Finite mathematics & Review exercises

Review exercises: Systems of linear equations and matrices

Copyright © 2000–2018 Stefan Waner 1/12

Finite mathematics & Applied calculus Review exercises: Systems of linear equations and matrices

1. In each of the following, solve the given system of linear equations. Enter the solution in coordinate form, as shown (if working online):

Type ofsolution Example Enter this

if online

Unique x = 1, y = − (1,-1/2)

Non-unique x = y − 2, y arbitrary (y-2,y)

Non-unique x arbitrary, y = (x,(2x-3)/4)

Nosolution ns

(a) −x + y = 5

2x + y = −4

Solution: (-3,2)

(b) 5x − 2y = 2

x − y = 1

Solution: ((2+2y)/5,y)

(c) 4x − 5y = 2

2x − y = −2

Solution: ns

(d) x − y =

0.4x + 0.8y = −5

Solution: (-9/2,-4)

(e) −0.6x + 0.4y = 0.6

x − y = −1

Solution: ((-3+2y)/3,y)

21

42x − 3

−3, 2( )

25

, y(5

2 + 2y )

25

ns

41

21 8

7

− , −4(29 )

32

, y(3

−3 + 2y )



2. A factory makes both basketballs and soccer balls. Each basketball requires 2 minutes on the forming machine while each soccer ball requires 1 minute.Further, each basketball requires 1 second on the inflating machine, while each soccer ball requires 1.5 seconds. Yesterday, the forming machine ran for only 60minutes before breaking down while the inflating machine exploded after exactly 60 seconds. What is the total number of basketballs and soccer balls that weremade yesterday?

Solution: 15 basketballs

30 soccer balls

Let x be the number of basketballs and let y be the number of soccer balls. First, set up the given data in a table:

basketballs soccerballs Total

Formingmachine

(Minutes)2 1 60

Inflatingmacine

(Seconds)1 1.5 60

This leads to the system2x + y = 60 x + 1.5y = 60

which has solution (15,30).



3. Your portfolio manager has suggested the following two companies for investment purposes: Gamma Gum and Beta Bank. Gamma Gum shares cost $36 pershare, while Beta Bank shares cost $45 per share. You have $12,600 to invest and wish to hold twice as many Beta Bank shares as Gamma Gum shares. Howmany shares of each company should you buy?

Solution: 100 shares of Gamma Gum

200 shares of Beta Bank

Let x be the number of Gamma Gum and let y be the number of Beta Bank. The cost data can be set up in a table:

GammaGum

BetaBank Total

Cost (Dollars) 36 45 12,600This gives one equation

36x + 45y = 12, 600Dividing by 9 gives

4x + 5y = 1, 400For the second equation, use the fact that you wish to hold twice as many Beta Bank shares as Gamma Gum shares. Rephrasing this as per the textbook gi

The number of Beta Bank shares is twice the number of Gamma Gum shares: y = 2x ⇒ − 2x + y = 0

Thus we have a system of two equations in two unknowns:4x + 5y = 1, 400 −2x + y = 0




4. At the beginning of last year, you purchased Epsilon NanoTech and Gamma Gum. The Epsilon NanoTech shares cost you $14 per share and paid 1% individends for the year, while Gamma Gum shares cost you $4 per share and paid 3% in dividends for the year. If you invested a total of $3,500 and earned $63 individends at the end of the year, how many shares of each company did you purchase?

Solution: 150 shares of Epsilon NanoTech

350 shares of Gamma Gum

Let x be the number of Epsilon NanoTech and let y be the number of Gamma Gum. The cost data can be set up in a table:

EpsilonNanoTech

GammaGum Total

Cost(Dollars) 14 4 3500

This gives one equation14x + 4y = 3, 500

Dividends:Epsilon NanoTech: 1% of a total cost of $ 14x = (0.01)(14x) = 0.14x Gamma Gum: 3% of a total cost of $ 4y = (0.03)(4y) = 0.12y

This gives:0.14x + 0.12y = 63

Thus we have a system of two equations in two unknowns:14x + 4y = 3, 500 0.14x + 0.12y = 63




5. In each of the following, find the augmented matrix of the system. Note Successive steps in the process will only become visible as you progress (if working online).

(a) Consider the system

6x + 3y + 3z = −8

8x + 3y + 4z = −11

−24x − 9y − 16z = 37.

The augmented matrix of the system is

6 3 3 -8

8 3 4 -11

-24 -9 -16 37

(b) Consider the system

6x + 3y − z = −5

8x + 3y − 2z = −7

−8x − 3y + 2z = 7.

The augmented matrix of the system is

6 3 -1 -5

8 3 -2 -7

-8 -3 2 7



6. In each of the following, solve the given system of linear equations. Enter the solution in coordinate form, as shown (if working online):

Type ofsolution Example Enter this

if online

Unique x = 1, y = − .z = 4 (1,-1/2,4)

Non-unique x = z − 2, y = 2(z + 4), z arbitrary (z-2,2(z+4),z)

Non-unique x = y − z + 2, y and z arbitrary (y-z+2,y,z)

Nosolution ns

(a) −6x + y = −13

12x − 3y + 4z = 26

−18x + 4y − 8z = −35

Solution: (3/2,-4,-1)

(b) −6x + y = −13

12x − 3y + 3z = 30

−18x + 4y − 3z = −43

Solution: (1/2(3+z),-4+3z,z)

(c) −0.06x + 0.01y = −0.13

2x − y + z =

−3.6x + 0.8y − 1.6z = −7

Solution: (3/2,-4,-1)

(d) −6x + y − 13u = −8

6x − 2y + 3z + 17u = 4

y − 3z − 4u = 4

−6x + y − 13u = −8

Solution: (1/2(4+z-3u),4+7z,z,u)

21

, −4, −1(23 )

(3 + z), −4 + 3z, z(21 )

21 3

2 3

13

, −4, −1(23 )

(4 + z − 3u), 4 + 7z, z,u(21 )



7. Your nutritionist, Jeff Mongillo, has decided that you need 3000 I.U. of Vitamin A, 600 mg. of Vitamin C, and 800 I.U. of Vitamin D per day,† andrecommends the following supplements: MegaSupra containing 500 I.U. of Vitamin A, 100 mg. of Vitamin C, and 200 I.U. of Vitamin D per capsule,CrazyCaps, containing 500 I.U. of Vitamin A, 200 mg. of Vitamin C, and 200 I.U. of Vitamin D per capsule, and AC DFrees, containing 500 I.U. of Vitamin A,50 mg. of Vitamin C, but no vitamin D.†The current (2015) US Recommended Daily allowances for these supplements are: Vitamin A: around 3,000 I.U., Vitamin C: around 85 mg., and Vitamin D: around 650 I.U. Manynutritionists feel that the Vitamin C and D RDAs should at least be doubled.

How many of each should you take to obtain exactly Mongillo's recommended daily dosages?Setting up the given data in a table gives

MegaSupra CrazyCaps AC DFrees Total

Vit A (I.U.) 500 500 500 3000

Vit C (mg) 100 200 50 600

Vit D (I.U.) 200 200 0 800

Solve the associated system to get the solution.

3 MegaSupras

1 CrazyCaps

2 AC DFrees



8. The top eyeglass retailers last year were XrayVision, YesSpecs and ZoomGlasses. Together, they accounted for 66% of total market sales. The combined shareof XrayVision and YesSpecs was twice the share of ZoomGlasses, and the share of YesSpecs is less than the combined share of ZoomGlasses and XrayVision by2.

What was the market share of each of the three companies?Take x = market share (%) of XrayVision, y that of YesSpecs, and z that ofZoomGlasses. The statement "Together, they accounted for 66% of total market sales"tells us that

x + y + z = 66.The statement "The combined share of XrayVision and YesSpecs wastwice the share of ZoomGlasses" tells us that

x + y = 2zThe statement "the share of YesSpecs is less than the combined share ofZoomGlasses and XrayVision by 2" tells us that

y = z + x − 2Solve the associated system to get the solution.

XrayVision: 12 %

YesSpecs: 32 %

ZoomGlasses: 22 %



9. Your portfolio manager has suggested the following three companies for investment purposes: Epsilon NanoTech, Alpha Centauri, and MuMu Dairies.Epsilon NanoTech shares cost $20 per share, Alpha Centauri shares cost $25 per share, and MuMu Dairies shares cost $25 per share. You have $9,250 to invest,wish to hold twice as many Epsilon NanoTech shares as Alpha Centauri shares, and (for reasons too complicated to explain) a total of 410 shares.

How many shares of each company should you buy?Take x = number of shares of Epsilon NanoTech, y = the number ofshares of Alpha Centauri, and z = the number of shares of MuMuDairies. The statement "You have $9,250 to invest" tells us that

20x + 25y + 25z = 9, 250.The statement "You wish to hold twice as many Epsilon NanoTechshares as Alpha Centauri shares" tells us that

x = 2yThe statement that you want to hold a total of 410 shares tells us that

x + y + z = 410Solve the associated system to get the solution.

Solution: 200 shares of Epsilon NanoTech

100 shares of Alpha Centauri

110 shares of MuMu Dairies



10. You run a surf fashion store, D'Amico Surf, in South Park Mall. Two days ago you were notified of an upcoming vacancy in another part of the Mall, and youare seriously considering relocating D'Amico Surf there as you suspect that there is more traffic in that area. Unfortunately, you can't spare one of your staff tocount the total customer traffic going past the locations, but you do have partial information based on data provided by the South Park Mall management, asshown in the following diagram.

The mall has a curious policy (the mall supervisor is a control freak) that all shopper traffic has to be one-way in the direction of the arrows.

y

s

x

z

t

40

40 50

60?

?NewLocation

PresentLocation

Burger King

GNC

Apple

Soriana

Starbucks

Abercrombie& Fitch

Telcel

Walmart

Macy's

McDonalds

If we calculate traffic in and traffic out at each intersection, we get

Intersection Traffic in Traffic out

Near Burger King y+s 40+t

Near Walmart z x+s+40

Near Apple t+50 z+60

Equating traffic in and traffic out gives a system of three equations:

yb + sb − tb = 40 xb − zb + sb = −40 zb − tb = −10

Solve this system to get the general solution.

You first set up a system of linear equations with 5 unknowns, and solve it.

General Solution (based on reduced matrix with unknowns in alphabetical order): xb = -s+t-50

yb = -s+t+40

zb = t-10 s = s

tb = t

Change in traffic from present location to new location (may be negative): Answer: 90

Change in traffic from present location to new location= yb − xb = −sb + tb + 40 − (−sb + tb − 50) = 90



11.

We are given z = 0, and want the minimum value of x.

Substituting 0 for z in the general solution for x gives

x = 0 + w + 100,

which can have any value ≥ 100.

We are given w = 0, and want the maximum value of z.

Substituting 0 for w in the general solution for y gives

y = −z + 0 + 75,

which needs to be ≥ 0, meaning that z can be no more than 75.

The following system of one-way streets appears in Gigantic State University's Fraternity Row:

Rush St

The Pit

Pledge Ave

Fasttrack Ave

Depledge Blvd

Scholars Path

BrothersBlvd

IndependenceTpke

x y

z100

25

75

w

a. Enter linear equations representing hourly traffic flow at the given intersections:

Resulting linear equations in the form ax + by + cz = d :

Leftmost intersection:

Top intersection:

Rightmost intersection:

x+z-w=100

x-y=25

y+z-w=75

b. General solution: , , ( -z+w+100 , -z+w+75 z w )

c. Find the particular solution if 85 cars per hour drive down Fasttrack Ave and 10 cars per hour drive down Depledge Blvd.

, , Answer: ( 25 , 0 85 10 )

d. Determine the following from the general solution:

If there is no traffic along Fasttrack Ave, what is the minimum possible hourly traffic along Rush St?

Answer: cars/hour100

If there is no traffic along Depledge Ave, what is the maximum possible hourly traffic along Fasttrack Ave? Enter dne if there is no maximum traffic along that road.

Answer: cars/hour75



12. The following diagram shows part of an exclusive neighborhood in Utarek City on Mars. The arrows indicate direction oftraffic flow (the streets are one-way) and the boxed numbers measure traffic (in Mars Volta vehicles per hour).

200 50

50

100

Xavier

Yvette

Zach x

yy zYvette

Enter linear equations representing hourly traffic flow at the given intersections:

Resulting linear equations in the form ax + by + cz = d :

Corner Xavier and Yvette:

Corner Xavier and Zach:

Corner Zach and Yvette:

-x+y=200

-x+z=100

y-z=100

General solution: , , ( z-100 z+100 z )

We deduce the following from the general solution:

Minimum traffic along Xavier = Volta vehicles/hour

Minimum traffic along Yvette = Volta vehicles/hour

Minimum traffic along Zach = Volta vehicles/hour

0

200

100

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