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Section 3.3Applications: Systems of Equations Sum & Difference Problems F + S = T

First + Second = Total Amount Total-Value Problems Num · UC = TV

Number x Unit Price = Total Value Interest Problems Prin · Rate · T = Int

Principal x Rate x Time = Interest Purity Problems Num · Pct = Pure

Number x Percent = Pure Amount Motion Problems R · T = D

Rate x Time = Distance

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Application Problem H/W Format

#. Title First sentence (can be abbreviated)Answer stmt with ____ and variable names Table Formula A Equation B Equation work work work work work work to find the numbers to put into the answer stmt

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 $39 necklace …

____ (s)silver beads and ____ (g)gemstone beads Num · UC = TV SilversGemstones total

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Sum & Difference Example p155, #42 In 2008, there were 146 (t)hreatened plant species and 600 (e)ndangered plant species42. 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)

30102020)10(2

10303

3022

30

AchkxB

yy

yyAByxB

yxA

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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?

Sum & Difference Example #2

1. Endangered Species. …tripled in last 20 years.Threatened: 146 (t), Endangered 600 (e)

41504)600(146600146746746

146,73051644

7464

746

41

41

BchkteA

tsotetBetAetB

etA

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(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 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

Num · UC = TV

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Total Value Example1. 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 … Bought: 52 (s)ilver beads and 28 (g)emstone beads____

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Total Value Example #2s2. Purchasing. 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…They bought 45 (p)ostcard and 75 1st-class(f) stamps__ __

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Total Value Example #3

3. Blending teas. Sonya’s House …Dragon’s BlendUse 13.75 oz. of (L)apsang 6.25 oz. of (A)ssam___ ___

Blending Teas. Sonya’s House of Tea sells loose Lapsang Souchong tea for 95¢ anounce and Assam Gingia for $1.43 an ounce. Sonya wants to make a 20-oz mixtureof the two types, called Dragon Blend, that will sell for $1.10 an ounce. How muchtea of each type should Sonya use?

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Interest Example

6. Student Loans. Total was $9600, interest was $633 (p)Perkins loan = $4500 and (s)Stafford loan was = $5100____ ____

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?

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Purity Example

4. Mixing Fertilizers. Creating a 12% nitrogen fertilizer… The Co. should mix 27 L of (G)G with 63 L of (S)S__ __

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?

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Motion (same direction) Example

7. Train Travel. Boston to Washington, two trains…The Amtrak will catch up with the Freight 360 km from Boston

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?

___

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Motion (winds & currents) Example

The speed of the plane with no wind is 540 mph

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.

___

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What Next? Section 3.4 –

Systems of Three Equations