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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
3 Summer 2018, Maya Johnson
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
4 Summer 2018, Maya Johnson
⇒
- 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.
5 Summer 2018, Maya Johnson
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.
6 Summer 2018, Maya Johnson
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.
7 Summer 2018, Maya Johnson
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
8 Summer 2018, Maya Johnson
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.
9 Summer 2018, Maya Johnson
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.
Example 2: Solve the following system of equations
x+ y + z = 1
3x� y � z = 4
x+ 5y + 5z = �1
10 Summer 2018, Maya Johnson
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.
Example 3: Solve the following system of equations
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.
Example 4: Solve the following system of equations
14x+ 2y = �10
20x� 4y = 20
�6x+ 6y = �30
11 Summer 2018, Maya Johnson
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
12 Summer 2018, Maya Johnson
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
13 Summer 2018, Maya Johnson
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
#
14 Summer 2018, Maya Johnson
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
#
15 Summer 2018, Maya Johnson
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.
16 Summer 2018, Maya Johnson
-
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
17 Summer 2018, Maya Johnson
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
#
18 Summer 2018, Maya Johnson
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 $10 for adults. The attendance
for the Sunday matinee is given by the matrix
A =
Cinema I
Cinema II
Cinema III
Cinema IV
2
6666664
Children Students Adults
245 120 70
95 170 245
280 75 120
0 250 245
3
7777775.
Write a column vector B representing the admission prices.
Compute AB, the column vector showing the gross receipts for each theater.
Find the total revenue collected at the Cinema Center for admission that Sunday afternoon.
19 Summer 2018, Maya Johnson
3130
438034804450
3130 +4386 +3480 +4450 =$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?
20 Summer 2018, Maya Johnson
people x fund type4
fund type x Returns
Returns
AB = Aghgayy
;
Hyogo]
Best is Maria 1biggest returns )
Worst is Alan ( smallest returns )