## 2.3 Applications of Systems of Linear Equations

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

Some On-line Resources for This Topic:

The following discussion is based on Section 2.3 of Finite Mathematics and Finite Mathematics and Applied Calculus.

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

 General Strategy for Solving Systems of Linear Equations First:  Identify and label the unknowns. 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 numbers, so we should not say somethiong like "Let x = video games." Second:  Use the information given to set up equations in the unknowns. How to do this depends on the way the problem is worded. We will look at a few examples below to develop some strategies. Third:  Solve the system to obtain the values for the unknowns. 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?

Q There are unknowns in the problem.

Q The unknowns in the problem are:

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.

Q Now fill in the values:

 LimeOrange(x gallons) LimeLemon (y gallons) OrangeLemon (z gallons) Total Available Lime yogurt (quarts) Orange yogurt (quarts) Lemon yogurt (quarts)

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:

Q Equation for Orange Yogurt:
Q Equation for 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 :

 2 3 0 800 2 0 3 650 0 1 1 350

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.

Q Solving the system leads to the following solution.
The Softflow Yogurt company should make:

gallons of LimeOrange,
gallons of LimeLemon, and
gallons of OrangeLemon yogurt.

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?

Q The unknowns in the problem are:

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

Now look at the first piece of information you are given:

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,...

Q This information can be represented as the following equation in the three unknowns:

Now look at the next piece of informatiuon:

[You] purchased twice as many FarmersBooks.com shares as JungleBooks.com.

Q With the unknowns as above, this translates to:

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.

Q Solving the system leads to the following solution.
You originally purchased

shares of OHaganBooks.com,
shares of FarmersBooks.com, and
shares of JungleBooks.com.

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

Last Updated: March, 2006