# linear equations and problem solving

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• 8/10/2019 linear equations and problem solving

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Linear Equations and Problem Solving

Word Problems!!!

Keys to succeed!

Write down important information

Put the info in a chart if you can

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Find three consecutive integers whose sum is

162.

Consecutive Integers

Integer 1 x - 1

Integer 2 xInteger 3 x + 1

Total 162

1 1 162x x x

3 162x

54x

1 54 1 53x

1 54 1 55x

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A pair of hikers, 18 miles apart, begin at the same time to

hike toward each other. If one walks at a rate that is 1 mph

faster than the other, and if they meet 2 hours later, how

fast is each one walking?

Hiker 1 x 2 2x

Hiker 2 x+1 2 2(x+1)

Rate Time Distance = 18

2 2( 1) 18x x

2 2 2 18x x

4 2 18x

4 16x

4 mphx

1 4 1 5 mphx

Traveling

Hiker 1s distance + Hiker 2s distance = 18

Hiker 1s dist.

Hiker 2s dist.

18

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A pair of cars, 280 miles apart, begin at the same time to run

toward each other. If car A from city A runs at a rate that is 10

mph faster than car B from city B, and if they meet 2 hours later,how far is the place they meet away from city A?

Car A x + 10 2 2(x + 10)

Car B x 2 2xRate Time Distance = 280

2 2( 10) 280x x

2 2 20 280x x

4 20 280x

4 260x

65 mphx

10 75 mphx

Traveling

Car As distance + Car Bs distance = 280

Car As dist. Car Bs dist.

280

A B

2( 10) 2 75 150 mi.x

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The Yankee Clipper leaves the pier at 9:00am at 8 knots (nauticalmiles per hour). A half hour later, The Riverboat Rover leaves

the same pier in the same direction traveling at 10 knots. Atwhat time will The Riverboat Rover overtake The YankeeClipper?

YankeeClipper

9:00 ~ 9:30

Traveled

4 nt. miles

8 x hours after9:30

8x 8x + 4

RiverboatRover

9:00 ~ 9:30

Traveled

0nt. miles

10 x hours after9:30

10x 0 + 10x

rate time dist. total

4 8 0 10x x

4 2x

2 hr.x

Yankee Total = Riverboat Rover Total

Traveling

4 nt. mi.

RR

YC

RR

YC

10x nt. mi.

8x nt. mi.YC

9:00 9:30 x hr. after9:30

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The school play sold 450 tickets for a total of \$1160. If

student tickets are \$2.00 and adult tickets are \$4.00,

how many of each type were sold?

Student 2 x 2x

Total ----- 450 1160

2 4(450 ) 1160x x

2 1800 4 1160x x

2 640x

320 tksx

2 1800 1160x

450 450 320 130 tksx

Tickets

Student tickets sales + Adult tickets sales = 1160

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Fred is selling tickets for his home movies. Tickets for friends are 3.00 and everyone else must pay

5.00 per ticket. If he sold 72 tickets and made 258 how many of each type did he sell?

Friend 3 x 3x

Non-Friend 5 72x 5(72x)

Total ---- 72 258

3 5(72 ) 258x x

3 360 5 258x x

2 102x

51 tksx

2 360 258x

72 72 51 21 tksx

Tickets

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Barney has \$450 and spends \$3 each week.

Betty has \$120 and saves \$8 each week. Howmany weeks will it take for them to have thesame amount of money?

Barney 450 3 x 4503x

Betty 120 8 x 120 + 8x

initial wk spend wk end total

450 3 120 8x x

450 120 11x

3 wkx

330 11x

Accounting

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Fred 100 4 x 100 + 4xWilma 28 10 x 28 + 10x

initial wk sp wk end total

100 4 28 10x x

100 28 6x

12 wkx

72 6x

AccountingYou Try This!

28 10 28 10 12 \$148x

Fred has \$100 and saves \$4 each week. Wilma has \$28 andsaves \$10 each week. How long will it take for them to have thesame amount of money? What is that amount?

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More on Traveling

A driver averaged 50mph on the highway and 30mph

on the side roads. If the trip of 185 miles took a total

of 4 hours and 30 minutes, how many miles were on

the highway.

Highway 50 x x/50

Total

185 4.5185

4.550 30

x x My God! It is so complicated!!!

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More on Consecutive Integers

Find three consecutive integers that the difference of

the product of two larger ones and the product of twosmaller ones is 30.

Integer 1 x - 1

Integer 2 xInteger 3 x + 1

Prod. of Larger 2 x(x + 1)

Prod. of Smaller 2

x(x - 1)( 1) ( 1) 30x x x x 2 2 30x x x x

2 30x

15x

1 16x

1 14x

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Highway 50 x 50x

Total 4.5 185

50 30(4.5 ) 185x x

50 135 30 185x x

20 135 185x

20 50x

2.5 hr.x

50 50(2.5) 125 mi.x

A driver averaged 50mph on the highway and 30mph on the sideroads. If the trip of 185 miles took a total of 4 hours and 30

minutes, how many miles were on the highway.

More on Traveling

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Weighted Averages

You have 32 coins made up of dimes and nickels.You have a total of \$2.85. How many of eachtype of coin do you have?

Dime

10 x 10xNickel 5 32x 5(32x)

Total 32 285

10 5(32 ) 285x x

10 160 5 285x x

5 160 285x

5 125x 25x

32 7x

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The Quick Mart has two kinds of nuts. Pecanssell for \$1.55 per pound and walnuts sell for\$1.95 per pound. How many pounds of walnutsmust be added to 15 pounds of pecans to make

a mixture that sells for \$1.75 per pound.

Pecans 1.55 15 15 1.55

Walnuts 1.95 x 1.95x

Mixture 1.75 x +15 1.75(x + 15)

1.55 15 1.95 1.75( 15)x x

23.25 1.95 1.75 26.25x x

23.25 0.2 26.25x

0.2 3x

15 lb.x

Weighted Averages

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A druggist must make 20 oz of a 12% saline

solution from his supply of 5% and 15% solutions.How much of each should he use?

12%solution

12% 20 2012%

5%solution

5% x x 5%

15%solution

15% 20x (20x) 15%

20 0.12 0.05 (20 ) 0.15x x

2.4 0.05 3 0.15x x

0.6 0.1x

6 oz.x 2.4 0.1 3x

Mixture

20 20 6 14 oz.x