chapter 2 section 2.1 systems of linear equations: an ...mayaj/m141_chapter2_sum18101.pdfsection 2.2...

Chapter 2

Section 2.1 Systems of Linear Equations: An Introduction

Definition A system of linear equations is a collection of multiple linear equations which are meant to

be solved at the same time or simultaneously.

Question What does it mean to solve a system of linear equations?

Answer: To solve a system of equations means to find all for the unknowns that

satisfy EVERY equation. To solve a system of linear equations, where all the equations are lines,

means to find every that the share.

We actually saw in the last section that solving a system of only two linear equations and two unknowns

(or variables) is referred to as the intersection of two lines.

Before continueing with solving systems of equations, we will first discuss how to setup a system of

equations from a word problem.

For the following three examples, we will setup but not solve the resulting system of equations.

Example 1: An insurance company has three types of documents to process: contracts, leases, and

policies. Each contract needs to be examined for 2 hours by the accountant and for 3 hours by the

attorney, each lease needs to be examined for 4 hours by the accountant and 1 hour by the attorney,

and each policy needs to be examined for 2 hours by the accountant and 2 hour by the attorney. The

company processes twice as many policies as contracts and leases combined. If the accountant has

40 hours and the attorney has 30 hours each week to spend working on these documents, how many

documents of each type can they process each week?

Note: ALWAYS define your variables when setting up a problem

Example 2: The Johnson Farm has 500 acres of land allotted for cultivating corn and wheat. The cost

of cultivating corn and wheat (including seeds and labor) is $44 and $28 per acre, respectively. Jacob

Johnson has $15, 600 available for cultivating these crops. If he wishes to use all the allotted land and

his entire budget for cultivating these two crops, how many acres of each crop should he plant?

Example 3: The management of Hartman Rent-A-Car has allocated $2.43 million to buy a fleet of new

automobiles consisting of compact, intermediate-size, and full-size cars. Compacts cost $18, 000 each,

intermediate-size cars cost $27, 000 each, and full-size cars cost $36, 000 each. If Hartman purchases

twice as many compacts as intermediate-size cars and the total number of cars to be purchased is 100,

determine how many cars of each type will be purchased. (Assume that the entire budget will be used.)

2 Summer 2018, Maya Johnson

±"

# of compact ears purchased"

y="

# of intermediate - sized cars purchased"

z=u # of full-sized cars purchased"

18000 " [email protected]

, , , , |•o

X = ZyX=2y → Twree as many

Compacts as intermediates

Let’s return to solutions of a system of equations.

Only Three Possible Outcomes for a system of Linear Equations

a) The system has one and only one solution. (Unique solution)

b) The system has infinitely many solutions.

c) The system has no solution.

Unique Solution:

3x+ 3y = 6

�2x+ y = 2

Infinitely Many Solutions:

2x+ 2y = 4

4x+ 4y = 8


No Solution:

2x+ 3y = 6

�2x� 3y = 2

Example 4: Determine whether the system of linear equations has one and only one solution, infinitely

many solutions, or no solution.54x� 2

3y = 614x+ 5

3y = 12

Example 5: Determine the value of k for which the system of linear equations below has no solution.

3x� y = 3

9x+ ky = 6


⇒

- syloxisst

* 8. Y= -9+158187=6

-3¥- Egx -3¥ . ,¥×

⇒ rises .4FF8 X=8,y=6(8,=f

Section 2.2 Systems of Linear Equations: Unique Solutions

A is an ordered rectangular array of numbers.

Augmented Matrices The system of equations

2x+ 4y � 8z = 22

3x� 8y + 5z = 27

x� 7z = 33

can be represented as the following augmented matrix

2

642 4 �8

3 �8 5

1 0 �7

��

22

27

33

3

75

Example 1: What value is in row 1, column 2 of the above matrix?

Example 2: Find the augmented matrix for the following system of equations.

9x+ 5y � 10z = 11

4x� 12y + 17z = 37

x� 2y = 45

Example 3: Find the system of equations for the following augmented matrix.

2

6410 0 �6

30 �9 0

1 19 �12

��

29

31

10

3

75

In order to solve the system, we need to “reduce” the matrix to a form where we can readily identify

the solution.


A Matrix is in Row-Reduced Form when:

1. Each row consisting entirely of zeros lies below all rows having nonzero entries

2. The first nonzero entry in each (nonzero) row is a 1 (called a leading 1).

3. In any two successive (nonzero) rows, the leading 1 in the lower row lies to the right of the leading

1 in the upper row.

4. If a column in the coe�cient matrix has a leading 1, then the other entries in the column are

zeros.

Example 4: Which of the matrices below are in row-reduced form?

2

641 0 �6

0 1 8

0 0 0

��

9

1

0

3

75

2

641 0 �6

0 0 0

0 1 �12

��

2

0

6

3

75

Row Operations

1. Interchange any two rows.

2. Replace any row by a nonzero constant multiple of itself.

3. Replace any row by the sum of that row and a constant multiple of any other row.

Notation for Row Operations Letting Ri denote the ith row of a matrix, we write:

Operation 1. Ri $ Rj Interchange row i with row j.

Operation 2. cRi to mean: Replace row i with c times row i.

Operation 3. Ri + aRj to mean: Replace row i with the sum of row i and a times row j.

Unit Column A column in a coe�cient matrix is called a unit column if one of the entries is a 1

and the other entries are zeros.

Note: If you transform a column in a coe�cient matrix into a unit column then this is called pivotting

on that column.


Example 5: Pivot the matrix below about the entry in row 1, column 12

643 6 12

2 2 1

�4 5 2

��

9

3

�8

3

75

The Gauss-Jordan Elimination Method

1. Write the augmented matrix corresponding to the Linear system.

2. Begin by transforming the entry in row 1 column 1 into a 1. This is your first pivot element.

3. Next, transform every other entry in column 1 into a zero using the (3) row operations. (Make

column 1 a unit column)

4. Choose the next pivot element (usually element in row 2 column 2)

5. Transform this 2nd pivot element into a 1, and every other entry in that column into a zero.

6. Continue until the final matrix is in row-reduced form.

You can determine the solution from the row-reduced matrix by turning it back into a

system of equations.


Example 6: Solve the following system of linear equations using the Gauss-Jordan elimination method.

a) 2x+ 6y = 1

�6x+ 8y = 10

From this moment on, you may use the calculator function “rref” to perform the

gauss-jordan elimination method to put a matrix into row-reduced form, and thus

solve the system of equations!!! Calculator steps for using “rref” can be found in a link

directly under these lecture notes on the course webpage.

b) 2x+ 2y = 4

�3x+ 6y = 5


c) 2x1 + x2 � x3 = 3

3x1 + 2x2 + x3 = 8

x1 + 2x2 + 2x3 = 4

Example 7: A person has four times as many pennies as dimes. If the total face value of these coins

is $1.26, how many of each type of coin does this person have? (Use gauss-jordan )

Example 8: Cantwell Associates, a real estate developer, is planning to build a new apartment complex

consisting of one-bedroom units and two- and three-bedroom townhouses. A total of 168 units is

planned, and the number of family units (two- and three-bedroom townhouses) will equal the number

of one-bedroom units. If the number of one-bedroom units will be 3 times the number of three-bedroom

units, find how many units of each type will be in the complex.


Diskin. EMI

⇐"

# of one - bedroom units "

y=" # of two - bedroom units

"

Z=" # of three - bedroom

units"

X + y t Z = 168 Xty + z = 168

ytz = × =) - × + y +2=0

- 3z = 0

X =3 Z X

Hoiseth :eEn

84oue-bedro•mun'=56 two - bedroom unitszgthme-bedroomun€)

Section 2.3 Systems of Linear Equations: Underdetermined and Overdetermined Systems

Infinitely Many Solutions: If an augmented coe�cient matrix is in row-reduced form and there

is at least one row which consists entirely of zeros, then,in most cases , the system has infinitely

many solutions and we use parameter t and/or s to write the solution.

Note: The case when this assumption is not always true is when the system is overdetermined or

underdetermined.

Example 1: Solve the following system of equations

x+ 2y � 3z = �2

3x� y � 2z = 1

2x+ 3y � 5z = �3

No solution If an augmented coe�cient matrix is in row-reduced form and there is at least one row

which consists entirely of zeros to the left of the vertical line and a nonzero entry to the right of the

line (the very last entry on that row), then the system has no solution.


x+ y + z = 1

3x� y � z = 4

x+ 5y + 5z = �1


Underdetermined System A system is underdetermined if there are equations

than there are variables.

Note: An underdetermined system can have no solution or infinitely many solutions.


x+ 2y + 8z = 6

x+ y + 4z = 3

Overdetermined System A system is overdetermined if there are equations than

there are variables.

Note: An overdetermined system can have a unique solution, no solution or infinitely many

solutions.


14x+ 2y = �10

20x� 4y = 20

�6x+ 6y = �30


Example 5: Solve the following systems of equations. (If there are infinitely many solutions, enter a

parametric solution using t and/or s).

a) 3y + 2z = 1

2x� y � 3z = 4

2x+ 2y � z = 5

b) 3x� 2y + 4z = 23

2x+ y � 2z = �1

x+ 4y � 8z = �25

c) 2x+ 2y + 2z = 10

8x+ 8y + 8z = 33

4x+ 5y + 3z = 23


t.it#Ed*Eio:oMx=3=) II?n+2t

y -27=-7 z=t

z=t ,at any

real mmW(z,n+2#

t.mu#toEktxNosolut@

Section 2.4 Matrices

What is a Matrix? A matrix is an ordered rectangular array of numbers. A matrix with m rows

and n columns has size m ⇥ n. The entry in the ith row and jth column of a matrix A is denoted by

aij.

Note: If A is an n⇥ n matrix, then we say A is a square matrix.

Example 1: Given the matrix

A =

2

66664

2 4 �8 �5

3 �8 5 2

1 0 �7 6

9 18 7 �10

3

77775

a) what is the size of A?

b) find a14, a21, a31, and a43

Equality of Matrices Matrices A and B are equal if and only if they have the same size and they

have the same corresponding entries (i.e. aij = bij for all values of i and j).

Example 2: Are the two matrices below equal?

A =

2

642 4 �8

3 �8 5

1 0 �7

3

75 B =

2

642 (5� 1) �8

3 �8 (2 + 3)

(12� 11) 0 �7

3

75

Example 3: If we know the matrices below are equal, find x, y, and z.

2

64x 9 2

3 5 y

10 z �6

3

75 =

2

6419 9 2

3 5 24� y

10 �z + 2 �6

3

75


Adding and Subtracting Matrices If matrices A and B have the same size (both m⇥ n matrices)

then:

1. The sum A+B is obtained by adding the corresponding entries in both matrices (aij + bij for all

values of i and j) and the resulting matrix is still an m⇥ n matrix.

2. The di↵erence A�B is obtained by subtracting the corresponding entries in both matrices (aij�bij

for all values of i and j) and the resulting matrix is still an m⇥ n matrix.

Note: You CANNOT add or subtract two matrices that have di↵erent sizes. Also, A + B = B + A

BUT A� B 6= B � A.

Scalar Product If c is a real number and A is an m⇥n matrix, then the scalar product cA is obtained

by multiplying every entry in A by c (caij for all values of i and j) and the resulting matrix is still an

m⇥ n matrix.

Example 4: Perform the indicated operations.

2

2

641 1 2

3 1 1

2 3 �2

3

75+ 3

2

6410 9 6

2 5 2

1 �3 �1

3

75

Transpose of a Matrix The transpose of a matrix A, denoted AT , is obtained by interchange the

rows and the columns of A. Therefore, if A is an m ⇥ n matrix with entries aij then AT is an n ⇥m

matrix with entries aji.

Example 5: Find the transpose of the matrix.

"8 6 0 �1

5 8 �1 9

#


Example 6: Matrix L is a 4⇥ 7 matrix, matrix M is a 7⇥ 7 matrix, matrix N is a 4⇥ 4 matrix, and

matrix P is a 7 ⇥ 4 matrix. Find the dimensions of the sums below, if they exist. (If an answer does

not exist, write DNE.)

a) L+M

b) L+ P T

c) M +N

d) N +N

Example 7: Find the values of a, b, c, and d in the matrix equation below.

"a b

c d

#+ 3

"2 4

3 5

#T

=

"0 9

20 10

#


Example 8:

The Campus Bookstore’s inventory of books is as follows.

Hardcover: textbooks, 5119; fiction, 1948; nonfiction, 2234; reference, 1514

Paperback: fiction, 2572; nonfiction, 1572; reference, 2223; textbooks, 1849

The College Bookstore’s inventory of books is as follows.

Hardcover: textbooks, 6298; fiction, 2054; nonfiction, 1986; reference, 1839

Paperback: fiction, 3033; nonfiction, 1719; reference, 2850; textbooks, 2477

a) Represent the Campus’s inventory as a matrix A.

b) Represent the College’s inventory as a matrix B.

c) The two companies decide to merge, so now write a matrix C that represents the total inventory

of the newly amalgamated company.


-

6298 2054 1986 1839

2477 3033 1719 2850

, | 11417 4002 4220 3353

]L 4326 5605 3291 5073

Section 2.5 Multiplication of Matrices

Matrix Product For an m⇥ p matrix A and a p⇥ n matrix B, the product AB is an m⇥ n matrix.

Note: If the number of columns of A are NOT the same as the number of rows of B then the product

AB is NOT defined.

Example 1: If A is a 5 ⇥ 8 matrix and B is a 8 ⇥ 6 matrix, find the sizes of AB and BA whenever

they are defined.

Multiplying a 1⇥ n and an n⇥ 1 Matrix Suppose A is a 1⇥ n matrix

A =ha11 a12 . . . a1n

i

and B is an n⇥ 1 matrix

B =

2

66664

b11

b21...

bn1

3

77775

then the product AB is a 1⇥ 1 matrix given by

AB =ha11 a12 . . . a1n

i

2

66664

b11

b21...

bn1

3

77775= a11b11 + a12b21 + · · ·+ a1nbn1

Example 2: Find the product AB if it is defined for

A =h2 �1 4

i, B =

2

64�3

0

6

3

75


Suppose

C = AB =

"a11 a12 a13

a21 a22 a23

#2

64b11 b12

b21 b22

b31 b32

3

75

then the entry c11 is the product of the row matrix composed of the entries in the first row of A and

the column matrix composed of the entries in the first column of B

c11 =ha11 a12 a13

i2

64b11

b21

b31

3

75 = a11b11 + a12b21 + a13b31.

The other entries of C can be computed similarly.

Example 3: Compute the indicated product.

"10 6 3

1 9 10

#2

64�6 6

5 9

3 2

3

75

Example 4: Compute the indicated product.

"9 2x

5y 7

#"3 9

13 14

#


Example 5: Find the values of x, y, and z.

"x 2 1

0 y 3

#2

641 1

3 z

4 2

3

75 =

"8 2

0 2

#

Matrix Representation We can use matrices to represent data and to compute desired quantities in

real world situations.

Example 6: The Cinema Center consists of four theaters: Cinemas I, II, III, and IV. The admission

price for one feature at the Center is $6 for children, $8 for students, and $10 for adults. The attendance

for the Sunday matinee is given by the matrix

A =

Cinema I

Cinema II

Cinema III

Cinema IV

2

6666664

Children Students Adults

245 120 70

95 170 245

280 75 120

0 250 245

3

7777775.

Write a column vector B representing the admission prices.

Compute AB, the column vector showing the gross receipts for each theater.

Find the total revenue collected at the Cinema Center for admission that Sunday afternoon.


3130

438034804450

3130 +4386 +3480 +4450 =$l5,4u@

Example 7: Three network consultants, Alan, Maria, and Steven, each received a year -end bonus of

$10, 000, which they decided to invest in a 401(k) retirement plan sponsored by their employer. Under

this plan, employees are allowed to place their investments in three funds: an equity index fund (I), a

growth fund (II), and a global equity fund (III). The allocations of the investments (in dollars) of the

three employees at the beginning of the year are summarized in the matrix

A =Alan

Maria

Steven

2

66664

I II III

4000 2000 4000

3000 5000 2000

3000 3000 2000

3

77775.

The returns of the three funds after 1 yr are given in the matrix

B =I

II

III

2

66664

Return

0.15

0.23

0.13

3

77775.

Which employee realized the best return on his or her investment for the year in question?

Which employee realized the worst return on his or her investment for the year in question?


people x fund type4

fund type x Returns

Returns

AB = Aghgayy

;

Hyogo]

Best is Maria 1biggest returns )

Worst is Alan ( smallest returns )

chapter 2 section 2.1 systems of linear equations: an ...mayaj/m141_chapter2_sum18101.pdfsection 2.2...

Documents