Objective – Understand Applications of Linear Systems

Types of Problems 1) Number and Value Problems2) Coin Problems3) Mixture4) Break-Even5) Comparison6) Boat or plane (current)

Number and Value Problem #1 -- EliminationJohn bought 15 items for $135. If the 15 items consisted of notebooks that cost $4.50 each andcalculators that cost $12.00 each, how many of each did he buy?Let x = # of notebooks Let y = # of calculators

x y 15 12x 12y 180 4.5x 12y 135

7.5x 457.5 7.5

x 6y 9

6 notebooks & 9 calculators

4.5x 12y 135

Number and Value Problem #1 -- SubstitutionDavid bought 15 items for $135. If the 15 items consisted of notebooks that cost $4.50 each andcalculators that cost $12.00 each, how many of each did he buy?Let x = # of notebooks Let y = # of calculators

x y 15 4.5x 12y 135 x x

y 15 x 4.5x 12 15 x 135

4.5x 180 12x 135 180 180

7.5x 45 7.5 7.5

x 6

y 15 x y 15 6 y 9

6 notebooks & 9 calculators

Number and Value Problem #2

Willa raised $24 by selling 40 baked items for the PTA. She sold cookies for 50 centseach and brownies for 75 cents each . How many of each did she sell?

Let c = # of cookies Let b = # of brownies

c b 40 .50c .75b 24

c b 40 c 1.5b 48

0.5b 8

24 cookies & 16 brownies

b 16

c b 40 c 16 40

16 16 c 24

World Famous Coin Problem

Elly had 400 coins worth $25. She only had nickels and dimes. How many of each did she have?

Let n = nickels Let d = dimes

n d 400 .05n .1d 25

n d 400 .5n d 250

0.5n 150

n 300

n d 400

d 100

World Famous Coin Problem – YOU DO!!

Emma had 140 coins worth $23. She only had quarters and dimes. How many of each did she have?

Let q = quarters Let d = dimes

3( )

Mixture Problem #1A 12 pound mixture of peanuts and cashews sellsfor $6.50/lb. If peanuts sell for $3 per pound and cashews sell for $8 per pound, how many pounds of peanuts are in the mix?Let p = peanuts (in lbs.) Let c = cashews(in

lbs.)p c 12 3p 8c 6.5 12

3p 3c 36 3p 8c 78

5c 425 5c 8.4

p c 12 p 8.4 12

8.4 8.4 p 3.6 3.6 lbs. peanuts &

8.4 lbs. cashews

Mixture Problem #2A 10% acid solution is mixed with a 60% acidsolution to produce 120 liters of a solution thatis 40% acid. How much of each solution was used to create the mixture?

Let x = liters of 10% sol. Let y = liters of 60% sol.x y 120

.10x .60y .40(120) y 120 x

.10x .60 120 x 48 .10x 72 .60x 48

72 72 .50x 24 .50 .50

x 48

y 120 48 y 72

48 L 10% Sol. &72 L 60% Sol.

Comparison ProblemThe population of Bentville is 50,000 but is growing at 2500 people per year. Payton Valley has a population of 26,000 but is growing at 4000 people per year. When will both towns have equal population.

Let x = # of years Let p = populationBentville PaytonValley

p 50,000 2500x p 26,000 4000x 50,000 2500x 26,000 4000x

2500x 2500x 50,000 26,000 1500x 26,000 26,000 24,000 1500x

16 x in 16 years

Breakeven ProblemSam is starting a candied yam business. Sam buyseach yam for $1.25 and spend $5000 on wages and$2500 on Utilities. If Sam sells each candied yamfor $4, how many yams will Sam need to sell toBreakeven?

Keys to Breakeven ProblemsNot Really a System (if you combine the problem)Set Up 1 Big Equation Equal to 0 (breakeven)Be Aware of Good Things (money made) and Bad Things

(Money Spent)

4x – 1.25x – 5000 – 2500 = 0

Solve for X !!!

Boat (current or jetstream) ProblemPassi and a friend take 3.2 hrs to canoe downstream12 miles. Going back upstream took 4.8 hrs. If they paddled at the same rate, what was their speed in still water and the current’s speed?

Key to Motion Problems – Use an RTD Table(and divide by time to simplify)

Rate Time Distance

x + y 3.2 12

x – y 4.8 12

Plane (jetstream) ProblemLaura and a friend flew 7 hrs to go 2800 miles fromMiami to Seattle. The return trip took 5.6hrs. IfThe airspeed was the same, what was the jetstream?

Key to Motion Problems – Use an RTD Table (thenDivide by time to simplify)

Rate Time Distance

x - y 7.0 2800

x + y 5.6 2800

x = 450 and y =50