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
Adult 4 450x 4(450x)
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
Side Road 30 185x (185x)/30
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
Side Road 30 4.5x 30(4.5x)
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
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