mat 124 finite mathematics page 1 sections 2.1-2.2 (lial...

MAT 124 – Finite Mathematics Sections 2.1-2.2 (Lial 11e)

[email protected] kradermath.jimdo.com 08/2016

NOTE

The textbook describes two methods for solving systems of linear equations. We will focus on the Gauss-Jordan Method, presented in Section 2.2.

Systems of Equations

In Chapter 1, we saw that many real-world situations can be described by a linear equation.

o EXAMPLES: Cost of parking, cost of manufacturing, number of baseball cards, etc.

Some real-world situations are described by a system of equations. o A system consists of two or more equations. o If the equations have n variables, solving the system of equations means

finding the values 1 2 3, , , ..., nx x x x that solve all of the equations in the

system. Systems of Two Linear Equations in Two Variables EXAMPLE: Dimes and Nickels I have seven coins – all dimes and nickels – that add up to 55 cents. How many dimes and how many nickels do I have?



EXAMPLE: Dimes and Nickels (cont’d)

What is the solution of this system of equations?

EXAMPLE: Inconsistent System of Linear Equations

What is the solution of this system of equations?

EXAMPLE: Dependent System of Linear Equations

What is the solution of this system of equations

Slide 5

Example: System of Linear EquationsDimes and Nickels

(4,3)

MAT 124 (Lial 10e) – 2.1-2.2 GHK 08/2012

Slide 6

Inconsistent systemof linear equations

MAT 124 (Lial 10e) – 2.1-2.2 GHK 08/2012Slide 7

Dependent systemof linear equations

MAT 124 (Lial 10e) – 2.1-2.2 GHK 08/2012



Systems of Three Linear Equations in Three Variables EXAMPLE: Ace Novelty Company STEP 1: Read the problem. The Ace Novelty Company wants to produce three types of souvenirs: keychains, magnets and pins. To manufacture a keychain requires 2 minutes on machine A, 1 minute on machine B, and 2 minutes on machine C. A magnet requires 1 minute on machine A, 3 minutes on machine B, and 1 minute on machine C. A pin requires 1 minute on machine A, 2 minutes each on machines B and C. There are 3 hours available on machine A, 5 hours available on machine B and 4 hours available on machine C. How many souvenirs of each type should Ace Novelty make in order to use all of the available time on the three machines? STEP 2: Identify the unknowns. STEP 3: Define one variable for each unknown. Write out the definition of each variable. Pay attention to the units associated with each variable; this may be helpful later on. STEP 4: Make a table with columns for each variable and a TOTAL column on the right. Put the variable at the top of each column. Use the problem data to fill in the remaining table entries. TOTAL



EXAMPLE: Ace Novelty Company (cont’d) STEP 5: Set up a system of equations based on the rows in the table. STEP 6: Solve the system of equations. STEP 7: Answer the original question in English.



Before we learn how to solve systems of equations in three (or more) variables, let’s talk about what those solutions “look like.” Solution of ONE First Degree-Equation

Number of variables

Sample equation Solutions and Geometrical representation

Equation with 1 variable

3 6x Solution set is a single number.

Represented as a point (0 dimensions) on a line (1-dimension).

Equation with 2 variables

3 6x y Solution set is a set of ordered pairs ,x y .

Represented as a line (1 dimension) in a plane (2-dimensions).

Equation with 3 variables

3 2 6x y z Solution set is a set of ordered triples , ,x y z .

Represented as a plane (2 dimensions) in a 3-dimensional space.

Do you see a pattern in the table, above? A lot of mathematics involves studying patterns! Solution of a SYSTEM of First-Degree Equations

Number of variables

Solutions and Geometrical representation

System of equations with 2 variables (e.g., Dimes and Nickels)

Solution of each equation is a line in a 2-dimensional space.

Solution of the system of equations has three possibilities, based on how two lines may or may not intersect.

System of equations with 3 variables (e.g., Ace Novelty Company)

Solution of each equation is a plane in a 3-dimensional space.

Solution of the system of equations has multiple possibilities, based on how three planes may or may not intersect.



Solution of a SYSTEM of First-Degree Equations (cont’d)

Is it possible to find a unique solution for a system of two first-degree (linear) equations in three variables?

Slide 13

System of three linear equations

Given an equation in three variables x, y and z:

Ax + By + Cz = D

the solution set is a plane in a 3-dimensional space.

MAT 124 (Lial 10e) – 2.1-2.2 GHK 08/2012

Slide 14

Systems of first-degree equations may have one solution, no solution or infinitely many solutions

If:

#equations #variables

then one of the following is true:

System has exactly one solution.

System has no solution.

System has infinitely many solutions.

If:

#equations < #variables

then one of the following is true:

System has no solution.

System has infinitely many solutions.

MAT 124 (Lial 10e) – 2.1-2.2 GHK 08/2012

If there are fewer equations than variables, there cannot be a unique solution!



EXAMPLE: Ace Novelty Company (cont’d) Keychain Magnet Pin TOTAL

Number of souvenirs

x y z

Time on Machine A (hrs.)

2 1 1 180

Time on Machine B (hrs.)

1 3 2 300

Time on Machine C (hrs.)

2 1 2 240

STEP 5: Set up a system of equations based on the rows in the table. STEP 6: Solve the system of equations. Represent the system of equations as a matrix.

A matrix (plural: matrices) is a rectangular array of numbers. Each row of the matrix represents an equation.

The dimensions of the matrix describe the number of rows and columns.

The matrix used to represent the system of equations is a _____X_____ matrix (fill in the dimensions).



Before you create the matrix, make sure each of the variables, the constants and the equal signs are lined up. EXAMPLE: Creating a Matrix to Represent a System of First Degree Equations Create a matrix for the following system of linear equations.

2 180

2 180

3

x y z

x z

z x y

Gauss-Jordan Method for Solving Systems of First-Degree Equations

Represent the system of equations as a matrix.

o Each row of the matrix represents one equation.

Use matrix “row operations” (which are analogous to the “Properties of Equality” used to solve linear equations in one variable) to transform the original matrix into a matrix that looks like the following (where m, n, and p are constants):

1 0 0

0 1 0

0 0 1

m

n

p

o The matrix row operations, like the Properties of Equality, do not affect the solution of the system of equations. Thus the matrix shown above is equivalent to the original matrix, i.e., both matrices represent the same solution.

o Translate the output matrix back into a system of equations to read the solution. For example, the first row of the matrix reads:

1 0 0x y z m or, simply, x m .

o Similarly, y n and , so the solution of our system of equations is

, ,m n p !

Manual Implementation of the Gauss-Jordan Method Transforming the original matrix to the output matrix:

Objective: Transform each column (except the rightmost column) of the original matrix into a unit column, i.e., a column of zeroes with a solitary “1” on the diagonal.

This process is called pivoting. Pivoting involves row operations which are analogous to the Properties of

Equality used to solve linear equations: o Interchange any two rows. o Multiply any row by a non-zero coefficient. o Add a non-zero multiple of one row to another row.



EXAMPLE: Ace Novelty Company (cont’d) STEP 6: Solve the system of equations (cont’d) Use the Gauss-Jordan method to solve the system of equations. Manual Solution Original matrix

Convert 1st pivot element (R1,C1) to “1”:

R1 (1/2)R1

Convert C1 into a unit column:

R2 R2 – R1 R3 R3 – (2)R1

Convert 2nd pivot element (R2,C2) to “1”

R2 (2/5)R2

1 1/ 2 1/ 2 90

0 1 3/ 5 84

0 0 1 60

Convert C2 into a unit column:

R1 R1 – (1/2)R2

3rd pivot element (R3,C3) is already a “1”

Convert C3 into a unit column.

R1 R1 – (1/5)R3 R2 R2 – (3/5)R3

1 0 0 36

0 1 0 48

0 0 1 60

Read the solution from the final matrix: x = 36 y = 48 z = 60 STEP 7: Answer the original question in English. “In order to use all of the available time, Ace should manufacture 36 keychains, 48 magnets and 60 pins.”

2 1 1 180

1 3 2 300

2 1 2 240

1 1/ 2 1/ 2 90

0 5 / 2 3/ 2 210

0 0 1 60

1 1/ 2 1/ 2 90

1 3 2 300

2 1 2 240

1 0 1/ 5 48

0 1 3/ 5 84

0 0 1 60



EXAMPLE: Ace Novelty Company (cont’d) STEP 6: Solve the system of equations (cont’d) Use the Gauss-Jordan method to solve the system of equations. Use the rref( ) command.

[2nd] [MATRIX] > MATH > B:rref > [ENTER]

[2nd] [MATRIX] > NAMES > Select matrix name and close parentheses > [ENTER]

When Doing the Homework Problems

Use the rref( ) command. You do not need to perform the individual row operations described in the textbook. (You’ll have an opportunity to learn them in Chapter 4!).

Instead of learning the tedious row operations, focus on: o Setting up the system of equations (and the original matrix). o Interpreting the output matrix, especially in situations (to be discussed

shortly) where there is no solution or infinitely many solutions.

Show your work, i.e., o Define your variables. Be specific, e.g., write “Let x=number of

keychains.” Just writing “x=keychains” is not specific. o Show the system of equations and the matrix you input to rref( ). o Show the output matrix. o Answer the question in English.



EXAMPLE: Soda Sales (Textbook P. 68, Example 6) A convenience store sells 23 sodas one summer afternoon in 12, 16 and 20 oz. cups (small, medium, large). The total volume of soda sold was 376 oz.

(a) Suppose that the prices for a small, medium and large soda are $1, $1.25 and $1.40 respectively, and that the total sales were $28.45. How many of each size did the store sell?



EXAMPLE: Soda Sales (Textbook P. 68, Example 6) (cont’d)

(b) Suppose the prices for small, medium and large sodas are changed to $1, $2 and $3 respectively, but the other information is kept the same. How many of each size did the store sell?



EXAMPLE: Soda Sales (Textbook P. 68, Example 6) (cont’d)

(c) Suppose the prices are the same as in part (b) but the total revenue is $48. Now how many of each size did the store sell?

(d) Give the solutions from part (c) that have the smallest and largest numbers of large sodas.



EXAMPLE: Investments (Textbook P. 71, No. 45) Katherine Chong invests $10,000 received from her grandmother in three ways. With one part, she buys US Savings Bonds at an interest rate of 2.5% per year. She uses the second part, which amounts to twice the first, to buy mutual funds that offer a return of 6% per year. She puts the rest of the money into a money market paying 4.5% annual interest. The first year her investments bring a return of $470. How much did she invest in each way?



EXAMPLE: Transportation (Textbook P. 72, No. 54) An auto manufacturer sends cars from two plants, I and II, to dealerships A and B located in a midwestern city. Plant I has a total of 28 cars to send, and Plant II has 8. Dealer A needs 20 cars and Dealer B needs 16. Transportation costs per car, based on the distance of each dealership from each plant, are:

$220 from Plant I to Dealer A, $300 from Plant I to Dealer B. $400 from Plant II to Dealer A, $180 from Plant II to Dealer B.

The manufacturer wants to limit transportation costs to $10,640. How many cars should be sent from each plant to each of the two dealerships?

mat 124 finite mathematics page 1 sections 2.1-2.2 (lial...

Documents