(Based on Section 2.3 in *Finite Mathematics* and *Finite Mathematics and Applied Calculus*)

**Some On-line Resources for This Topic:**

- On-Line Pivot & Gauss-Jordan Utility
- Free Mac Software (Including Gauss-Jordan Helper)
- Pivot Program for the TI-82 and TI-83

The following discussion is based on Section 2.3 of

In all applications of linear equations, we follow the same general strategy.

General Strategy for Solving Systems of Linear Equations
In other words, what are we asked to find? In answering this question, you should note down something like the following: - Let x be the number of video games.
Let y be the number of applications. Let z be the number of documents. Note that all the unknowns should be
How to do this depends on the way the problem is worded. We will look at a few examples below to develop some strategies.
How to do this depends on the way the problem is worded. We will look at a few examples below to develop some strategies. |

There are several kinds of applications generally found in textbooks.

- Applications in which the given information can be tabluated
- Applications in which some of the given information must be translated from words into equations
- Applications of specialized types, such as "transportation problems" and "traffic flow problems." These require special techniques for setting up the system of equations, and are discussed in the book.

Here is a typical application of the first type, based on an example in *Finite Math.*

The Softflow Yogurt Company makes three yogurt blends: LimeOrange, using 2 quarts of lime yogurt and 2 quarts of orange yogurt per gallon; LimeLemon, using 3 quarts of lime yogurt and 1 quart of lemon yogurt per gallon; and OrangeLemon, using 3 quarts of orange yogurt and 1 quart of lemon yogurt per gallon. Each day the company has 800 quarts of lime yogurt, 650 quarts of orange yogurt, and 350 quarts of lemon yogurt available. How many gallons of each blend should it make each day if it wants to use up all of the supplies?

Did you get those right? If so, write down all of the unknowns, and press here to see if your list is correct.

Here is the problem stated once again:

The Softflow Yogurt Company makes three yogurt blends: LimeOrange, using 2 quarts of lime yogurt and 2 quarts of orange yogurt per gallon; LimeLemon, using 3 quarts of lime yogurt and 1 quart of lemon yogurt per gallon; and OrangeLemon, using 3 quarts of orange yogurt and 1 quart of lemon yogurt per gallon. Each day the company has 800 quarts of lime yogurt, 650 quarts of orange yogurt, and 350 quarts of lemon yogurt available. How many gallons of each blend should it make each day if it wants to use up all of the supplies?

We can organize the given information in a table. To set up the table, do the following:

- Place the categories corresponding to the unknowns along the top.
- Add an extra column for the "Total Available"
- Place the "ingredients" down the side.

Now read across the first row of the table: it gives the amounts of lime yogurt needed for the three blends, and also the total available.

If Softflow makes x quarts of LimeOrange, y quarts of LimeLemon, and z quarts of OrangeLemon, it will need a total of

- 2x + 3y

quarts of lime yoghurt. Since Softflow has a total of 800 quarts of lime yogurt on hand, and it wants nothing left over, we must have

Amount used | = | Amount Available |

2x + 3y | = | 800 |

Similarly, we get two more equations for orange and lemon yogurt:

Now you have a system of three equations in three unknowns. You will notice, when you set it up in matrix form, that *the augmented matrix is exactly the same as the table we set up above *:

To solve the system, row-reduce the given matrix (you can either do it by hand or use the On-Line Pivot & Gauss-Jordan Utility.

The next example we look at is stated in a way so that not all the data can be tabulated..

Last year you purchased shares in three Internet companies: OHaganBooks.com, FarmersBooks.com, and JungleBooks.com. The OHaganBooks.com cost you $50 per share, FarmersBooks.com stocks cost you $45 per share, and JungleBooks.com cost you $30 per share. You spent a total of $24,400, and purchased twice as many FarmersBooks.com shares as JungleBooks.com. The OHaganBooks.com stocks appreciated by 20%, while the other two appreciated by 10%, and you sold all the stocks for $3,440 more than you originally paid. How many stocks of each company did you originally purchase?

Now look at the third piece of information:

The OHaganBooks.com stocks appreciated by 20%, while the other two appreciated by 10%, and you sold all the stocks for $3,440 more than you originally paid.

**Q **Select which (if any) of the following equations conveys this information.

0.20x + 0.10y + 0.10z = 3,440 | 20x + 10y + 10z = 3,440 | |||

10x + 4.5y + 3z = 3,440 | 5.5x + 2.5y + 3.5z = 3,440 | |||

Now you have a system of three equations in three unknowns. (Press here to bring up the list of the three correct equations.)

To solve the system, row-reduce the associated matrix (you can either do it by hand or use the On-Line Pivot & Gauss-Jordan Utility.

Now try the rest of the exercises in Section 2.3 of *Finite Mathematics* and *Finite Mathematics and Applied Calculus*.