Algebra Nspirations

Algebra Nspirations

Algebra Nspirations

Algebra Nspirations

Solving Systems of Equations

As an algebra student, you've learned by now that many day

to day operations in business, science, government, and

elsewhere can be described by equations

in one or more variables.

The variables represent quantities that change, and

the equations model relations among the variables.

But perhaps more important in all fields of study, are

problem situations involving, not only two or more

variables, but also two or more equations.

Such multi-variable and multi-relation mathematical

models are called systems.

Solving algebraic systems of equations will expand your

knowledge of variables, equations, and functions.

In the process, you'll also become more proficient with

the TI-Nspire technology.

We'll start by solving linear systems with the

Nspire in many ways.

Then, algebraically, using two classical methods.

And finally, we'll summarize the different types of

solutions you can encounter.

Without further delay let's go straight to our first

investigation.

Have you ever found yourself in the situation of having to

decide between two competitive cellphone plans?

Or having to choose between two rental car companies for a

short out-of-town trip?

Systems of equations really come in handy for these types

of problems.

Suppose that you and a few friends or relatives need to

make an out-of-town trip for the day.

You're debating between two companies.

EZ Ride charges $0.20 per mile for the compact car of your

choice, plus a fixed daily fee of $22.

For the same car, Just4U, Just charges for mileage at a rate

of $0.30 per mile.

Which offer would you choose?

First, we need to formulate equations for your total cost

y as a function of your number of miles x for each company.

$0.20 or $0.2 for each mile translates into 0.2 times x.

We add to that 22 for the fixed daily fee of $22.

For Just4U, it's just $0.30 times the number of miles x.

Jot down these equations for our next step.

In a technological world, it's not only important to know how

to use technology, but also, to use it efficiently.

The Nspire offers several representations you can use to

solve this problem.

I'd like to walk you through four of them, and then,

depending on the particular problem, you'll decide which

method is most efficient.

The first approach uses the spreadsheet application.

Turn on the TI Nspire, press Home, Open a New Document.

If another document is open, you'll be prompted to Save it.

Click to choose Yes or Tab, then click to choose No.

Create a New Spreadsheet Page.

Arrow up to the top of Column A and type x for miles, our

independent variable.

Arrow down one line to the Formula Line, press Menu, and

under Data, select Generate Sequence.

We're going to generate the sequence of every 10 miles

starting with zero.

Head down, and enter zero for the starting value.

Enter 30 for the maximum number of terms, which stands

for 300 miles.

Then tab down to OK and click.

You can see the first few multiples of 10

in the miles column.

For the cost of EZ Ride, arrow right and up to the top of

Column B, type y1 and press Enter.

That's the second way to move down to the formula line.

Type 2x plus 22.

Notice that the equal sign is inserted by default.

Press Enter.

Press the Down Arrow to resolve the Detected Conflict,

and select Variable Reference.

Head down and click OK when you're done.

These are the costs for the respective numbers of miles x.

Let's do the same for the second company, Just4U, arrow

right and up to the top of Column C, type y2 this time,

press Enter and type the formula point 3x.

Press Enter again, and make the same choice as before,

then tab down to OK.

You can now compare both companies.

Scroll down the first 20 lines or so, and notice Just4U is

the better deal.

But at line 23, both costs are equal to $66.

Thereafter, EZ Ride becomes more affordable.

So your choice between the two companies depends on the

length of your trip.

If your round-trip is more than 220 miles, EZ Ride is

your best choice.

If it's less than 220 miles, you'll pick Just4U.

This method is also called the Tabular or Numerical approach,

as the table of numbers gives you a good sense of

what's going on.

You can see the change in y variable or cost as a function

of the change in the x variable or

miles for each company.

You even found that at x equals 220 miles,

both costs are equal.

This specific fact helps illustrate the meaning of

solving a system of equations.

First, the notation.

To denote the system comprising our two rental car

equation, we use a left brace.

Here, the equations are in slope-intercept form,

rearranging the terms, and then multiplying the first

equation by five and the second by 10 to get integer

coefficients, you obtain the equivalent standard form.

I'll let you verify that.

Solving a system means finding the solution or solutions for

x, y that satisfy the equation simultaneously.

In other words, they must make both equations true.

From our spreadsheet, we saw that 220 comma 66 satisfies

both equations.

So as far as we could see, it is the unique solution of the

system of equations.

I said, as far as we could see, because the spreadsheet

was incomplete.

It did not show all the possible x and y values.

The graphical method we will explore next, will bring us

more certainty about the uniqueness of this solution.

But, once again, first, a quick review.

Since both equations are linear, they

graph a straight line.

But what are the possible relative positions of two

lines in the x,y plane?

The two lines can intersect at one point.

The lines can be parallel, or the two lines can be

coincident, meaning they're the same.

With that in mind, let's use the graphical method to solve

a slightly different problem.

What number of miles x will the cost of renting a compact

car from either EZ Ride or Just4U be the same?

In other words, you must solve this system.

While we may know the numerical answer, let's

explore a second technological approach onto the TI Nspire.

Press Home to add a Graphs in Geometry Page to our document.

Note the one point two at the top left of the screen

indicating the second page.

Type point two x plus 22 for f1 of x.

You see no graph, because the window needs to be adjusted.

We'll do so after defining f2.

Type point 3 x, then press Enter.

Press Menu, and under Window, select Window Settings.

Enter negative 10 and 300 for x, skip Auto, then negative 5

and 100 for y.

Skip Auto again, then tab down and click OK.

To find the coordinates of the point where these two graphs

intersect, press Menu and under Points and Lines select

Intersection Points.

You can see the intersection icon at the

top left of the screen.

Now, use the mouse pad to move the arrow cursor to either

line until it begins flashing.

Click or press Enter to select that line.

Notice that as you move away from it, it continues to

blink, but more slowly.

This confirms that it has been selected.

Do the same for the second graph.

Click or press Enter.

The Nspire generates the cornice of the intersection

point, of this pair of line graphs is the graphical

representation of our linear system.

And their intersection point, 220,66 is a graphical

representation of this system's solution.

Let's construct a function table to confirm our findings.

Press Menu, and under View, select Add Function Table.

To Edit, press Menu again, and under Function Table select

Edit Function Table Settings.

Insert 218 for Table Start, then tab down to OK.

Lastly, to fully view both columns, f1 and f2, under Page

Layout, pick Custom Split.

Arrow left until both columns are visible, press Enter.

The table confirms that both y values or function values are

equal for x equals 220.

For x less than 220, f1 is greater than F2.

For x greater than 220, it's the reverse.

Can you make the visual connection with the relative

position of the two graphs?

For exactly 220 miles, both rental car companies would

charge $66.

We'll assume that other optional costs, like

insurance, are the same.

This graphical method works for any system

two or more equations, two or more variables, and linear or

nonlinear graphs.

This method provides helpful information, even if you are

not too familiar with the types of equations involved.

A well known business application of systems of two

equations is the concept of equilibrium point.

Suppose the red graph represents the company supply

curve of some commodity, like sneakers, for example.

And the blue curve represents the consumers demand curve for

those particular sneakers.

Equilibrium occurs when the amount consumers wish to

purchase at price p, is the same as the amount producers

are willing to offer for sale at the same price.

So the equilibrium point is where supply meets demand, and

both parties are satisfied.