level for each model? What is the optimal
profit?
A) The optimal profit of $120,000 occurs when no units of model A and 1600 units
of of model B are produced.
B) The optimal profit of $112,000 occurs when no units of model A and 1000 units
of of model B are produced.
C) The optimal profit of $97,500 occurs when no units of model A and 600 units of
of model B are produced.
D) The optimal profit of $60,000 occurs when no units of model A and 0 units of of
model B are produced.
E) The optimal profit of $112,000 occurs when no units of model A and 750 units of
of model B are produced.
Ans: A
92. A company makes two models of doghouses. The times (in hours) required for
assembling, painting, and packaging are shown in the table.
Process Model A Model B
Assembling 2.5 3
Painting 2 1
Packaging 0.75 1.25
The total times available for assembling, Painting, and packaging are 4000 hours, 2500
hours, and 1500 hours, respectively. The profits per unit are $60 for model A and $75
for model B. what is the optimal production level for each model? What is the optimal
profit?
A) The optimal profit occurs when 571 units of model A and 857 units of model B
are produced. The optimal profit is $98,571.
B) The optimal profit occurs when 571 units of model A and 857 units of model B
C) The optimal profit occurs when 571 units of model A and 857 units of model B
D) The optimal profit occurs when 571 units of model A and 857 units of model B
E) The optimal profit occurs when 571 units of model A and 857 units of model B
Ans: B
93. An accounting firm charges $2500 for an audit and $350 for a tax return. Research and
available resources have indicated the following constraints.
The firm has 900 hours of staff time available each week.
The firm has 155 hours of review time available each week.
Each audit requires 75 hours of staff time and 10 hours of review time.
Each tax return requires 12.5 hours of staff time and 2.5 hours of review time.
What numbers of audits and tax returns will bring in an optimal revenue?
A) The optimal revenue of $0 will occurs when 0 audits and no tax returns are
processed.
B) The optimal revenue of $27,200 will occurs when 5 audits and no tax returns are
processed.
C) The optimal revenue of $21,700 will occurs when 62 audits and no tax returns are
processed.
D) The optimal revenue of $30,000 will occurs when 12 audits and no tax returns are
processed.
E) The optimal revenue of $27,200 will occurs when 42 audits and no tax returns are
processed.
Ans: D
94. An accounting firm charges $2500 for an audit and $350 for a tax return. Research and
available resources have indicated the following constraints.
The firm has 900 hours of staff time available each week.
The firm has 155 hours of review time available each week.
Each audit requires 75 hours of staff time and 10 hours of review time.
Each tax return requires 12.5 hours of staff time and 2.5 hours of review time.
The accounting firm lowers its charge for an audit to $2000. What numbers of audits
and tax returns will bring in an optimal revenue?
A) The optimal revenue will be $24,700 if the firms does 5 audits and 42 tax returns
each week.
B) The optimal revenue will be $24,000 if the firms does 12 audits and 0 tax returns
each week.
C) The optimal revenue will be $21,700 if the firms does 0 audits and 62 tax returns
each week.
D) The optimal revenue will be $0 if the firms does 0 audits and 0 tax returns each
week.
E) The optimal revenue will be $30,000 if the firms does 12 audits and 0 tax returns
each week.
Ans: A
95. A fruit juice company makes two drinks by blending apple and pineapple juices. The
percent of apple juice and pineapple juice in each drink are shown in the table.
Mixture Drink A Drink B

Apple juice 30% 60%
Pineapple juice 70% 40%
There are 1000 liters of apple and 1500 liters of pineapple juice available. The profit for
drink A is $0.70 per liter and the profit for drink B is $0.60 per liter. What is the optimal
production level for each type of drink? What is the optimal profit?
A) 2 2
.The optimal profit is $1,666.67 when 1, 666 liters of Drink A and 1, 666
3 3
liters Drink B are produced.
B) 2 1
The optimal profit is $1,666.67 when 1, 666 liters of Drink A and 833 liters
3 3
Drink B are produced
C) 1 1
The optimal profit is $1,666.67 when 833 liters of Drink A and 833 liters
3 3
Drink B are produced.
D) 1 2
The optimal profit is $1,666.67 when 833 liters of Drink A and 1, 666 liters
3 3
E) 6
The optimal profit is $1,666.67 when 2,142 liters of Drink A and 0 liters
7
Ans: B
96. Maximize the object function= z 15 x + y subject to the constraints
x + 5 y ≤ 25, x + y ≤ 9, 4 x + 2 y ≤ 32, x ≥ 0, and y ≥ 0 .
A) The maximum value is 5 at ( 0,5 ) .
B) The maximum value is 107 at ( 7, 2 ) .
C) The maximum value is 79 at ( 5, 4 ) .
E) The maximum value is 120 at ( 2, 7 ) .
Ans: D
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XXII
Two or three days later, she was back at Fairholme Court.
Burks had been sent on to join the other servants, and by the
time Anne reached the house, everything was in its usual spotless
order.
As she sat looking into the fire the afternoon following her return,
Anne felt that it was good to be home. She glanced round the
charming room, and experienced a thrill of pleasure. The fresh
curtains at the windows with their rose garlands, pleased her eye.
The inlaid cabinets, the tables, the dainty bookcases, shining and
spotless from the maids’ energetic ministrations, reflected the
firelight at every angle. The pictures she loved seemed even more
beautiful for her absence, and the pots of lilies and hyacinths about
the room filled the air with sweet scent.
Anne looked from them to her books, as one glances from one
loved face to another. It was good to be home, and she felt happy,
and at rest. Painful misgivings had disappeared, and her mind was
filled with contented thoughts of her friends.
From Sylvia, inexpressibly relieved, she had just received a letter
of girlish effusion and gratitude.
It was in her hand as she sat smiling into the fire, glad to
remember the girl as she had yesterday seen her, pretty once more,
gay, and full of extravagantly noble resolutions for the future.
Madge Dakin, who with her husband had returned a few days
previously, she had already seen.
She looked thin and pale still, but Anne was satisfied to hear that
Harry was the dearest and best of men, and that she had never been
so much in love with any one in her life.
To-day Anne found the human comedy agreeable. A spectacle to
be viewed with a smile from which tears of pity and sympathy were
not very far removed. But the smile came first. She reflected that she
must see the Vicar, and she was making up her mind to leave the
fireside for that purpose, when the door opened, and he was
announced.
She rose quickly with an exclamation of pleasure, and went to
meet him.
He took her outstretched hand, but let it drop again immediately,
and glancing at him with half-defined surprise, she saw that he wore
his pulpit expression of slightly pompous gravity.
“Sit down,” begged Anne, cordially. “I was just coming up to see
you.”
“Thank you,” said the Vicar, dropping heavily into the chair she
indicated.
“I saw Sylvia only yesterday. She sent many messages to you,
and to her mother.”
“Thank you,” repeated the Vicar. “It is distressing to me, but I am
constrained to say I’m sorry you saw her,” he added after a
moment’s hesitation.
Anne looked at him in silence, and Mr. Carfax cleared his throat.
“Miss Page,” he began, “I am here to speak on a very painful
subject, and I think the sooner I mention it the better.”
“Certainly,” said Anne, drawing herself back against the cushions
of her chair.
“I repeat, I am sorry you have seen my child, because in future, I
say it with great reluctance, I wish her acquaintance with you to
cease.”
Anne still waited in silence, and again the Vicar cleared his
throat. It was difficult to talk with her eyes upon him, and his carefully
prepared speeches seemed a trifle ridiculous.
“I’d better tell you the history of this affair from the beginning,” he
broke out abruptly. “Shortly it is this. Some two or three weeks ago I
received a private letter from a lady whose name I will not mention
——”
“Madame Didier,” interrupted Anne quietly.
The Vicar paused.
“Madame Didier, since you seem to know my correspondent. It
was a letter written to me as the vicar of the parish, begging me to
warn Dr. Dakin against your influence with his wife.”
Anne did not speak.
“Madame Didier gave reasons for this interference,” he went on
after a moment. “Reasons which seemed to me to be based on false
and scandalous charges. The letter, however, so intimately
concerned my friend, that I was compelled to show it to him. It was
burnt in my presence, and such was my implicit confidence in you
that I wrote a strong, I may say a threatening letter to the lady,
forbidding her to circulate libellous reports.”
“I am grateful to you,” Anne said.
The Vicar glanced at her.
“I have since regretted that letter,” he added deliberately.
“A fortnight ago, business called me to London, and I spent an
evening with my wife’s friends, the Lovells. Madame Didier, whose
stay in England has been protracted, was with her aunt. I did not
know this when I went to see the Lovells,” he added, “or I should
naturally have avoided the chance of an unpleasant encounter.
“However, in spite of my protestations, and my refusal to hear
your name spoken by her, the lady insisted, and to avoid entering
upon unpleasant details, I may say at once that she gave me
incontrovertible evidence as to the truth of her assertions.”
There was a pause which Anne did not break. She sat quite still,
looking into the fire.
“I need not say,” pursued the Vicar stiffly, “that though I was
constrained to offer an apology to Madame Didier for my somewhat
intemperate letter, I repeated my warning to her with regard to the
danger of spreading this story.”
“Thank you,” said Anne again.
The Vicar moved uncomfortably.
“Under any other circumstances—had Madame Didier, I mean,
merely reported gossip or hearsay, I should immediately have come
to you for an explanation, and I should have accepted your bare
word against what might to others appear grave suspicion. But
unfortunately, as I said, her evidence is incontrovertible. I have seen
letters. In short, to put it plainly, Miss Page, to ask for an explanation
from you would be the merest farce. It therefore becomes my painful
duty——”
“An explanation of what?” asked Anne, turning to him with a
deliberate movement, and again the Vicar fidgeted under her gaze.
“Of—of—a mode of life which proves you to have been unworthy
of the position you have held in our midst.”
The Vicar gathered himself together; it was time for the
peroration, and from force of habit his voice grew full and deep. He
reminded himself vigorously of the sanctity of the home, the
preservation of the family, and in sonorous tones continued——
“You have been loved and trusted by pure and innocent women.
You have been esteemed as a friend by myself, as well as by many
another upright and honourable man. And I say it with pain, you have
deceived us. My own child has made you her confidante——”
Anne rose, and the stream of the Vicar’s eloquence suddenly ran
dry.
There was a moment’s silence, during which he felt a prey to
greater and more paralyzing nervousness than he had experienced
since the preaching of his first sermon.
The pause was broken by the opening of the door, and the
appearance of Burks with a letter on a tray.
“This is sent down from the Vicarage, ma’am,” she said,
addressing her mistress, “and the maid says will Mr. Carfax kindly
read it at once.”
She handed the tray to the Vicar, who took the letter, and with a
murmured apology, broke the envelope. A note from his wife
dropped out first. He picked it up, and hurriedly glanced through its
contents.
“I am wild with anxiety. I send you the enclosed, which has
just come from Mrs. Lovell, so that you may read it while you
are with Miss Page. She may perhaps be able to throw some
light upon the matter. At any rate, ask her advice. She is so
good and wise.”
The Vicar snatched up the other letter, which mechanically, in a
dazed voice, he began to read aloud.
“My dear Mary,
“I hasten to tell you, though I fear too late, of something
I have just heard about your dear Sylvia. She has signed a
contract to go to America with a theatrical travelling company,
and I am told that she has already sailed. The manager I
understand to be a man of bad character, as indeed he must
be to induce a girl to leave England without her parents’
consent. This has come to my knowledge in a roundabout
way through a chorus girl who happens to be related to my
maid. I should have telegraphed, but Simpkins has just
showed me the announcement of the company’s departure
from Liverpool, and in that case a telegram is useless.
“All my sympathy, dear. In haste,
“Your affectionate
“Laura Lovell.”
Mr. Carfax dropped the letter.
In the waning light, Anne saw that his face was white.
“You must have known of this!” he broke out fiercely. “You must
have known, I say!”
Anne moved swiftly to his side, and laid her hand on his arm.
“It’s all right,” she whispered hurriedly. “I did know. I stopped it.
Sylvia is quite safe, at Carlisle House. If I had guessed that such
news would reach you, I would have told you at once. I was going to
tell you when you came in. But you put it out of my head,” she added
simply.
The Vicar’s colour had not returned. He stood mopping his
forehead slowly with his handkerchief, his face working so painfully
that Anne, her eyes full of tears, turned away.
She opened her writing-table, and rang the bell.
“What are you going to do?” stammered her companion.
“Send a note to your wife. I can’t bear to think of her anxiety.”
“True,” murmured the Vicar. “You are very kind. It’s like you—to
think of everything,” he added, still in a dazed voice.
He began to pace the room with uneven steps.
“If the maid has gone, run up as quickly as you can to the
Vicarage with this note,” said Anne, sealing the envelope, as Burks
entered.
“She’s still here, ma’am.”
“Then give it to her, and tell her to go at once, please, Burks. It’s
important. Don’t keep her a moment longer talking.”
The maid disappeared, and Anne lighted the candles on the
mantelpiece, quietly, one by one.
“You need have no anxiety,” she said without looking at the Vicar.
“Sylvia has been very imprudent, but she realizes it, and is sorry.
She had arranged with me to come home and tell you all about it, as
soon as I had first spoken to you. She seemed to think that I might
have some—some little influence. But I must now leave her to tell
her own story. I only want you to understand that she’s safe. I went
to my solicitor about the matter, and as she is under age, he had no
difficulty in settling the whole affair.”
“But—this man?” demanded Mr. Carfax in an unsteady voice.
“The man Mrs. Lovell mentions?”
For the first time she glanced at him, and saw the fear in his
eyes.
“Be quite easy. Sylvia had no idea of any evil intention on the
man’s part. She is only utterly ignorant and inexperienced. She is
one of the pure and innocent women you mentioned just now.”
Her voice was gentle, and had not a trace of bitterness.
The Vicar continued for a moment his perambulation of the room.
Then he stopped abruptly and raised his head.
“Thank you,” he said in a husky tone. “I owe you a debt I can
never repay. I——” he hesitated painfully. “I wish to God——” he
broke out again, and again paused. She looked at him steadily.
All the pompous self-importance had died out of his face; all the
arrogance of the priest who denounces the sinner. His was the very
human face of a man still gasping with relief from deadly fear, still
unable to believe that the threatened danger is over. And with this
expression of scarcely assured safety there was mingled real sorrow,
a look of real affection for the woman to whom he owed his escape
from a crushing blow.
“You spoke of an explanation,” said Anne in a low voice. “A
moment ago I should have asked you to leave me, because of the
manner in which you spoke of it.
“Now I have changed my mind, and I think I should like to give
you an explanation—my explanation.”
She was still standing, still looking at him steadily.
“You were kind enough to say that people here had loved and
trusted me. I am glad if that is the case—very glad.” She waited a
moment.
“If as you say they have been good enough to give me their love
and confidence, it is because I have understood them; because they
have never been afraid to tell me their inmost thoughts. Well, you will
not believe me, perhaps,—that power of understanding would never
have been mine but for the ‘mode of life’ to which you have alluded.
“Twenty years ago, Mr. Carfax, I was a self-doubting, colourless
woman. My youth, as I thought, was past. It had brought me nothing.
No love, no human experience, no joys, no very deep sorrows even.
Nothing but the grey hopeless depression of a woman who has
never taken her part in the world, who has always stood outside,
who knows nothing of life; the sort of woman who ignorant to begin
with, grows narrower and more prejudiced as the years pass, till at
last in the bitter sense of the word, she is an old maid by nature,
useless as a friend, helpless as a comforter, of no account in a world
of men and women she cannot understand.
“Well, before that happened to me, before I was old in heart at
least, I met a man who loved me, and whom I loved. I might have
married him. I chose not to marry him, because——” She smiled a
little. “I need not trouble you with my reasons. They seemed good
reasons to me, and I have never regretted them. I lived with him for
three years. The memory of those three years has lasted with me to
this day, and has made me a woman so proud and happy that if my
deep content has overflowed, and reached the lives of others, it is no
credit to me. I simply can’t help caring for people, because by the
mercy of Heaven, I have loved and been loved. Nothing else, for me
at least, would have made that understanding and caring possible.
Not the money that came to me, nor the opportunities it afforded for
what is called ‘doing good.’ It was a change in me, that was needed,
a personal experience of loving and suffering. Well! I have loved and
I have suffered, and now I understand.
“That’s my little story. It’s a story I would not have told you ten
minutes ago. But—well, you made me feel just now that you were
human.
“Don’t imagine you see before you the sinner that repenteth. She
has never repented. She never will repent, though it’s an old white-
haired woman who is talking—to a man years younger than herself!”
Her eyes met his, and beneath their smiling gaze, half wise, half
whimsical, the Vicar dropped his own, and reddened like a school-
boy.
The gentle reproof, implied rather than spoken, went home.
Suddenly, in the presence of this dignified gracious woman, he
felt raw and awkward, very young, more than a little ashamed. He
was confused moreover, with the sense that there existed possibly
whole realms of experience which no code of morals he had ever
preached seemed adequate to cover.
Here was a woman who certainly possessed the fairest of the
Christian virtues. She was gentle, tolerant, generous (with a twinge
of compunction he realized how great a part the anticipated loss of
her donations had played in his reflections during the walk from the
Vicarage to Fairholme Court). She was patient, longsuffering,—the
Vicar ran through the whole gamut of spiritual gifts, and
acknowledged her richly endowed.
Could it be that there were other paths to the Kingdom of Heaven
than the strait way and the narrow gate that alone were said to lead
to salvation?
The very useful brain of Mr. Carfax, unaccustomed to be
exercised in unusual directions, began to feel the strain, and its
possessor wisely took the hint, and abandoned the fatiguing labour
of original research.
In any case Miss Page was a charming woman, and by however
amazing process the result had been achieved, a good one also.
He looked at her, and with a sudden frank movement, held out
his hand.
“Forgive me,” he said simply. “You—you have shown me I had no
right to judge. I beg your pardon.”
Anne put her hand into his with a very sweet smile.
“My dear friend,” she replied, “you must do what you think right,
and Dymfield will not be behind the judgment of most of the world in
this matter. You know I love the place, but I can’t stay here when the
people no longer look upon me as a friend. Well, the world is wide,
and fortunately for me I’m not a poor woman.”
“You mustn’t leave us! You won’t leave us!” begged the Vicar.
“There will be no occasion. The position is unchanged. The only two
people who know anything of—of the matter, are your friends. Even
if through malice or carelessness a breath of scandal should reach
others, surely you can trust us to treat the rumour with the——” He
hesitated.
“With the contempt it doesn’t deserve?” suggested Anne gently.
Greatly to his surprise, and somewhat to his horror, the Reverend
George Carfax was betrayed into an answering smile.
He hastened to efface it, but the deed was done.
“And Sylvia?” asked Anne tentatively. “I wanted her to stay with
me for a few days. You have only to say if you would rather she did
not, and I won’t ask her.”
“If after all the trouble she has given you, Sylvia will be welcome,
I can answer for her delight,” returned Mr. Carfax promptly.
Anne put out her hand with an impulsive gesture.
“You are quite a dear!” she observed, and her sudden smile still
further illuminated the dusky corners of the Vicar’s strictly limited
imagination.
The entrance of Burks with the tea-things gave him a moment to
recover from the shock of a series of mental and emotional
upheavals to which he was unaccustomed.
“You will stay, of course?” begged Anne. “My note to your wife
was quite explicit,” she added. “She won’t be anxious now.”
“Thank you,” said Mr. Carfax. “I want to hear particulars about
Sylvia, and I feel I should be all the better for a cup of tea.”
Five minutes later, Mrs. Carfax entered a room bright with fire
and candle-light, in which her husband sat comfortably ensconced in
an arm-chair opposite to Miss Page, who was passing him hot cakes
of a delicious crispness.
Anne went quickly across the room.
“It’s quite right. Don’t worry,” she hastened to say, as she kissed
her visitor. “I’m just telling your husband all about it.”
“Sylvia must come home!” declared her mother, after Anne’s
recital. Her hand was still trembling as she put down her tea-cup.
“She’s not fit to be left alone in a great wicked city. I always said to
George it was madness to let her go away from us!”
“It’s so difficult to get women to take broad views,” complained
the Vicar, turning to Anne, “it requires the masculine mind, free from
prejudice and indifferent to common opinion, to see the wider
outlook.”
Anne laid her hand on his wife’s arm.
“Dear Mrs. Carfax, do let her finish her training,” she urged. “The
child acknowledges her foolishness. I quite agree that she ought not
to be alone, and before you came in, I was suggesting a plan to your
husband.
“Let her go to my brother and his wife. They lost their little girl
some years ago, and Alice has always longed for a daughter. She’s
such a nice kind little woman, and she would treat Sylvia as her own
child. I spoke to her of the possibility of this, before I left London, and
she was delighted with the idea.”
“It would be a splendid thing for her, Mary, if it can be arranged.
It’s so like Miss Page to have thought of such a plan.”
Mrs. Carfax hesitated.
“We must think about it. I wouldn’t give my consent for her to go
anywhere else. But if it’s a case of your relations, dear, it’s different. I
should feel safe and happy about her, of course. We must talk about
it, George.”
Anne leant back against her sofa cushions with a satisfied
expression.
When her visitors rose to go, she followed them to the door.
While his wife was being helped into her goloshes by Burks,
outside in the hall, the Vicar lingered a moment to hold her hand in a
tight grasp.
“I can never thank you enough,” he murmured. “You are the best
woman I ever met,” he added, looking her straight in the face.
Anne flushed a little; there were tears in her eyes.
“Good-bye,” she said. “I shall look forward to having Sylvia here
next week.”
When the hall door had closed, she drew a deep breath of
exhaustion and relief.
She had won peace with honour. She knew it, and was thankful.
But she was glad to be alone.
She walked round the room, bending over the pots of lilies of the
valley, touching the waxen bells of the hyacinths with gentle fingers.
They had been grown for her home-coming, and they welcomed her
delicately. She stirred the fire to a brighter blaze, and smiled to see
its glow spreading to the furthest corner of the room.
Never had her home seemed so sweet, so inviting, so restful.
“It would have broken my heart to leave it!” she thought with
sudden conviction.
She looked at the bookcases filled with books all the more
precious, because for three months she had not touched them.
Finally she reached for a volume on one of the upper shelves,
and taking it to the sofa, turned to a poem she loved.
“Mère des souvenirs, maîtresse des maîtresses,

O toi, tous mes plaisirs! O toi, tous mes devoirs!
Tu te rappelleras la beauté des caresses,
La douceur du foyer et le charme des soirs,
Mère des souvenirs, maîtresse des maîtresses!
“Les soirs illuminés par l’ardeur du charbon,

Et les soirs au balcon, voilés de vapeurs roses.
Que ton sein m’était doux! que ton cœur m’était bon!
Nous avons dit souvent d’impérissables choses,
Les soirs illuminés par l’ardeur du charbon.”
Anne let the book slip into her lap. “Nous avons dit souvent
d’impérissables choses,” she repeated softly.
It was of these “imperishable things” she was thinking, the things
of the spirit, that persist when as with her the desire of the flesh is
dead, and the lust of the eyes. The imperishable things that last into
the evening of life, when the stars come out, and ever nearer and
nearer draw the “murmurs and scents of the infinite sea.”
XXIII
It was two years before François Fontenelle re-visited Fairholme
Court. Again it was June, and Anne had taken him to the garden, full
of pride to show him her roses in the height of their beauty.
They strolled round its paths talking of a thousand things, and
finally sat down under the arch, over which there poured a cascade
of snowy bloom. The table in front of the bench was littered with
papers, which François began idly to examine.
“The New Thought!” he exclaimed, holding up one of the leaflets
between his finger and thumb. “What on earth are you doing with this
latter day product?”
Anne laughed. “A strenuous young thing who is spending her
holiday in the village brought a heap of papers this morning, and
begged me to read them. She said it was scandalous that such an
intelligent woman as I appeared to her, should be ignorant of the
‘movement,’” she added demurely.
“Whatever the modern young woman lacks, it isn’t cheek,” he
returned.
“Well! What do you think of the ‘no property’ idea in the eternally
boring sex question? Let me see, there are to be state babies, aren’t
there? Have state lovers been suggested yet, or is that a figment of
my imagination?”
Anne sighed. “Perhaps I’m too old for it,” she said. “I know I don’t
understand it. It all seems to me so terribly business-like, and I was
never a business woman.”
François laughed. “I should as soon expect one of these roses to
start company promoting.”
“One thing I feel quite sure about,” she went on, drawing her lace
shawl round her shoulders. “The men and women who write some of
these letters have never loved.”
“Love has gone out of fashion in England, and the new wisdom
has taken its place,” observed François. “Its professors are
gentlemen who live on grape nuts, and are occupied municipally.
They don’t believe in love, partly because a diet of grape nuts is not
conducive to the emotion, partly because they are afraid of
disagreeing with Mr. Bernard Shaw. You have saved me from
belonging to the latter class, but only as a brand is snatched from the
burning.”
“The simile is ill-chosen,” declared Anne serenely. “There’s no fire
about any of the new doctrines. They are all eminently cool,
calculating and dull. Dull as ditch-water, and quite as appetizing.”
François smiled. “You are a very old-fashioned woman, Anne,” he
declared, “and the sight of these things near you is absurd, and even
indecent.”
He swept them from the table.
“Go and fetch your Herrick, and read me how roses first came
red, and lilies white.”
“My lord shall be obeyed—another time,” said Anne laughing.
“How is Mrs. Dakin?” asked François suddenly, lighting a
cigarette.
Anne was engaged in pushing the end of a trailing green branch
through one of the spaces in the lattice work.
“She and the baby, who is six weeks old to-day, are away on a
visit to her mother. She is very well, and exceedingly happy,” she
added after a moment spent in arranging the branch to her
satisfaction.
“I’m glad to hear it.”
She turned to him. “I believe you are, François.”
“I’m also glad to hear she’s away, since because of that
circumstance presumably I was honoured with an invitation to-day.”
“Why haven’t I seen you for so long?” inquired Anne irrelevantly.
“I was afraid to come,” he said, looking at her with a smile.
“Why?”
“Oh, not because I dreaded a scene with you. Have you ever
made a scene in your life, Anne? You ought to have done it once at
least, to prove your affinity with the sex you adorn. But I don’t believe
you ever have. No. I was afraid of your eyes.”
“What’s the matter with my eyes?” she asked, with a smile
concealed in them.
“Anne Page, if you’re going to flirt with me I give you due warning
that I’m a poor weak man, and I can’t answer for the consequences.”
She laughed. “The baby is a darling, and I’m its godmother.
They’ve called her Anne.”
“They may, but they needn’t flatter themselves she’ll ever be as
attractive as Anne Page.”
“Her father already thinks her the most lovely creature in the
world—except his wife.”
“And how are all the other worthies? Still at your feet, I suppose?”
“They are all charming to me. My little friend Sylvia, the Vicar’s
daughter, sang at her first concert the other night, and had a great
success. The vicarage is standing on its head with pride, in
consequence.”
“And the pastoral life still amuses you?”
“Very much.”
“Wonderful woman!”
“Dear François, why not?” she asked. “You know I am a very
simple person.”
“Yes. Though you were once the queen of quite a brilliant salon.”
She was silent.
“When are you coming over to see your picture?”
“This autumn.” For a moment she paused. “You know my wishes,
François? I have left René’s pictures to the Luxembourg. The two we
like best—you know them—are to hang on either side of the portrait.
It’s in my will, of course.”
He smoked a moment without speaking.
“I wonder if he’ll come and look at them?” he said at last. “I think
he will, and you’ll smile at him out of the portrait.”
“I’m so glad he liked it,” she answered softly, after a long pause.
“He only saw it once. I never dared show it to him again. That’s
why I put it away.”
The birds had begun their evening song, and the garden rang
with the voices of blackbirds and thrushes.
“Well! I must get back to The Chase,” declared François, glancing
at his watch. “I shall be late for dinner as it is. This is good-bye till
September. Not a moment later mind, and then you will stay in Paris
a decent time?”
He looked at her, as she got up and stood for a moment
embowered in the roses, her lace shawl hanging from her arms, her
figure still beautiful and gracious.
“The gods have granted you the gift of eternal youth, Anne,” he
declared. “I want to paint another portrait.”
She laughed, and shook her head.
“There will be no more portraits,” she said.
She went with him as far as a little gate which gave upon the
meadows, through which a field path led to The Chase.
After he had gone she wandered into the lavender-garden, and in
the gathering summer twilight paced the path between the grey-
green borders.
In the west, the sky was still flushed with sunset. The air, so quiet
that not a leaf trembled, was sweet with the scent of flowers.
Anne walked slowly, her mind occupied with pleasant trifles. She
decided that the lilies in the south border must this autumn be
divided. She must tell Davis to plant more daffodils in the orchard
under the apple trees. There was the village children’s treat to be
considered, and she must not forget to talk it over with the Vicar.
Suddenly she remembered that Dr. Dakin was coming in to smoke
his pipe and talk. Madge and the baby were returning on Thursday.
He would therefore be in excellent spirits.
The roses on the hedge round the sundial breathed a sweet
strong fragrance into the dusk.
Anne picked one of them, and tucked it into the front of her gown,
before she turned towards the house.
THE END
PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, LONDON AND
BECCLES.
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The Morals of Marcus Ordeyne

At the Gate of Samaria
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pen. It is impossible to read and not marvel at the skilled
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Derelicts
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2. Determine which ordered pair is a solution of the system.

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9. Solve the system below by method of substitution, if possible.

10. Solve the system by the method of substitution.

21. Solve the system by elimination.

22. Solve each system of equations by the elimination method.

24. Solve the system by the method of elimination.

25. Solve the system by the method of elimination.

 p + 0.55 x = 1484 Demand

Points: (1, –1) , ( 3, –7 ) , ( 6, –15 ) , ( 7, –19 )

36. Use back-substitution to solve the system of linear equations.

38. Solve the system of equations below, if possible.

39. Solve the system of linear equations.

41. Solve the system of equations below:

42. Solve the system of equations below:

44. Solve the system of linear equations.

49. Find the equation of the circle

Combination I : 21 ounces of food X and 22 ounces of food Y

A) $100,000 in type A (5%), $200,000 in type B (7%)

A) The minimum value is 0 at ( 0, 0 ) and the maximum value is 32 at ( 0, 4 ) .

84. Maximize the object function = z 4 x + 3 y subject to the constraints

85. Maximize the object function= z 10 x + 8 y subject to the constraints

A) Answer will vary. Sample answer: z= x + 3 y .

A) Answer will vary. Sample answer is z=0

Mixture Drink A Drink B

“Mère des souvenirs, maîtresse des maîtresses,

“Les soirs illuminés par l’ardeur du charbon,

The Morals of Marcus Ordeyne

The Belovéd Vagabond

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