Algebra Applications: Systems of Equations
[Music]
[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.