Algebra Applications: Systems of Equations

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[Music]

[Music]

Title: Algebra Applications: Systems of Equations

Title: Profit and Loss

Title: Profit and Loss

Narrator: A business needs one thing more than anything

else in order to survive, profits.

A business is profitable when its revenue is greater

than its expenses.

Whether your business is a fruit stand, a fancy boutique,

or a multinational bank, profits are what make the

business survive and thrive.

Without yearly profits a company will eventually go out

of business.

But at the beginning all businesses have very high

expenses compared to the amount of revenue

they generate.

At the start, a business is not profitable, so one of the

first goals of a new company is to reach what's called the

break-even point.

Let's look at an example.

Suppose a company sells a product for twenty dollars.

The linear function y equals twenty x can be used to track

sales where x represents the number of units sold.

Now suppose it costs the company ten thousand dollars

to create this product and it costs two dollars to

manufacture each unit.

Then the linear function y equals ten thousand plus 2x is

used to track expenses.

Let's use the TI-Nspire to explore this simple scenario.

Turn on the TI-Nspire and create a new document.

You may need to save a previous document.

Create a graphs and geometry window.

At the function entry line for f one of x input twenty x and

press the DOWN ARROW to go to the f two of x entry line.

Input ten thousand plus two x.

Press ENTER.

You won't see both graphs so you'll have to change the

WINDOW SETTINGS.

Press MENU, and under WINDOW select WINDOW SETTINGS.

Change xMin to zero, xMax to one thousand, yMin to zero,

and yMax to fifteen thousand.

Try to get your screen to look like this.

Notice that the graphs intersect.

To find the coordinates of the intersection point press MENU

and under POINTS AND LINES select INTERSECTION POINTS.

Use the nav pad to place the pointer above one

of the graphs.

Press ENTER, then move the pointer above the other graph.

Press ENTER again.

Try to get your screen to look like this.

The intersection point means that when the company sells

five hundred and fifty-six units the money from sales

will equal the expenses.

The intersection point is known as the break-even point,

and the graph of the two equations is known as a system

of equations.

This simple example relied on a linear system of equations

to show how a business achieves profitability.

But in reality businesses don't operate in a

linear fashion.

Linear models are helpful approximations of what

happens, but to get a more realistic understanding of how

a business achieves profitability let's take a

look at a real example.

Amazon.com is the world's largest online retailer.

Books, DVDs, electronic equipment, and many other

items are available for sale online.

The convenience of an online retailer is that you shop

using your computer without having to go to the store.

The rise of Amazon mirrors the rise of the Internet and our

dependence on it.

Amazon has only been in business since the mid-1990's.

Like most new businesses, Amazon spent its first few

years with expenses exceeding revenue.

This chart shows the sales and expenses for Amazon from its

first year through 2008.

As you can see from the chart, from 1995 to 2002 expenses

were greater than sales.

In 2003 sales exceeded expenses for the first time,

and since then Amazon has shown a profit each year.

Let's analyze this data set on the Nspire.

Returning to the previous document, press the HOME key

and create a spreadsheet window.

Go to the top of the column A and input the label year.

Tab to the top of column B and input the label sales, then

tab to the top of column C and input the label expenses.

Go to cell A1 and input the data from the chart.

Pause the video to input the data.

After you have input the data press the HOME key and create

a statistics window.

Use the nav pad to move the pointer to the horizontal axis

until you see the pop-up text 'click to select variable'.

Select year and press ENTER.

Now move the pointer to the vertical axis and

select sales.

Since there are two data series you can create a second

scatter plot.

Press MENU and under plot properties select add

y variable.

Select the expenses data series and press ENTER.

You can also change the appearance of the scatter plot

by connecting the points.

Press MENU and under PLOT TYPE select XY LINE PLOT.

Try to get your screen to look like this.

The two graphs overlap so it's hard to see the

break-even point.

Zoom in to get a better view.

Press MENU and under WINDOW zoom select zoom in.

Move the pointer to the area corresponding to 2003.

Press ENTER.

On average it takes a business five to seven years to

achieve profitability.

This chart summarizes Amazon's rise to profitability.

It took Amazon six years to reach the break-even point.

If we overlay the performance of Amazon's stock price you

can see that 2003 was also an important year for the

stock price.

In general, the stock price of a profitable company

will go up.

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Narrator: We live in a world of coded information.

DNA is nature's code for creating new life.

Language is humanity's means of coded communication.

But DNA and language are public codes.

They are not encrypted.

Anyone who knows these languages can decipher strands

of DNA or sentences on a page.

But sometimes it's necessary to encode a message.

In World War II encrypted messages were constantly used

to communicate to troops in the field.

Today we use encrypted messages on the Internet so

often that we take it for granted.

The reason that information is encrypted is because it is

private information that is conveyed through public

information systems like telephone lines or

Internet connections.

Private information can include anything from banking

data to top-secret government intelligence.

So the practice of encoding messages has always been

important, but it has become a growing branch of mathematics

and linear systems are a way of encrypting information.

A cipher is a means of encrypting a message.

One of the simplest types of ciphers, called a substitution

cipher, is to replace each letter of the alphabet

with a number.

This chart shows how each letter is replaced

with a number.

The @ symbol is used as a substitution for the

space symbol.

So the message 'invasion at dawn' becomes the following

string of numbers.

Let's use the TI-Nspire to decipher some messages.

Create a new document.

You may need to save a previous document.

Create a spreadsheet.

Go to the top of column A and add the column heading

substitution.

This represents the numerical substitutions that will

be used.

Press ENTER.

We want to generate sequence of numbers from zero to

twenty-six.

The zero will represent a word space, and the numbers one

through twenty-six will represent the letters of

the alphabet.

Press MENU and under DATA select GENERATE SEQUENCE.

At the dialog box input this expression, which will

generate a set of consecutive numbers.

Press TAB to set the starting value at zero.

Press TAB again to set the maximum value.

Tab to OK and press ENTER.

If you scroll down you'll see the list of numbers from zero

to twenty-six.

Now move the cursor to the top of column B.

We want to assign the letters of the alphabet to the numbers

in column A.

In fact, we'll be using the numbers already assigned to

each character on the keyboard by the operating system.

Every computer uses a character set where number

sixty-four to eighty-nine are assigned the letters of

the alphabet.

At the top of column B input the heading alphabet and

press ENTER.

At the formula line input this formula.

The formula takes each value in column A and uses it to

generate the characters from code sixty-four to

eighty-nine.

Press ENTER and you will see twenty-seven characters.

This is your substitution cipher.

Now suppose you get the following encoded message.

Using the substitution cipher the message becomes this.

You can think of the cipher as the alphabet mapped to a

number line.

But this cipher is easy to break.

The numbers and letters are in sequential order.

Once you figure out one of the terms of the sequence then you

can figure out the rest very easily.

So this type of substitution cipher offers low security.

To make it a bit more complex you could take the values in

column A of the spreadsheet and assign them to a function

that generates the same numbers, zero to twenty-six,

but in a scrambled order.

So move to the top of column C and input the

heading encryption.

Press ENTER.

At the formula line you will be inputting a function that

deals with modular arithmetic.

In other words, the function will only generate numbers

from zero to twenty-six.

Input the following formula.

The formula finds all the remainders of twenty-seven

when divided by the term two a plus three.

Press ENTER and you will see the numbers zero to twenty-six

but in a non-sequential order.

Notice that each number in column C is used just once.

This is because of the way the function was defined.

Create a scatter plot of the data in columns A and C.

Press control-I and select a graphs and geometry window.

Press MENU and under graph type select scatter plot.

For the x variable select substitution, and for the y

variable select encryption.

Press MENU, and under WINDOW select zoom fit.

Try to get your screen to look like this.

The graph is beginning to look like a linear system.

Unlike the number line shown earlier this is a

two-dimensional mapping of the alphabet to the

coordinate grid.

The graph shows that the cipher does offer a bit more

security since the numbers are not in sequential order, but

even this encryption can be broken.

For example, take a look at the following

encrypted message.

Even if you didn't have the encryption cipher but

suspected that a substitution cipher was used you could

analyze the message.

Highlight the terms that occur most often.

These will correspond to the most frequently

occurring letters.

The more encrypted messages you could analyze that use the

same cipher the more you could eventually decipher

the message.

So, an additional layer of security could be added and

this involves using a three-dimensional mapping of

the letters to x y z coordinates on the 3-D

coordinate system.

Rather than mapping each letter of the alphabet to a

function, clusters of letters are multiplied by a three by

three matrix.

Let's start with a message that needs to be encoded.

Let's use the encryption cipher to convert the text to

a string of numbers.

Now convert this string of numbers into a series of three

by one matrices, as shown.

Note the last matrix has a three in the final position, a

blank space, since the last letter of the message was in

the second position.

Our additional layer of informational security comes

from taking each of these three by one matrices and

multiplying them by a three by three encoding matrix.

The encoding matrix adds the additional layer of security

that makes it extremely difficult to decipher

the message.

For our encoding matrix we will use this three by

three matrix.

Let's use this matrix to encode our message matrices.

Press control-I to create a new document.

Select one to create a calculator window.

Each of the matrices will be defined as a variable.

Press the catalog button and four to activate the fourth

tab within the catalog.

Use a nav pad to select the icon in the second row that

looks like a three by three matrix.

Press ENTER.

At the dialog box define a matrix that is a one column

and three rows.

Tab to OK and press ENTER.

Input the numbers for the first matrix.

Press the RIGHT ARROW key.

Assign this matrix to variable M1.

Press CONTROL and the VAR button.

You will see an arrow pointing to the right.

Input M1 and press ENTER.

This matrix is now stored in memory.

Continue this process with the other three by one matrices.

You will end up with matrices M1, M2, M3, up to M8.

Pause the video to enter these matrices.

After you have defined the eight message matrices, define

the encoding matrix.

Once again click on the catalog key, select the matrix

icon, but this time create a three by three matrix.

Tab to OK and press ENTER.

Fill out the matrix with the numbers from the

encoding matrix.

Store this matrix in a variable called encode.

You are now ready to encode the matrices.

You will be multiplying each three by one matrix by the

encoding matrix in this form.

Furthermore, you will assign the product to a new matrix in

this form.

Encode all eight matrices and assign them the

appropriate variables.

Pause the video to complete the multiplication.

Once you have multiplied the matrices and stored the

products you can see the result of your work.

Your results should look like this.

So now you know how to encode a message with a high degree

of security based on using an encoding matrix, but how do

you use it to decode a message?

Suppose you got the following encoded message.

How do you work backward to decode it?

First, take the message and turn it into a series of three

by one matrices.

Start each matrix in the calculator as a variable.

Use a different variable, as shown.

Rather than encoding the matrix with the three by three

matrix used earlier, you will use the inverse of the

encoding matrix.

So define a new variable as shown.

Now multiply the three by one matrices by the decoding

matrix, as shown.

Pause the video to multiply the encoded message matrices by

the decoding matrix.

Your decoded message yields this set of numbers.

Using the original substitution cipher yields

this message.

The matrix-based encoding system maps the text of the

message to a linear system in three-dimensional space, a

very effective means of encrypting information.

[Music]

Narrator: During World War II the Germans rained down V2

rockets on many civilian targets especially in England.

Thousands were killed through this devastating attack.

The V2 was one of the first ballistic missiles.

Its rocket fuel propelled it to a height of fifty miles,

where the rocket would descend to Earth hitting the ground at

incredible speed.

Little could be done to defend against the V2 rocket except

to hide in underground bunkers.

Thus began the age of the ballistic missile.

During the Cold War the US and the former Soviet Union had

thousands of intercontinental ballistic missiles, ICBM's,

aimed at each other.

These missiles could travel long distances, reach

altitudes of over a hundred miles, and had the potential

for far more devastating damage because these missiles

were nuclear.

It was only the prospect of what was termed mutually

assured destruction, known by the acronym MAD, that prevented

a nuclear war.

But during the 1980's President Reagan proposed a

ballistic missile defense system that would stop these

missiles in mid-air.

Since that time the technology for a ballistic missile

defense has steadily improved.

While the technological challenges were significant,

the mathematics behind this technology relies on a

quadratic system of equations.

Basically, the path of a missile is a parabola, and the

path of the intercepting missile is also a parabola.

Where the missiles intersect is the solution to the

quadratic system.

Let's explore a quadratic system graphically using the

TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a graph window.

Let's graph two parabolas.

One parabola will represent the ballistic missile, and the

other parabola will represent the anti-ballistic missile.

At the F1 entry line, input this function.

Press the DOWN ARROW, and at the F2 entry line input

this function.

Press ENTER.

As you can see the graphs intersect.

To find the intersection point press MENU and under POINTS

AND LINES select INTERSECTION POINT.

Use the nav pad to move the pointer above one of the

graphs and press ENTER.

Then use the nav pad again to hover over the second graph.

Press ENTER again.

The coordinates of the intersection point

will appear.

The intersection point is a solution to the quadratic

system of equations.

This intersection point is where the anti-ballistic

missile will impact the ballistic missile.

In reality the path of the ballistic missile varies from

the parabolic path.

There are a number of reasons for this.

Such factors as weather, the missile guidance system, and

other factors will change the path of the missile from one

parabola to another parabola.

What the anti-ballistic missile does is read the

coordinates of the path of the missile and projects where the

missile will be.

This involves instantaneous reading of data in order to

make adjustments.

But the underlying principle still holds.

When the anti-ballistic missile hits the ballistic

missile a solution to our quadratic system has

been determined.

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