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[ 3 4 10 17    |    10 439 ]

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[ 3 2 0 −1 −9 4 8 5 7    |    3 −1 8 ]

3 x + 2 y = 13 x −9 y + 4 z = 53 8 x + 5 y + 7 z = 80

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[ 8 29 1 −1 7 5 0 0 3    |    43 38 10 ]

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[ 4 5 −2 0 1 58 8 7 −3    |    12 2 −5 ]

4 x + 5 y −2 z = 12          y + 58 z = 2 8 x + 7 y −3 z = −5

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For the following exercises, solve the system by Gaussian elimination.

[ 1 0 0 0    |    3 0 ]

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[ 1 0 1 0    |    1 2 ]

No solutions

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[ 1 2 4 5    |    3 6 ]

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[ −1 2 4 −5    |    −3 6 ]

( −1 , −2 )

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[ −2 0 0 2    |    1 −1 ]

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   2 x 3 y = 9 5 x + 4 y = 58

( 6 , 7 )

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6 x + 2 y = −4 3 x + 4 y = −17

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2 x + 3 y = 12    4 x + y = 14

( 3 , 2 )

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−4 x −3 y = −2    3 x −5 y = −13

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−5 x + 8 y = 3 10 x + 6 y = 5

( 1 5 , 1 2 )

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   3 x + 4 y = 12 −6 x −8 y = −24

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−60 x + 45 y = 12    20 x −15 y = −4

( x , 4 15 ( 5 x + 1 ) )

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11 x + 10 y = 43 15 x + 20 y = 65

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  2 x y = 2 3 x + 2 y = 17

( 3 , 4 )

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−1.06 x −2.25 y = 5.51 −5.03 x −1.08 y = 5.40

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3 4 x 3 5 y = 4 1 4 x + 2 3 y = 1

( 196 39 , 5 13 )

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1 4 x 2 3 y = −1 1 2 x + 1 3 y = 3

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[ 1 0 0 0 1 1 0 0 1    |    31 45 87 ]

( 31 , −42 , 87 )

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[ 1 0 1 1 1 0 0 1 1    |    50 20 −90 ]

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[ 1 2 3 0 5 6 0 0 8    |    4 7 9 ]

( 21 40 , 1 20 , 9 8 )

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[ −0.1 0.3 −0.1 −0.4 0.2 0.1 0.6 0.1 0.7    |    0.2 0.8 −0.8 ]

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  −2 x + 3 y 2 z = 3        4 x + 2 y z = 9       4 x 8 y + 2 z = −6

( 18 13 , 15 13 , 15 13 )

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       x + y 4 z = −4    5 x 3 y 2 z = 0    2 x + 6 y + 7 z = 30

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       2 x + 3 y + 2 z = 1    −4 x 6 y 4 z = −2   10 x + 15 y + 10 z = 5

( x , y , 1 2 ( 1 −2 x −3 y ) )

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     x + 2 y z = 1 x 2 y + 2 z = −2 3 x + 6 y 3 z = 5

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     x + 2 y z = 1 x −2 y + 2 z = −2   3 x + 6 y −3 z = 3

( x , x 2 , −1 )

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    x + y = 2     x + z = 1 y z = −3

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x + y + z = 100      x + 2 z = 125 y + 2 z = 25

( 125 , −25 , 0 )

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1 4 x 2 3 z = 1 2 1 5 x + 1 3 y = 4 7 1 5 y 1 3 z = 2 9

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1 2 x + 1 2 y + 1 7 z = 53 14     1 2 x 1 2 y + 1 4 z = 3      1 4 x + 1 5 y + 1 3 z = 23 15

( 8 , 1 , −2 )

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1 2 x 1 3 y + 1 4 z = 29 6     1 5 x + 1 6 y 1 7 z = 431 210 1 8 x + 1 9 y + 1 10 z = 49 45

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Extensions

For the following exercises, use Gaussian elimination to solve the system.

x −1 7 + y −2 8 + z −3 4 = 0                       x + y + z = 6       x + 2 3 + 2 y + z −3 3 = 5

( 1 , 2 , 3 )

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x −1 4 y + 1 4 + 3 z = −1     x + 5 2 + y + 7 4 z = 4           x + y z −2 2 = 1

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       x −3 4 y −1 3 + 2 z = −1 x + 5 2 + y + 5 2 + z + 5 2 = 8                       x + y + z = 1

( x , 31 28 3 x 4 , 1 28 ( −7 x −3 ) )

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x −3 10 + y + 3 2 −2 z = 3   x + 5 4 y −1 8 + z = 3 2 x −1 4 + y + 4 2 + 3 z = 3 2

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      x −3 4 y −1 3 + 2 z = −1 x + 5 2 + y + 5 2 + z + 5 2 = 7                       x + y + z = 1

No solutions exist.

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Real-world applications

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution.

Every day, a cupcake store sells 5,000 cupcakes in chocolate and vanilla flavors. If the chocolate flavor is 3 times as popular as the vanilla flavor, how many of each cupcake sell per day?

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At a competing cupcake store, $4,520 worth of cupcakes are sold daily. The chocolate cupcakes cost $2.25 and the red velvet cupcakes cost $1.75. If the total number of cupcakes sold per day is 2,200, how many of each flavor are sold each day?

860 red velvet, 1,340 chocolate

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You invested $10,000 into two accounts: one that has simple 3% interest, the other with 2.5% interest. If your total interest payment after one year was $283.50, how much was in each account after the year passed?

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You invested $2,300 into account 1, and $2,700 into account 2. If the total amount of interest after one year is $254, and account 2 has 1.5 times the interest rate of account 1, what are the interest rates? Assume simple interest rates.

4% for account 1, 6% for account 2

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Bikes’R’Us manufactures bikes, which sell for $250. It costs the manufacturer $180 per bike, plus a startup fee of $3,500. After how many bikes sold will the manufacturer break even?

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A major appliance store is considering purchasing vacuums from a small manufacturer. The store would be able to purchase the vacuums for $86 each, with a delivery fee of $9,200, regardless of how many vacuums are sold. If the store needs to start seeing a profit after 230 units are sold, how much should they charge for the vacuums?

$126

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The three most popular ice cream flavors are chocolate, strawberry, and vanilla, comprising 83% of the flavors sold at an ice cream shop. If vanilla sells 1% more than twice strawberry, and chocolate sells 11% more than vanilla, how much of the total ice cream consumption are the vanilla, chocolate, and strawberry flavors?

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At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up 12% of total ice cream sales. This year, the same three ice creams made up 16.9% of ice cream sales. The rocky road sales doubled, the banana sales increased by 50%, and the pumpkin sales increased by 20%. If the rocky road ice cream had one less percent of sales than the banana ice cream, find out the percentage of ice cream sales each individual ice cream made last year.

Banana was 3%, pumpkin was 7%, and rocky road was 2%

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A bag of mixed nuts contains cashews, pistachios, and almonds. There are 1,000 total nuts in the bag, and there are 100 less almonds than pistachios. The cashews weigh 3 g, pistachios weigh 4 g, and almonds weigh 5 g. If the bag weighs 3.7 kg, find out how many of each type of nut is in the bag.

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A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. 30% of the almonds, 20% of the cashews, and 10% of the pistachios were eaten, and now there are 770 nuts left in the bag. Originally, there were 100 more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.

100 almonds, 200 cashews, 600 pistachios

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Practice Key Terms 7

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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