(Based on Section 2.3 in Finite Mathematics and Finite Mathematics and Applied Calculus)
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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.
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.
Third: Solve the system to obtain the values for the unknowns.
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There are several kinds of applications generally found in textbooks.
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:
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
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.