3 In Example 1, we set up the problem as a linear system and then give the solution; the emphasis in this section is on modeling a scenario by a system of linear equations and then interpreting the solution that results, rather than on obtaining the solution. For practice, you should do the row reduction necessary to get the solution.
4 Example 1 – Resource Allocation The Arctic Juice Company makes three juice blends: PineOrange, using 2 quarts of pineapple juice and 2 quarts of orange juice per gallon; PineKiwi, using 3 quarts of pineapple juice and 1 quart of kiwi juice per gallon; and OrangeKiwi, using 3 quarts of orange juice and 1 quart of kiwi juice per gallon. Each day the company has 800 quarts of pineapple juice, 650 quarts of orange juice, and 350 quarts of kiwi juice available. How many gallons of each blend should it make each day if it wants to use up all of the supplies?
5 Example 1 – Solution The first step is to identify and label the unknowns. Looking at the question asked in the last sentence, we see that we should label the unknowns like this: x = number of gallons of PineOrange made each day y = number of gallons of PineKiwi made each day z = number of gallons of OrangeKiwi made each day.
6 Example 1 – Solution Next, we can organize the information we are given in a table: Notice how we have arranged the table; we have placed headings corresponding to the unknowns along the top, rather than down the side, and we have added a heading for the available totals. This gives us a table that is essentially the matrix of the system of linear equations we are looking for. cont'd
7 Example 1 – Solution Now we read across each row of the table. The fact that we want to use exactly the amount of each juice that is available leads to the following three equations: 2x + 3y = 800 2x + 3z = 650 y + z = 350. The solution of this system is (x, y, z) = (100, 200, 150), so Arctic Juice should make 100 gallons of PineOrange, 200 gallons of PineKiwi, and 150 gallons of OrangeKiwi each day. cont'd