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Final Exam Review - Questions 4, 5, and 6

!±8± Final Exam Review - Questions 4, 5, and 6

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Solve for Three Unknowns Using Matrices and Row Operations

!±8± Solve for Three Unknowns Using Matrices and Row Operations

A system of linear equations can be solved by (1) preparing an Augmented Matrix, and then (2) using row operations to convert the Augmented Matrix into a triangular form. Once the Augmented Matrix has been converted to triangular form, the solution to every variable can often be read directly from the matrix with no further work.

This can be illustrated by solving for the variables x, y, and z in the following system of linear equations:

-4x - 3y - z = -12
-5x + 6y + 4z = -24
-3x - y - 6z = -22

First, arrange all three equations in standard form.

Ax + By + Cz = D

This has already been done in the statement of the problem:

-4x - 3y - z = -12
-5x + 6y + 4z = -24
-3x - y - 6z = -22

Prepare an Augmented Matrix of 3 rows and 4 columns.

Each row in the Augmented Matrix will represent one of the three equations.

The first number in each row is the coefficient for the x variable, the second number in each row is the coefficient for the y variable, the third number in each row is the coefficient for the z variable, and the last number will be the constant:

| -4 -3 -1:-12 |
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |

THIS IS the GOAL: Convert the Augmented Matrix into a triangular form. Once this is done, we are finished.

The triangular form will look like this: a diagonal pattern of 1's with 0's everywhere else but the last column.

| 1 0 0:? |
| 0 1 0:? |
| 0 0 1:? |

Once we have converted the matrix into the triangular form, the? shown in row 1 will be the solution value of x, the? shown in row 2 will be the solution value of y, and the? shown in row 3 will be the solution value of z.

To convert the augmented matrix into a triangular matrix, we will perform various row operations, one at a time.

To convert the convert the Augmented Matrix to a triangular form, the following operations can be used:

1. switch two of the rows
2. multiply any row by a number which is not zero
3. Replace any row with the result of adding that row to another row.

| -4 -3 -1:-12 |
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |

Note:

There are many calculations involved, which means there is a high probability of making errors in arithmetic. For this reason I recommend using a calculator.

| -4 -3 -1:-12 |
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |

Multiply R 1 by -1/4

| 1 3/4 1/4: 3|
| -5 +6 +4: -24 |
| -3 -1 -6:-22 |

add 5 times the 1st row to the 2nd row

| 1 3/4 1/4: 3|
| 0 39/4 21/4: -9|
| -3 -1 -6:-22 |

add 3 times the 1st row to the 3rd row

| 1 3/4 1/4: 3|
| 0 39/4 21/4: -9|
| 0 5/4 -21/4:-13|

multiply the R 2 by 4/39

| 1 3/4 1/4: 3|
| 0 1 7/13: -12/13|
| 0 5/4 -21/4:-13|

add -5/4 times the 2nd row to the 3rd row

| 1 3/4 1/4: 3|
| 0 1 7/13: -12/13|
| 0 0 -77/13:-154/13|

multiply the R 3 by -13/77

| 1 3/4 1/4: 3|
| 0 1 7/13: -12/13|
| 0 0 1: 2|

add -7/13 times the 3rd row to the 2nd row

| 1 3/4 1/4: 3|
| 0 1 0: -2|
| 0 0 1: 2|

add -1/4 times the 3rd row to the 1st row

| 1 3/4 0: 5/2|
| 0 1 0: -2|
| 0 0 1: 2|

add -3/4 times the 2nd row to the 1st row

| 1 0 0: 4|
| 0 1 0: -2|
| 0 0 1: 2|

This is the final triangular form of the Augmented Matrix (the row echelon form):

| 1 0 0: 4|
| 0 1 0: -2|
| 0 0 1: 2|

The solution to the three simultaneous equations is:

From R 1, x = 4
From R 2, y = -2
From R 3, z = 2

The final answer is:

x = 4
y = -2
z = 2

This answer can be verified by substituting the numerical values of x, y and z into the original equations:

-4x - 3y - z = -12
-4*4 - 3*(-2) - 2 = -12, correct

-5x + 6y + 4z = -24
-5*4 + 6*(-2) + 4*2 = -24, correct

-3x - y - 6z = -22
-3*4 - (-2) - 6*2 = -22, correct

Since the numerical values of x, y, and z work in all three of the original equations, the solutions are correct.

Solve for Three Unknowns Using Matrices and Row Operations

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Teach Your Kids Arithmetic - Division Shortcuts

!±8± Teach Your Kids Arithmetic - Division Shortcuts

The famous Greek mathematician Pythagoras (you know the one with that theorem) said, "Numbers have a way of taking you by the hand and leading you down the path of reason." What Pythagoras was getting at, I think, is that numbers--by their very nature--permit us to do things which enable an understanding of the very universe and its intricate laws. Numbers have their roots in arithmetic, and a mastery of this field, particularly the operations of addition, subtraction, multiplication, and--yes that monster operation of division--will certainly pave a smooth road down that path of reason.

For most children, the tedious task of memorizing their number facts evokes long-winded yawns and puts pain-riddled expressions on their cherubic faces. However, rote tasks such as these (see my article Mastering Arithmetic and Singapore - What's the Connection?) are what insure that children have a chance to progress up the ladder in mathematics. Of the four arithmetic operations, division is the one that children find most uncomfortable as this is the one operation which produces "uneven" results in the form of remainders. Such remainders thrust children into the world of fractions, percents, and decimals (see my article on this topic) and we all know how painful those things called fractions can be.

Given the difficulties that division presents to children learning their arithmetic basics, it would be both expedient and practical to present an approach which would lead them down this path of reason. Of the myriad number tricks and math shortcuts that I have developed over the years, the method of division by ten-multiples is one I am quite fond of. Basically, this method hinges on understanding the basic multiplication facts. After all, mathematics is a building blocks discipline, and this method, by being dependent on having mastered multiplication, clearly shows this fact.

This method shows an effective way to divide a three digit number by a one digit number. Take the example 306/6. A ten-multiple of 6 is 60. What we are trying to do is see how many ten-multiples of 6 go into 306 without going over. To do this effectively, we need to know our 6-times table. We know that 6x5 = 30 and 6x6 = 36. The respective ten-multiples of 30 and 36 are 300 and 360. Since 300 is less than 306, we know that 306 divided by 6 is going to be "50-something." After we divide out the 300, we are left with 6, which gives another 1 as part of the quotient. Thus 306/6 is 51.

Play with this method a little and teach it to your kids. The more exposure they have to arithmetic techniques, the more they will come to master mathematics. Who knows, they might even become whiz kids.

See more at my cool math site Arithmetic Magic and Fractions Troubleshooter.

Teach Your Kids Arithmetic - Division Shortcuts

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For students in any field, but especially math and sciences, the Casio CFX-9850GC PLUS graphing calculator is a problem-solving tool with over 900 features for storing, graphing, and analysis. You can execute evaluation tables on a split screen (which allows you to trace the graph and scroll table values simultaneously) to display graphs and tables. With a long list of features, you can master list-based, one- and two-variable statistical calculations, a variety of regressions, statistics, hypothesis testing, and advanced financial calculations.

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The GED Math Test: About the Casio fx-260 Calculator

!±8± The GED Math Test: About the Casio fx-260 Calculator

To score well on the GED math test, it's important to learn how to use the Casio fx-260 calculator, since it's the calculator used for the official General Education Development Test. To prepare well for the test, it's a good idea to get hands-on practice with this Casio, and to understand which calculator functions the test expects you to know.

The Casio fx-260 is used for Part 1 of the two-part math test, which covers basic algebra and geometry, data analysis and basic number operations. Each one of the two parts has 25 questions, with 45 minutes slotted, or 90 minutes total allowed for the entire GED math test. The Casio fx-260 is distributed by the test site center for Part 1 of the test, and then collected before part 2. Calculators can't be used for Part 2.

About the Casio Calculator

The Casio fx-260 is a scientific calculator. It's more advanced than the simpler or basic calculator models most adults use to balance their checkbooks or to add a grocery bill while shopping. Many of today's high schoolers and even college graduates aren't familiar with the advanced calculators and multiple functions of scientific calculators used in today's technology, science fields and for advanced financial operations. So calculator skills aren't just important for GED students; knowledge of scientific calculators is important to everyone engaged in today's rapidly progressive technological society.

Here are some basics to understand about the Casio fx-260:
Learn the location of the keys. Learn the functions that the keys perform. Use the 'On' button to reset the calculator, or to clear the memory. How to use the 'Clear' and 'All Clear' buttons or functions to clear the last number entered or memory. Using number keys 0-9, and basic operation keys for addition, multiplication, subtraction and division. Learning the location of the decimal point key, equals and percent, and how and when each is used. Using the 'Shift' key -- to change other keys to alternate functions. How the 'Change Sign' and 'Fraction' keys work, and when to use them. How the 'Square' and 'Square Root' keys work, and when to use them. The 'Parenthesis' keys are important, since these keys are used to control the order of mathematical operations. Understanding the keys to raise numbers to another power, and for exponentials -- 'EXP' key -- used in scientific notation.

Does it look complicated? Sure, and the Casio calculator used for the GED test has additional functions that can be used for highly complex mathematical functions.

But don't worry. While it seems like a lot to learn -- and to understand -- the test requires only basic knowledge and application of a few functions. And this knowledge will actually help test candidates solve the test's more complicated problems.

The Casio fx-260 is worth learning. Understanding goes a long way toward reducing "math anxiety" and should improve the final GED Test math score.

For additional GED study tips and math test tips, test information and free resources on the GED test, official testing sites, financial aid and student support, visit http://www.passGED.com. The website also provides links to federal agencies and nonprofits that serve GED students, instructors, corrections students and workforce development programs.

The GED Math Test: About the Casio fx-260 Calculator

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The dangers of trillions of dollars of debt

!±8± The dangers of trillions of dollars of debt

We are not out of the woods yet. The stock market has begun to take measures for the experimental aspect of the end of the bear market, Chrysler and General Motors are steps on the road to a more streamlined and, hopefully, more prosperous and credit markets begin to take steps to attempt a opening up of the credit.

But while this is good news, apparently, the problem is that the TARP, TRAP, GM, Chrysler, the reform of social security and the warthe rest of the federal budget is necessary, the costs out of control.

The federal government has a deficit of 1,700 billion U.S. dollars this year because of spending more than income.

Our national debt is now estimated at about $ 11 billion.

Most people have no idea of ​​the difference between a trillion and a trillion dollars. All most people know is that it sounds like a lot of money, but nothing that can not be addressed.

When Istarted college, we worked some problems with "scientific notation", which show a way to enter numbers, if it is too big for a pocket calculator. Sun $ 1.7 trillion is $ 1,700,000,000,000 or 1.7 x 1012th That is, a point seven times ten to the twelfth power.

To understand a bit 'better, try this for a million dollars is to return would pay a dollar a second for the next 12 days.

For the return of a billion dollars, you take a second to the U.S. dollar for the next 31Years.

A trillion dollars for one U.S. dollar, it takes a second for the next 31 thousand years.

The government needs the growth of debt and the deficit over the next three years for the control or we are all responsible for the collapse of our system as we know it now. This means that forcing by a ghost, our elected politicians to take responsibility for their actions to be taken to get into questions.

They are thinking that there are too many lawyers in Congress and not enoughAccountants, we need more people who understand the calculation of budgets and fewer people who write the offer for lack of legislation for special interests to the left or right.

We're Americans. All of us, whether we look at each point or less. We are citizens of the United States of America. Now is the time for us to go back, to maintain the principles that underpin the country and start our representatives accountable for their actions do, andfind solutions to these problems, rather than up or placing the sidewalk cracks.

The dangers of trillions of dollars of debt

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Post Date : Aug 07, 2011 23:15:06 | N/A

Item Name: HP 48GX (128KB) graphing calculator - used. Cosmetic Condition: Good Perfect Working ConditionHP 48GX Expandable RPN Graphing Calculator with two expansion card slots This item has been discontinued by the manufacturer. We have a limited number of used calculators available at the price listed on this page; once they are gone we'll likely stop carrying this model. These used units have been verified by us to be in good condition and in working order. Includes cable and batteries. Does not include: Manuals or manufacturer warranty. Returns will not be accepted for this item, but we will repair or replace defective units for an extended period of 60 days.

• Over 2,300 functions including over 300 for scientific and engineering applications
• Built-in equation library with over 300 formulas and constants
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• 128 KB memory; 2 card ports for expandable RAM and ROM
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Vedic Mathematics

!±8± Vedic Mathematics

Due to the incredibly large Indian history, mathematics has always used the best place in the scientific heritage of India. Mathematics is generally regarded as the science of all sciences. You may not mathematics as a scientific discipline, not much progress - the coronation of Indian mathematics was the invention of zero and the introduction of decimals. Indian mathematics (Vedic Mathematics) is a vibrant discipline with great potential for diversificationapplications of the modern world.

According to Sir Herbert JF Everything that had made the greatest minds ever to understand the types of concepts is a great science, mathematics, popularly known as the crops.

And 'uncertain when the Vedic literature really began, but in modern historian, began the Vedas (Rig-Veda, Sama Veda, Yajur Veda and Atharva Veda), to be developed in about 1600BC. Over time, most of the VedicThe tradition fell into stagnation. Then, in the 19 th Centuary recovered scholars interested in the Vedas. Then Sri Krishna Bharti Tirathji after long and careful investigation produced a reconstruction of the ancient Indian mathematics based on 16 sutras, along with a subsutra.

Vedic Mathematics is not only a sophisticated tool, but also an introduction to ancient civilizations. It takes us back to millennia of cultural and scientific heritage of India. It has roots in the VedicSources, which marked the dawn of human history, and illuminated by their scientific exegesis. Vedic Mathematics deals mainly with various mathematical formulas and their applications to perform long and complicated algebraic equations and for the most part try to run these spiritual formulas. Vedic Mathematics is a unique system of calculations based on simple rules and principles with which any mathematical problem-based - can be resolved - there are arithmetic, algebra, geometry, trigonometry ororally. It offers us the possibility to do calculations with different methods and ways too. The calculation can be done in both directions, from left to right, and can also be obtained from right to left and at the same end result. In this way, a general development of the child easily and permanently, and their creativity levels are raised above the roof.

Calculated with the help of this old system, such as entry into Playway school, where every step is taken after 'Heads of the youngest and not a rule. In this system, multiplication and additions to a game of subtraction is done with the addition. Division can be a process of multiplication and addition. The simplicity of the method makes the top-heavy processes, our logic-driven world.

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