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.)
" # of " compact ears purchased " "
# of - sized y= intermediate cars purchased " z=u of # full-sized cars purchased
, , , |•o ± , X = Zy
→ X=2y Twree as many
Compacts as intermediates
2 Summer 2018, Maya Johnson 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. 5 2 4 x 3 y =6 1 5 4 x + 3 y =12
* 8. Y= -9+158187=6
- syloxisst ⇒
⇒ rises .4FF8 X=8,y=6(8,=f - -3¥ . -3¥ Egx ,¥× 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 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
24 8 22 2 3 85 27 3 10 7 33 6 7 4 5 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.
10 0 6 29 2 30 90 31 3 119 12 10 6 7 4 5
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 1intheupperrow.
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?
10 6 9 10 6 2 2 01 8 1 3 2 00 0 0 3 00 0 0 01 12 6 6 7 6 7 4 5 4 5
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. R R Interchange row i with row j. i $ 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 Acolumninacoe cientmatrixiscalledaunit 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 1 3612 9 2 221 3 3 45 2 8 6 7 4 5
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) 2x + x x =3 1 2 3 3x1 +2x2 + x3 =8 x1 +2xDiskin2 +2x3 =4 . EMI 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.
" " # of one - bedroom units ⇐ "
" of two - bedroom units y= # " units " - bedroom Z= # of three
+ z = 168 X + t Z = 168 Xty y + +2=0 - × = × =) y ytz = 0 - 3z X =3 Z X Hoiseth :eEn
units - bedroom zgthme-bedroomun€)84oue-bedro•mun'=56 two
9 Summer 2018, Maya Johnson 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 ,thesystemhasinfinitely 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 t.it#Ed*Eio:oMx=3
=) II?n+2t
- 27=-7 y z=t at real mmW(z,n+2# z=t , any
c) 2x +2y +2z =10 8x +8y +8z =33 4x +5y +3z =23 t.mu#toEktxNosolut@
12 Summer 2018, Maya Johnson 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.Theentryintheith 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 24 8 5 2 3 85 23 A = 6 10 767 6 7 6 9187 10 7 6 7 4 5 a) what is the size of A?
b) find a14, a21, a31,anda43
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?
24 8 2(51) 8 A = 3 85 B = 3 8(2+3) 2 3 2 3 10 7 (12 11) 0 7 6 7 6 7 4 5 4 5
Example 3: If we know the matrices below are equal, find x, y,andz.
x 92 19 9 2 35 y = 3524y 2 3 2 3 10 z 6 10 z +2 6 6 7 6 7 4 5 4 5
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)andtheresultingmatrixisstillanm n matrix. ⇥ 2. The di↵erence A B is obtained by subtracting the corresponding entries in both matrices (a b ij ij for all values of i and j)andtheresultingmatrixisstillanm n matrix. ⇥ Note: You CANNOT add or subtract two matrices that have di↵erent sizes. Also, A + B = B + A BUT A B = B A. 6 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.
11 2 10 9 6 2 2 31 13 +32 25 23 23 2 1 3 1 6 7 6 7 4 5 4 5
Transpose of a Matrix The transpose of a matrix A,denotedAT ,isobtainedbyinterchangethe rows and the columns of A.Therefore,ifA is an m n matrix with entries a then AT is an n m ⇥ ij ⇥ matrix with entries aji.
Example 5: Find the transpose of the matrix.
86 0 1 " 58 19#
14 Summer 2018, Maya Johnson Example 6: Matrix L is a 4 7matrix,matrixM is a 7 7matrix,matrixN is a 4 4matrix,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,andd in the matrix equation below.
T ab 24 09 +3 = " cd# " 35# " 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.
6298 2054 1986 1839 2850 2477 3033 1719 c) The two companies decide to merge, so now write a matrix C that represents the total inventory of the newly amalgamated company.
4220 3353 11417 4002 , | 4326 5605 3291 5073 ] L
16 Summer 2018, Maya Johnson Section 2.5 Multiplication of Matrices
Matrix Product For an m p matrix A and a p n matrix B,theproductAB 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 8matrixandB is a 8 6matrix,findthesizesofAB and BA whenever ⇥ ⇥ they are defined.
Multiplying a 1 n and an n 1 Matrix Suppose A is a 1 n matrix ⇥ ⇥ ⇥
A = a11 a12 ... a1n h i and B is an n 1matrix ⇥
b11 2 b21 3 B = . 6 . 7 6 7 6 b 7 6 n1 7 4 5 then the product AB is a 1 1matrixgivenby ⇥
b11 2 b21 3 AB = a a ... a = a b + a b + + a b 11 12 1n . 11 11 12 21 ··· 1n n1 6 . 7 h i 6 7 6 b 7 6 n1 7 4 5 Example 2: Find the product AB if it is defined for
3 A = 2 14 ,B= 0 2 3 h i 6 6 7 4 5
17 Summer 2018, Maya Johnson Suppose
b11 b12 a11 a12 a13 C = AB = 2 b21 b22 3 " a21 a22 a23 # b b 6 31 32 7 4 5 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
b11 c11 = a11 a12 a13 2 b21 3 = a11b11 + a12b21 + a13b31. h i b 6 31 7 4 5 The other entries of C can be computed similarly.
Example 3: Compute the indicated product.
66 10 6 3 2 593 " 1910# 32 6 7 4 5
Example 4: Compute the indicated product.
92x 39 " 5y 7 #" 13 14 #
18 Summer 2018, Maya Johnson Example 5: Find the values of x, y,andz.
11 x 21 82 2 3 z 3 = " 0 y 3 # " 02# 42 6 7 4 5
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 $10 for adults. The attendance for the Sunday matinee is given by the matrix
Children Students Adults Cinema I 2 245 120 70 3 A = Cinema II 95 170 245 . 6 7 6 7 Cinema III 6 280 75 120 7 6 7 Cinema IV 6 02502457 6 7 4 5 Write a column vector B representing the admission prices.
Compute AB,thecolumnvectorshowingthegrossreceiptsforeachtheater.
3130
4380 3480 4450 Find the total revenue collected at the Cinema Center for admission that Sunday afternoon.
3130 +4386 +3480 +4450 =$l5,4u@19 Summer 2018, Maya Johnson 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
IIIIII Alan 2 4000 2000 4000 3 A = people. x fund Maria 6 3000 5000 2000 7 6 7 type Steven 6 3000 30004 2000 7 6 7 4 5 The returns of the three funds after 1 yr are given in the matrix
Return I 2 0.15 3 B = . fund x Returns II 6 0.23 7 type 6 7 III 6 0.13 7 6 7 4 5 Which employee realized the best return on his or her investment for the year in question? Returns HyogoBest is Maria 1biggest returns )
AB = Aghgayy ; ]
Which employee realized the worst return on his or her investment for the year in question?
Worst is Alan ( smallest returns )
20 Summer 2018, Maya Johnson