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unit 7 review for final

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 1. Which linear system has the solution x = –2 and y = 6?

a. x + 3y = 16

4x + 4y = 16

c. x + 2y = –2

2x + 4y = –4

b. x + 3y = 17

2x + y = 15

d. 2x + y = –2

x + y = 16

____ 2. Which linear system has the solution x = 4 and y = –2?

a. x + 4y = 15

4x = –17

c. 4x + y = 14

–2x = –16

b. 2x + 4y = 4

–2x + y = 14

d. x + 4y = 4

2x + 4y = 8

____ 3. Which linear system has the solution x = 8 and y = 2.5?

a. 2x + 2y = 21

2x – 2y = 11

c. 2x + 2y = 8

x – y = 21

b. x + 2y = 8

2x – 4y = 16

d. x + 3y = 22

2x – y = 10

____ 4. Create a linear system to model this situation:

The perimeter of an isosceles triangle is 36 cm. The base of the triangle is 9 cm longer than each equal side.

a. s + b = 36

b – 9 = s

b. 2s + b = 36

b + 9 = s

c. 2b + s = 36

s + 9 = b

d. 2s + b = 36

s + 9 = b


A collection of nickels and dimes contains four times as many dimes as nickels. The total value of the

collection is $20.25.

a. d = 4n

5n + 10d = 2025

b. d = 4n

5d + 10n = 2025

c. n = 4d

5n + 10d = 2025

d. d + n = 15

5n + 10d = 2025


In a board game, Judy scored 3 points more than twice the number of points Ann scored.

There was a total of 39 points scored.

a. j = 3 + 2a

j + a = 39

b. j – 3 = 2a

j + 2a = 39

c. j + 3 = 2a

j + a = 39

d. a = 3 + 2j

j + a = 39


A woman is 3 times as old as her son. In thirteen years, she will be 2 times as old as her son will be.

a. w = s + 3

w + 13 = 2s

c. w = 3s

w = 2s

b. w = 3s

w + 13 = 2(s + 13)

d. w = 3s

s + 13 = 2(w + 13)


Cheri operates a grass-cutting business. She charges $19 for a small lawn and $29 for a large lawn. One

weekend, Cheri made $287 by cutting 13 lawns.

a. s + l = 13

19s + 29l = 287

c. s + l = 13

29s + 19l = 287

b. s + l = 287

19s + 29l = 13

d. s + l = 287

29s + 19l = 13


A length of outdoor lights is formed from strings that are 5 ft. long and 11 ft. long. Fourteen strings of lights

are 106 ft. long.

a. 5x + 11y = 14

x + y = 106

c. x + y = 14

5x + 11y = 106(14)

b. x + y = 14

5x + 11y = 106

d. x + y = 14

x + 2y = 106


A rectangular field is 35 m longer than it is wide. The length of the fence around

the perimeter of the field is 290 m.

a. l + 35 = w

2l + 2w = 290

b. l = w + 35

2l + 2w = 290

c. l = w + 35

l + w = 290

d. l = w + 35

lw = 290


Tickets for a school play cost $8 for adults and $4.75 for students.

There were ten more student tickets sold than adult tickets, and a total of $1399 in ticket sales was collected.

a. 8a + 4.75s = 1399

s = a + 10

c. 8a + 4.75s = 1399

a = s + 10

b. 8a + 4.75s = 1399

a + s = 10

d. 4.75a + 8s = 1399

s = a + 10

____ 12. Match each situation to a linear system below.

A. The perimeter of a rectangular playground is 163 m. The length is 6 m less than double

the width.

B. The perimeter of a rectangular playground is 163 m. The width is one-half the length

decreased by 6 m.

C. The perimeter of a rectangular playground is 163 m. The length decreased by 6 m is

double the width.

i) ii) iii)

a. A-i, B-ii, C-iii c. A-ii, B-i, C-iii

b. A-iii, B-i, C-ii d. A-i, B-iii, C-ii

____ 13. Which graph represents the solution of the linear system:

y = –2x

y + 6 = 2x

0

(2, –2)

Graph A

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

0

(2.2, –0.5)

Graph C

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

0

(1, –2)

Graph B

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

0

(1.4, –0.8)

Graph D

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

a. Graph B c. Graph C

b. Graph A d. Graph D

____ 14. Which graph represents the solution of the linear system:

–3x – y = –5

4x – y =

0

(1, 2)

Graph A

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

0

(–1, –2)

Graph C

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

0

(0, 0)

Graph B

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

0

(–0.8, –2.7)

Graph D

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

a. Graph A c. Graph C

b. Graph B d. Graph D

____ 15. Use the graph to solve the linear system:

y = –3x – 5

y = 3x

0

y = –3x – 5

y –1 =3 x

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

a. (1, –2) c. (1, 0)

b. (–1, 0) d. (–1, –2)

____ 16. Use the graph to solve the linear system:

y = –5x

y + = 2x

0

y = –5x –2

y +2 = 2 x

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

a. (2, 0) c. (0, 0)

b. (2, –2) d. (0, –2)

____ 17. Use the graph to approximate the solution of the linear system:

0 2 4 6 8–2–4–6–8 x

2

4

–2

–4

y

a. (–3, 0.2) c. (0.2, –3)

b. (0, –2.8) d. (–2.8, 0)

____ 18. Car A left Calgary at 8 A.M. to travel 500 mi. to Regina, at an average speed of 63 mph.

Car B left Regina at the same time to travel to Calgary at an average speed of

37 mph. A linear system that models this situation is:

d = 500 – 63t

d = 37t,

where d is the distance in miles from Regina, and t is the time in hours since 8 A.M. Which graph would you

use to determine how far the cars are from Regina when they meet? What is this distance?

Dis

tan

ce f

rom

Reg

ina (

mi.)

100

200

300

400

500

Time (h)

0 2 4 6 8

Car A

Car B

(5, 185)

Graph A

10 12 14

Dis

tan

ce f

rom

Reg

ina (

mi.)

100

200

300

400

500

Time (h)

0 2 4 6 8

Car A

Car B

(2.5,92.5)

Graph B

10

Dis

tan

ce f

rom

Reg

ina (

mi.)

100

200

300

400

500

Time (h)

0 2 4 6 8

Car A

Car B

(4.1,195.8)

Graph C

10 12

Dis

tan

ce f

rom

Reg

ina (

mi.)

100

200

300

400

500

Time (h)

0 2 4 6 8

Car A

Car B

(3.8,200)

Graph D

10

a. Graph C:

195.8 mi.

b. Graph D:

200 mi.

c. Graph A:

185 mi.

d. Graph B:

92.5 mi.

____ 19. Which linear system is represented by this graph?

a) x – y = 3

6x + 5y = 14

b) x + y = 5

6x + 5y = 14

c) x + y = 7

7x + 5y = 14

d) x + y = 9

5x + 6y = 14 0 2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

a. System a b. System b c. System c d. System d


a) x – y = 5

5x + 6y = 18

b) x – y = 7

5x + 6y = 18

c) x – y = 9

6x + 6y = 18

d) x – y = 11

6x + 5y = 18 0 2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

a. System d b. System b c. System a d. System c


a) 2x – 5y = –16

x = 1

b) 2x + 5y = 16

2x – 5y = 16

c) 2x – 5y = 16

x – 2

5y = –1

d) 2x + 5y = 16

x = –1

0 2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

a. System a b. System d c. System b d. System c

____ 22. Determine the solution of the linear system represented by this graph.

a) (2, 3.8)

b) (3.8, 2)

c) (–3, 3.8)

d) (–2, 3.8)

0 2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

a. b b. a c. d d. c

____ 23. Two life insurance companies determine their premiums using different formulas:

Company A: p = 2a + 24

Company B: p = 2.25a + 13, where p represents the annual premium, and a represents the client’s age.

Use the graph to determine the age at which both companies charge the same premium.

Pre

miu

m (

$)

50

100

150

200

250

Age (years)

10 20 30 40

Company A

Company B

50 60 700

a

p

80 90

a. 62 years b. 24 years c. 59 years d. 44 years

____ 24. At a skating rink, admission is $4.00 for a student and $8.00 for an adult.

Tuesday evening, 20 people used the skating rink and a total of $132 in admission fees was collected. A linear

system that models this situation is:

4s + 8a = 132

s + a = 20

where s represents the number of student admissions, and a represents the number of adult admissions

purchased.

Use the graph to solve this problem:

How many students used the skating rink on Tuesday evening?

Nu

mb

er

of

ad

ult

ad

mis

sio

ns

10

20

30

40

50

Number of student admissions

0 10 20 30 40 50

s

a

a. 19 students b. 20 students c. 13 students d. 7 students

____ 25. Use the graph to approximate the solution of this linear system:

6x – 7y = –4

– 3

5y = 3x + 7

0 2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

a. (–0.1, 3.8) b. (–2.1, –1.2) c. (–1.2, 3.8) d. (–2.1, –0.1)

____ 26. Use substitution to solve this linear system.

y = – x

13x + 5y = 178

a. (6, –20) b. (6, 20) c. (–6, –20) d. (–6, 20)


x = 2y – 56

5x + 13y = 410

a. (4, –30) b. (–4, 30) c. (4, 30) d. (–4, –30)

____ 28. Identify two like terms and state how they are related.

–10x + 20y = 460

30x + 60y = 1620

a. –10x and 30x; by a factor of –3 c. 30x and 60y; by a factor of 2

b. –10x and 20y; by a factor of –2 d. –10x and 460; by a factor of 46

____ 29. Identify two like terms and state how they are related.

a. 7x and –5y; by a factor of

5

7

c. 8x and –4y; by a factor of

1

2

b. 8x and –96; by a factor of 12 d. 8x and 7x; by a factor of

7

8


x = 4 + y

4x + 16y = –264

a. (–14, –14) b. (–10, –10) c. (–10, –14) d. (–14, –10)

____ 31. Use substitution to solve this problem:

The perimeter of a rectangular field is 276 m. The length is 18 m longer than the width.

What are the dimensions of the field?

a. 58 m by 80 m b. 68 m by 70 m c. 78 m by 60 m d. 48 m by 90 m

____ 32. For each equation, identify a number you could multiply each term by to ensure that the coefficients of the

variables and the constant term are integers.

(1) 5

4x +

1

6y =

47

12

(2) 4

5x –

6

7y = 16

a. Multiply equation (1) by 35; multiply equation (2) by 12.

b. Multiply equation (1) by 12; multiply equation (2) by 35.

c. Multiply equation (1) by 2; multiply equation (2) by 3.

d. Multiply equation (1) by 3; multiply equation (2) by 2.

____ 33. Write an equivalent system with integer coefficients.

3

7x + 3y =

438

7

5

6x + 5y =

310

3

a. 3x + 21y = 438

5x + 30y = 620

c. 3x + 21y = 438

30x + 5y = 620

b. 21x + 3y = 438

5x + 30y = 620

d. 3x + 21y = 1

5x + 30y = 1

____ 34. Write an equivalent system with integer coefficients.

5x + 3

2y = 14

5

6x + 5y =

755

6

a. 10x + 3y = 1

5x + 30y = 1

c. 10x + 3y = 28

30x + 5y = 755

b. 3x + 10y = 28

5x + 30y = 755

d. 10x + 3y = 28

5x + 30y = 755

____ 35. The solution of this linear system is (–3, y). Determine the value of y.

x – 3y = 33

6

7x – y =

88

7

a. 20 b. 30 c. 10 d. 40

____ 36. Use an elimination strategy to solve this linear system.

a. and c. and

b. and

d. and


a. and c. and

b. and

d. and

____ 38. Write an equivalent linear system where both equations have the same x-coefficients.

a. and c. and

b. and d. and

____ 39. Write an equivalent linear system where both equations have the same y-coefficients.

a. and c. and

b. and d. and

____ 40. Model this situation with a linear system:

Frieda has a 13% silver alloy and a 31% silver alloy. Frieda wants to make 26 kg of an alloy that is 47%

silver.

a. and c. and

b. and d. and


a. and c. and

b. and d. and


a. and c. and

b. and d. and

____ 43. Without graphing, determine which of these equations represent parallel lines.

i) –6x + 6y = 12

ii) –4x + 6y = 12

iii) –2x + 6y = 12

iv) –6x + 6y = 14

a. ii and iii b. i and ii c. i and iv d. i and iii

____ 44. Determine the number of solutions of the linear system:

2x – 5y = 23

–6x + 15y = 21

a. one solution c. two solutions

b. no solution d. infinite solutions


14x – 5y = 123

14x – 5y = 73

a. no solution c. two solutions

b. infinite solutions d. one solution


14x + 7y = 315

16x – 2y = 610

a. no solution c. two solutions

b. one solution d. infinite solutions


5x + 7y = 76

–25x – 35y = –380

a. 2 solutions c. infinite solutions

b. one solution d. no solution

____ 48. The first equation of a linear system is 2x + 3y = 52. Choose a second equation to form a linear system with

infinite solutions.

i) 2x + 3y = –260 ii) –10x – 15y = –260 iii) –10x + 3y = –260 iv) –10x + 3y = 255

a. Equation iii b. Equation iv c. Equation i d. Equation ii

____ 49. The first equation of a linear system is 8x + 13y = 166. Choose a second equation to form a linear system with

exactly one solution.

i) 8x + 13y = –830 ii) –40x – 65y = –830 iii) –40x + 13y = –830 iv) –40x – 65y = 0

a. Equation iii b. Equation i c. Equation ii d. Equation iv

____ 50. The first equation of a linear system is –6x + 12y = –42. Choose a second equation to form a linear system

with no solution.

i) –6x + 12y = 126 ii) 18x – 36y = 126 iii) 18x + 12y = 126 iv) 18x + 36y = 0

a. Equation iv b. Equation ii c. Equation iii d. Equation i

____ 51. Two lines in a linear system have the same slope, but different y-intercepts.

How many solutions does the linear system have?

a. two solutions c. infinite solutions

b. no solution d. one solution

Short Answer

52. Quincy used this linear system to represent a situation involving a collection of $5 bills and $10 bills:

f + t = 70

5f + 10t = 575

a) What problem might Quincy have written?

b) What does each variable represent?

53. Solve this linear system by graphing.

–3x – 2y = 16

–x + y = –8

0 2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

54. a) Write a linear system to model this situation:

A hockey coach bought 25 pucks for a total cost of $70. The pucks used for practice cost

$2.50 each, and the pucks used for games cost $3.25 each.

b) Use a graph to solve this problem:

How many of each type of puck did the coach purchase?

Pu

cks u

sed

fo

r g

am

es

10

20

30

40

50

Pucks used for practice

0 10 20 30 40 50

p

g

55. Use substitution to solve this linear system:

56. Use substitution to solve this linear system:

x + 7

8y = –34

–3x + 4y = –4

57. Create a linear system to model this situation. Then use substitution to solve the linear system to solve the

problem.

At the local fair, the admission fee is $8.00 for an adult and $4.50 for a youth. One Saturday, 209 admissions

were purchased, with total receipts of $1304.50. How many adult admissions and how many youth

admissions were purchased?

58. Determine the number of solutions of this linear system.

7x – 3y = 43

7x – 3y = 13

59. Determine the number of solutions of this linear system.

15x + 30y = –240

17x + 21y = 53

Problem

60. a) Write a linear system to model this situation:

The coin box of a vending machine contains $23.75 in quarters and loonies. There are 35 coins in all.


How many of each coin are there in the coin box?

61. a) Write a linear system to model this situation.

Mrs. Cheechoo paid $155 for one-day tickets to Silverwood Theme Park for herself, her husband, and 3

children. Next month, she paid $285 for herself, 3 adults, and 5 children.


What are the prices of a one-day ticket for an adult and for a child?

62. a) Write a linear system to model the situation:

For the school play, the cost of one adult ticket is $6 and the cost of one student ticket is $4. Twice as

many student tickets as adult tickets were sold. The total receipts were $2016.

b) Use substitution to solve the related problem:

How many of each type of ticket were sold?

63. Use an elimination strategy to solve this linear system. Verify the solution.

64. a) Model this situation with a linear system:

To rent a car, a person is charged a daily rate and a fee for each kilometre driven. When Chena rented a

car for 15 days and drove 800 km, the charge was $715.00. When she rented the same car for 25 days and

drove 2250 km, the charge was $1512.50.

b) Determine the daily rate and the fee for each kilometre driven. Verify the solution.

65. Use an elimination strategy to solve this linear system. Verify the solution.

66. Explain what happens when you try to solve this linear system using an elimination strategy. What does this

tell you about the graphs of these equations?

67. Explain what happens when you try to solve this linear system using a substitution strategy. What does this

indicate about the graphs of these equations?

unit 7 review for final

Answer Section

MULTIPLE CHOICE

1. ANS: A PTS: 1 DIF: Easy

REF: 7.1 Developing Systems of Linear Equations LOC: 10.RF9

TOP: Relations and Functions KEY: Conceptual Understanding

2. ANS: C PTS: 1 DIF: Easy






4. ANS: D PTS: 1 DIF: Easy



5. ANS: A PTS: 1 DIF: Moderate






7. ANS: B PTS: 1 DIF: Moderate






9. ANS: B PTS: 1 DIF: Easy













REF: 7.2 Solving a System of Linear Equations Graphically LOC: 10.RF9














18. ANS: C PTS: 1 DIF: Moderate

























REF: 7.4 Using a Substitution Strategy to Solve a System of Linear Equations

LOC: 10.RF9 TOP: Relations and Functions KEY: Conceptual Understanding





























REF: 7.5 Using an Elimination Strategy to Solve a System of Linear Equations

LOC: 10.RF9 TOP: Relations and Functions KEY: Procedural Knowledge










40. ANS: D PTS: 1 DIF: Moderate










REF: 7.6 Properties of Systems of Linear Equations LOC: 10.RF9


























SHORT ANSWER

52. ANS:

a) There are 70 bills in a collection of $5 bills and $10 bills.

The value of the collection of bills is $575.

How many $5 bills and $10 bills are in the collection?

b) Variable f represents the number of $5 bills, and variable t represents the number of $10 bills.

PTS: 1 DIF: Moderate REF: 7.1 Developing Systems of Linear Equations


53. ANS:

(0, –8)

0 2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

PTS: 1 DIF: Easy REF: 7.2 Solving a System of Linear Equations Graphically


54. ANS:

a) b)

p + g = 25

2.5p + 3.25g = 70

The team purchased 15 pucks for practice and

10 pucks for games.

Pu

cks p

urc

hased

fo

r g

am

es

10

20

30

40

50

Pucks purchased for practice

0 10 20 30 40 50

p

g

PTS: 1 DIF: Moderate REF: 7.2 Solving a System of Linear Equations Graphically


55. ANS:

x = –55; y = –18

PTS: 1 DIF: Moderate



56. ANS:

x = –20; y = –16




57. ANS:

Let a represent the number of adult admissions, and y represent the number of youth admissions purchased.

a + y = 209

8a + 4.5y = 1304.5

104 adult admissions and 105 youth admissions were purchased.




58. ANS:

No solutions

PTS: 1 DIF: Easy REF: 7.6 Properties of Systems of Linear Equations


59. ANS:

One solution

PTS: 1 DIF: Easy REF: 7.6 Properties of Systems of Linear Equations


PROBLEM

60. ANS:

a) Let q represent the number of quarters, and l represent the number of loonies.

The value of q quarters is 25q cents, and the value of l loonies is 100l cents.

Then, a system of equations is:

q + l = 35

25q + 100l = 2375

b)

Nu

mb

er

of

loo

nie

s

10

20

30

40

50

Number of quarters

0 10 20 30 40 50

q

l

Since the intersection point is at (15, 20), there are 15 quarters and 20 loonies in the coin box.


LOC: 10.RF9 TOP: Relations and Functions KEY: Problem-Solving Skills

61. ANS:

a) Let a represent the cost in dollars for a one-day adult ticket, and c represent the cost in dollars for a

one-day child ticket.

Then, a system of equations is:

2a + 3c = 155

4a + 5c = 285

b)

Co

st

of

ch

ild

's t

icket

($)

10

20

30

40

50

Cost of adult's ticket ($)

0 10 20 30 40 50

a

c

Since the intersection point is at (40, 25), the cost of a one-day adult ticket is $40, and the cost of a

one-day child ticket is $25.



62. ANS:

a) Let a represent the number of adult tickets sold, and s represent the number of student tickets sold.

There were twice as many student tickets as adult tickets.

The first equation is:

2a = s

The total receipts were $2016.

The second equation is:

6a + 4s = 2016

The linear system is:

2a = s (1)

6a + 4s = 2016 (2)

b) Solve for s in equation (1).

2a = s (1)

s = 2a

Substitute s = 2a in equation (2).

6a + 4s = 2016 (2)

6a + 4(2a) = 2016

6a + 8a = 2016

14a = 2016

a =

a = 144

Substitute a = 144 in equation (1).

2a = s (1)

2(144) = s

288 = s

144 adult tickets and 288 student tickets were sold.



LOC: 10.RF9 TOP: Relations and Functions

KEY: Problem-Solving Skills | Communication

63. ANS:

Multiply equation ‚ by 2, then subtract to eliminate x.

2 equation :

Subtract equation ƒ from equation .

Substitute in equation .

Verify the solution.

In each equation, substitute: and

For each equation, the left side is equal to the right side, so the solution is: and




KEY: Communication | Problem-Solving Skills

64. ANS:

a) Let d dollars represent the daily rate and let k dollars represent the fee for each kilometre driven.

The linear system is:

b) Multiply equation by 25 and equation by 15, then subtract to eliminate d.

25 equation :

15 equation :

Subtract equation from equation .




So, the daily rate is $29 and the fee for each kilometre driven is $0.35.

PTS: 1 DIF: Difficult



65. ANS:

Multiply equation by 7, then add to eliminate y.

7 equation :

Add:




For each equation, the left side is equal to the right side, so the solution is: and





66. ANS:

Eliminate x first.

Multiply equation by 3, then add.

3 equation :

When I try to eliminate one variable, I eliminate the other variable and the constant term, so the equations

must be equivalent. This indicates that the graphs of these equations are coincident lines. So, the linear system

has infinite solutions.

PTS: 1 DIF: Moderate REF: 7.6 Properties of Systems of Linear Equations



67. ANS:

Solve equation for y:


does not equal , so the linear system has no solution. This tells me that the graphs of these equations

are parallel.

PTS: 1 DIF: Difficult REF: 7.6 Properties of Systems of Linear Equations



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