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Section 3.3 Applications: Systems of Equations Sum & Difference Problems F + S = T Total-Value Problems Num · UC = TV Principal x Rate x Time = Interest Purity Problems Num · Pct = Pure Number x Unit Price = Total Value Interest Problems Prin · Rate · T = Int First + Second = Total Amount Number x Percent = Pure Amount Motion Problems R · T = D Rate x Time = Distance 3.3 1 Application Problem H/W Format In your textbook exercises: 1. Jewelry Design. To make a necklace, a jewelry designer bought 80 beads for a total of $39. Some were silver beads (40 cents each) and the rest were gemstone beads (65 cents each). How many beads of each type did the designer buy? On your homework paper: 1. Jewelry Design. Making a(can $39 necklace … #. Title First sentence be abbreviated) Answer stmtbeads withand ____ and variable beads names Table· Formula Num UC = TV ____ (s)silver ____ (g)gemstone Silvers A Equation Gemstones B Equation total work work work work work work to find the numbers to put into3.3the answer stmt 2 Sum & Difference Example p155, #42 In 2008, there were 146 (t)hreatened plant species and 600 (e)ndangered plant species 42. The sum of two numbers is 30. The first number is twice the second number. What are the numbers? 42. The sum of two numbers is 30. The numbers are 20 (x) and 10 (y) A x y 30 B x 2y B A 2 y y 30 3 y 30 y 10 B x 2(10) 20 chk A 20 10 30 3.3 3 Sum & Difference Example #2 1. Endangered Species. The number of plant species listed as threatened or endangered has more than tripled in the last 20 years. In 2008 there were 746 plant species listed as either threatened or endangered. The number threatened was 4 less than ¼ the number endangered. How many plant species were endangered and how many were threatened? 1. Endangered Species. …tripled in last 20 years. A t e 746 Threatened: 146 (t), Endangered 600 (e) B t 14 e 4 A t e 746 4 B 4t e 16 5t 730, so t 146 A e 746 t 746 146 600 chk B 146 1 4 (600) 4 150 4 3.3 4 (Your textbook does not have this overview) Setting up Tables to Solve Problems Read over the problem, write Original Problem & Answer Stmt Determine the appropriate formula Num · UC = TV Draw a table: Make the columns match the formula Make one row for each item (or each situation), and label it Add a Total or Mixture row, if appropriate Use two meaningful letters for the unknowns (see answer stmt) Fill in two of the columns with letters and numbers, and compute the expressions for the other column or for the Total row Find 2 different equations from the table relationships 3.3 5 Total Value Example 1. Jewelry Design. To make a necklace, a jewelry designer bought 80 beads for a total of $39. Some were silver beads (40 cents each) and the rest were gemstone beads (65 cents each). How many beads of each type did the designer buy? 1. Jewelry Design. Make a necklace of 80 beads … __ (s)ilver beads and 28 __ (g)emstone beads Bought: 52 3.3 6 Total Value Example #2 2.sPurchasing. Recently the Woods County Art Center purchased 120 stamps for $33.90. If the stamps were a combination of 23¢ postcard stamps and 37 ¢ first-class stamps, how many of each type were bought? 2. Purchasing. Woods County Art Center stamps… __ (p)ostcard and 75 __ 1st-class(f) stamps They bought 45 3.3 7 Total Value Example #3 Blending Teas. Sonya’s House of Tea sells loose Lapsang Souchong tea for 95¢ an ounce and Assam Gingia for $1.43 an ounce. Sonya wants to make a 20-oz mixture of the two types, called Dragon Blend, that will sell for $1.10 an ounce. How much tea of each type should Sonya use? 3. Blending teas. Sonya’s House …Dragon’s Blend Use 13.75 ___ oz. of (A)ssam ___ oz. of (L)apsang 6.25 3.3 8 Interest Example 6. Student Loans. Ranjay’s student loans totaled $9600. Part was a Perkins loan made at 5% interest and the rest was a Stafford loan made at 8% interest. After one year, Ranjay’s loans accumulated $633 in interest. What was the original amount of each loan? 6. Student Loans. Total was $9600, interest was $633 (p)Perkins loan = $4500 ____ ____and (s)Stafford loan was = $5100 3.3 9 Purity Example 4. Mixing Fertilizers. Sky Meadow Gardening, Inc., carries two brands of liquid fertilizer containing nitrogen and water. “Gently Green” is 5% nitrogen and “Sun Saver” is 15% nitrogen. The company needs to combine the two types of solutions to make 90 L of a solution that is 12% nitrogen. How much of each brand should be used? 4. Mixing Fertilizers. Creating a 12% nitrogen fertilizer… __ L of (S)S __ L of (G)G with 63 The Co. should mix 27 3.3 10 Motion (same direction) Example 7. Train Travel. A Vermont Railways freight train, loaded with logs, leaves Boston, bound for Washington D.C. at a speed of 60 km/h. Two hours later, and Amtrak Metroliner leaves Boston bound for Washington D.C., on a parallel track at 90 km/h. At what point will the Metroliner catch up to the freight train? 7. Train Travel. Boston to Washington, two trains… The Amtrak will catch up with the Freight 360 ___km from Boston 3.3 11 Motion (winds & currents) Example 8. Jet Travel. A Boeing 747-400 jet flies 4 hours west with a 60 mph tailwind. Returning against the wind takes 5 hours. Find the speed of the plane with no wind. The speed of the plane with no wind is 540 ___mph 3.3 12 What Next? Section 3.4 – Systems of Three Equations 3.3 13