matrices and systems packet

7/28/2019 Matrices and Systems Packet

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Algebra 2 Level 1 Name: ________________________________

Unit Six: Matrices & Systems of Linear Inequalities Date: _________________________________

Matrix Package!

Day One: Applications of & Practice with Matrix Addition and Multiplication

Both Kate and Chris tutor students in math. They tutor individual students and groups. We can organize

this information using matrices, where the columns are the days of the week and the rows are the two

types of tutoring sessions. Here is their data from last week.

Kate M T W Th F Chris M T W Th FIndividual Students

Groups of StudentsK =

25500250

01020010

Individual Students

Groups of StudentsC =

2500050

0020300

How much did they make together each day last week?

K + C =

25500250

01020010+

2500050

0020300=

_______________

______403010

** This is an example of matrix addition.

They each used to charge $10/hr for individual students and $25/hr for small groups. Again, this can be

represented with a matrix.

Cost Dollars ($)

Individual Students

Groups of StudentsD =

25

10

They both decide to double the amount they charge. How much will they charge now?

New charges, N = 2*D =

50

20

25

102 ** This is an example of scalar multiplication.

Now Kate and Chris want to know how much money they made last week. So, first we have to total the

number of individuals and groups each saw.

Total students Individual students Group Sessions I G

Kate

ChrisT =

7550

10040

25000500020300

2550025001020010


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Now, we need to multiply this total matrix, T, by the new charges, N, to determine their earnings.

Earnings, E = T * N =

7550

10040*

50

20=

580050*10020*40

Multiplying the top row of matrix T by the first (and only) column of matrix N gives Kates earnings.

Earnings, E = T * N =

7550

10040*

50

20=

475050*7520*50

5800

Multiplying the bottom row of matrix T by the first (and only) column of matrix N gives Chris earnings.

** This is an example of matrix multiplication.

Part of the fee Kate and Chris charge is for the supplies they use during tutoring. Supplies cost $1 for

individuals and $3 for small groups. Lets create a matrix which stores the information for N, new charge;

S, the cost of supplies; and, P, the profit made.

N S P

Individuals

Small GroupsM =

22325

9110

So, to figure out how much each tutor charges in total, how much they each spend on supplies, and their total

profit, we do the following matrix multiplication.

T*M =

7550

10040*

22325

9110=


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Practice!

Simplify.

1)

371

641

063

724

2)

76

25

09

26

35

14

3)

280

471

231

*3

530

145

361

*4

4)

7

135

5)

711

346*

26

17

35

Lets check our answers with our calculators!!


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Day Two: Solving Systems using Inverse Matrices

Consider how we solve the equation: .To isolate the variable x, we need to eliminate the coefficient, A, either by dividing each

side by or multiplying each side by , the reciprocal or the multiplicative inverse.

Since

and

are multiplicative inverses, their product is 1.

Since 1 is the multiplicative identity and multiplying by it doesnt really change the

value, we dont write it as a coefficient. Voila were done! Our variable is the product

of the reciprocal of the coefficient and the constant.

The use of the multiplicative identity is very important here. As we advance in our study of mathematics and our

equations become more complicated and this identity becomes more complicated. Using the equation above, lets

let and . To find the multiplicative identity, consider a 2X2 matrix equation:

Intuitively we know that the variable matrix, , is the multiplicative identity because when we multiplied anothermatrix by it, nothing changed! Lets multiply the left-hand side of this matrix equation to get a system of equations that

we can then solve for a, b, c, and d.

Now we have the identity matrix, which is denoted by [] and we can confirm that [] [] [] with our calculators. Now we want the inverse of matrix A, which is denoted []. (Hey, where have we seen that kind of notation

before?) Since we know that [] [] [] [] [] we can use the inverse of a matrix to solve a realmatrix equation. To find this inverse, consider the matrix equation: [] [] []. Lets multiply the left-hand sideof this matrix equation to get a system of equations that we can then solve for a, b, c, and d, which now represent

the entries for the [].

Now that we have the inverse matrix, we can confirm that [] [] [] using our calculator.


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Not every square matrix has an inverse. Find the inverses of each of these matrices, if one exists. Make aconjecture about the type of 2X2 matrix that does not have an inverse.

a)

b)

c)

d)

Finally, lets apply all of this to solving systems of equations! {

More Practice!1) {

2)

3)

4)

5) {

Whats odd about these three systems?

6) {

7)

8) {


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Day Three: Graphing Systems Of Linear Inequalities

Review:

What is an inequality?

What is a compound sentence?

What is the intersection of two sets?

What is the union of two sets?

Graph the following inequalities.

Write a compound sentence that describes each graph.

Solve the following inequalities.


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Review: Graphing Systems of Equations

Is (-4, 4) a solution to this system? Justify your answer.

Graphing Systems of Linear Inequalities

Graph Graph

Graph

{


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Graphing Linear Inequalities on your calculator -

http://mathbits.com/mathbits/tisection/Algebra1/linearinequ2.htm

Example 1: Graph Enter 2x+ 1 into Y1 Arrow to the far left side ofY1 Hit ENTERuntil the shade above symbol is displayed. Hit ZOOM #6 ZStandard (for a 10x10 window) Graph NOTE: You will have to determine whether to draw a solid line

or a dotted line for y = 2x+ 1. This problem uses a solid line

because of the "less than or equal to" sign. The calculator will

display a solid line at all times. (The Application program

called Inequalities will display solid and dotted lines on the

graphs.)

Example 2: Graph Notice the "shade below" symbol to the far left of the Y1. Again, you must determine whether to draw a solid or dotted

line. This problem uses a dotted line because it is strictly "less

than" (no "equal to").

Example 3: Solve the system: { You will need to isolate the yvariable in the second equation

so that it can be entered into the calculator with the "shadeabove" indicated.

Solving algebraically, becomes and finally, (be careful of the direction of theinequality symbol in this problem.)

The answer is the double shaded region on the graph.

Example 4: Solve the system:

Enter all three equations with appropriate "shade above" or"shade below" symbol.

Answer is the darkest shading and is in Zoom Decimal view.


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Day Four: Linear Programming Problems

1) Translate the problem into mathematical language.a) Organize the data (a table may be helpful).b) Identify the unknown quantities and assign them variables.c) Translate the restrictions or constraints into linear inequalities.d) Form the Objective Function(what youre trying to maximize or minimize).

2) Graph the feasible set.a) Put the inequalities in slope-intercept form.b) Graph the lines corresponding to each inequality.c) Shade on the appropriate side.

3) Find the vertices of the feasible set (the Fundamental Theorem of Linear Programming states that the maximum (orminimum) value of the objective function is achieved at one of the vertices of the feasible set).

a) Find the intersections of the lines.4) Substitute each vertex into the objective function to determine the optimum point.5) Translate your optimum point into a sentence that answers the question.

EXAMPLE:

A furniture manufacturer makes two types of furniture- chairs and sofas. There are three distinct operations in

the production process- carpentry, finishing, and upholstery. The amount of labor required for each operation

varies. Manufacture of a chair requires 6 hours of carpentry, 1 hour of finishing, and 2 hours of upholstery.

Manufacture of a sofa requires 3 hours of carpentry, 1 hour of finishing, and 6 hours of upholstery. Owing to

limited availability of skilled labor as well as of tools and equipment, the factory has available each day 96

man-hours for carpentry, 18 man-hours for finishing, and 72 man-hours for upholstery. The profit per chair is

$80 and the profit per sofa is $70. How many chairs and how many sofas should be produced each day in

order to maximize the project?

1a)

Chair Sofa Available Labor

Carpentry 6 hours 3 hours 96 hours

Finishing 1 hour 1 hour 18 hours

Upholstery 2 hours 6 hours 72 hours

Profit $80 $70 xxxxxxxxxxxxx

1b) Let x= # of chairs produced per day. Let y= # of sofas produced per day.

1c) 6x+3y < 96 (Carpentry)

x + y < 18 (Finishing)

2x + 6y < 72 (Upholstery)

x < 0, y < 0 (because the fewest chairs or sofas possible is 0).

1d) We wish to maximize profit: P= 80x + 70y. In other words, what (x,y) satisfying the conditions above will

make P as big as possible?


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2a) 6x+3y = 96 y= -2x + 32

x + y = 18 y= x + 18

2x + 6y = 72 y = -(1/3)x + 12

2b-c)

Note: each point in the shaded region, called the feasible set,

fits within all three restrictions. The maximum profit will occur at

one of the vertices (labeled A-D below).

3) Point A- solve (14,4)

418)14(

14

32218

322

18

y

x

xx

xy

xy

Point D- solve

16

0322

322

0

x

x

xy

y

4)

VERTEX PROFIT = 80X + 70Y

(14,4) 80(14) + 70(4) = 1400

(9,9) 80(9) + 70(9) = 1350

(0,12) 80(0) + 70(12) = 840

(16,0) 80(16) + 70(0) = 1280

5) Solution: The factory should produce 14 chairs and 4 sofas each day in order to achieve maximum profit,

which would be $1400 per day.

carpentry

finishing

upholstery

C (0, 12)

B (9, 9)

A (14, 4)

D (16, 0)


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PRACTICE PROBLEMS:

1) A truck traveling from New York to Baltimore is to be loaded with two types of cargo. Each crate of cargo A

is 4 cubic feet in volume, weighs 100 pounds, and earns $13 for the driver. Each crate of cargo B is 3 cubic feet

in volume, weighs 200 pounds, and earns $9 for the driver. The truck can carry no more than 300 cubic feet of

crates and no more than 10,000 pounds. Also, the number of crates of cargo B must be less than or equal to

twice the number of crates of cargo A. How many crates of each cargo should be shipped in order to satisfy

the shipping requirements and yield the greatest earnings?

2) A coal company owns mines in two different locations. Each day mine 1 produces 4 tons of anthracite (hard

coal), 4 tons of ordinary coal, and 7 tons of bituminous (soft) coal. Each day mine 2 produces 10 tons of

anthracite, 5 tons of ordinary coal, and 5 tons of bituminous coal. It costs the company $150 per day to

operate mine 1 and $200 per day to operate mine 2. An order is received for 80 tons of anthracite, 60 tons of

ordinary coal, and 75 tons of bituminous coal. Find the number of days that each mine should be operated inorder to fill the order at the least cost.

A B Truck capacity

Volume

Weight

Earnings xxxxxxxxxxxx

Mine 1 Mine 2 Ordered

Anthracite

Ordinary

Bituminous

Daily Cost xxxxxxxxxxxx


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3) A contractor builds two types of homes. The first type requires one lot, $12,000 capital, and 150 man-days

of labor to build and is sold for a profit of $2400. The second type of home requires one lot, $32,000 capital,

and 200 man-days of labor to build and is sold for a profit of $3400. The contractor owns 150 lots and has

available for the job $2,880,000 capital and 24,000 man-days of labor. How many homes of each type should

she build in order to realize the greatest profit?

4) A nutritionist, working for NASA, must meet certain nutritional requirements and yet keep the weight of

food at a minimum. He is considering a combination of two foods which are packaged in tubes. Each tube of

food A contains 4 units of protein, 2 units of carbohydrate, 2 units of fat, and weighs 3 pounds. Each tube of

food B contains 3 units of protein, 6 units of carbohydrate, 1 units of fat, and weighs 2 pounds. The

requirement calls for 42 units of protein, 30 units of carbohydrate, and 18 units of fat. How many tubes of

each food should be supplied to the astronauts?

matrices and systems packet

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